From 3aa7074c22592645055f2373b6191e5eb9d05348 Mon Sep 17 00:00:00 2001 From: mmcky Date: Thu, 28 Mar 2024 15:42:07 +1100 Subject: [PATCH] MAINT: remove code-book --- .github/workflows/ci.yml | 105 +- .../downloads => book}/networks-book.pdf | Bin code_book/_config.yml | 49 - code_book/_toc.yml | 14 - code_book/appendix.md | 547 ------- code_book/ch_fpms.md | 178 --- code_book/ch_intro.md | 788 ---------- code_book/ch_mcs.md | 476 ------ code_book/ch_opt.md | 416 ----- code_book/ch_opt_julia.md | 118 -- code_book/ch_production.md | 416 ----- code_book/img/betweenness_centrality_1.png | Bin 64051 -> 0 bytes code_book/img/code.png | Bin 29941 -> 0 bytes code_book/intro.md | 15 - code_book/pkg_funcs.md | 359 ----- code_book/status.md | 17 - ipynb/appendix.ipynb | 776 +++++++++ ipynb/ch_fpms.ipynb | 273 ++++ ipynb/ch_intro.ipynb | 1399 +++++++++++++++++ ipynb/ch_mcs.ipynb | 799 ++++++++++ ipynb/ch_opt.ipynb | 728 +++++++++ ipynb/ch_opt_julia.ipynb | 209 +++ ipynb/ch_production.ipynb | 737 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auto-update-conda: true - auto-activate-base: true - miniconda-version: 'latest' - python-version: 3.11 - environment-file: environment.yml - activate-environment: networks - - name: Install latex dependencies - run: | - sudo apt-get -qq update - sudo apt-get install -y \ - texlive-latex-recommended \ - texlive-latex-extra \ - texlive-fonts-recommended \ - texlive-fonts-extra \ - texlive-xetex \ - latexmk \ - xindy \ - dvipng \ - cm-super \ - msttcorefonts - - name: Set up Julia - uses: julia-actions/setup-julia@v1 - with: - version: 1.9 - - name: Install IJulia and Setup Project - shell: bash - run: | - julia -e 'using Pkg; ENV["PYTHON"]="/usr/share/miniconda3"; Pkg.add(["JuMP","GLPK","PyPlot","IJulia"]); using PyPlot' - # - name: Install latex dependencies - # run: | - # sudo apt-get -qq update - # sudo apt-get install -y \ - # texlive-latex-recommended \ - # texlive-latex-extra \ - # texlive-fonts-recommended \ - # texlive-fonts-extra \ - # texlive-xetex \ - # latexmk \ - # xindy \ - # dvipng \ - # cm-super \ - # msttcorefonts - # - name: Display Conda Environment Versions - # shell: bash -l {0} - # run: conda list - # - name: Display Pip Versions - # shell: bash -l {0} - # run: pip list - # - name: Download "build" folder (cache) - # uses: dawidd6/action-download-artifact@v2 - # with: - # workflow: cache.yml - # branch: main - # name: build-cache - # path: _build - # # Build Assets (Download Notebooks and PDF via LaTeX) - # - name: Build PDF from LaTeX - # shell: bash -l {0} - # run: | - # jb build lectures --builder pdflatex --path-output ./ -n --keep-going - # mkdir -p _build/html/_pdf - # cp -u _build/latex/*.pdf _build/html/_pdf - # - name: Build Download Notebooks (sphinx-tojupyter) - # shell: bash -l {0} - # run: | - # jb build lectures --path-output ./ --builder=custom --custom-builder=jupyter - # mkdir -p _build/html/_notebooks - # cp -u _build/jupyter/*.ipynb _build/html/_notebooks - # Build HTML (Website) - - name: Build HTML - shell: bash -l {0} - run: | - jb build code_book --path-output ./ - - name: Upload Execution Reports - uses: actions/upload-artifact@v2 - if: failure() - with: - name: execution-reports - path: _build/html/reports - - name: Custom Front Page - shell: bash -l {0} - run: | - cp -r frontpage/assets _build/html/ - cp -r frontpage/images _build/html/ - cp frontpage/index.html _build/html/ - - name: Book PDF - shell: bash -l {0} - run: | - mkdir -p _build/html/_downloads - cp pdf/networks.pdf _build/html/_downloads/ - cp pdf/networks_chinese.pdf _build/html/_downloads/ - - name: Save Build as Artifact - uses: actions/upload-artifact@v1 - with: - name: _build - path: _build + uses: actions/checkout@v4 - name: Preview Deploy to Netlify - uses: nwtgck/actions-netlify@v1.1 + uses: nwtgck/actions-netlify@v2 with: - publish-dir: '_build/html/' + publish-dir: 'website' production-branch: main github-token: ${{ secrets.GITHUB_TOKEN }} deploy-message: "Preview Deploy from GitHub Actions" diff --git a/code_book/downloads/networks-book.pdf b/book/networks-book.pdf similarity index 100% rename from code_book/downloads/networks-book.pdf rename to book/networks-book.pdf diff --git a/code_book/_config.yml b/code_book/_config.yml deleted file mode 100644 index 0e923b4..0000000 --- a/code_book/_config.yml +++ /dev/null @@ -1,49 +0,0 @@ -# Book settings -title: | - Economic Networks - Theory and Computation -author: John Stachurski and Thomas J. Sargent -only_build_toc_files: true - -execute: - timeout: 100 - -sphinx: - extra_extensions: ["sphinx_design"] - config: - html_js_files: - - https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.4/require.min.js - html_theme: quantecon_book_theme - html_theme_options: - header_organisation_url: https://quantecon.org - header_organisation: QuantEcon - repository_url: https://github.com/quantecon/book-networks - twitter: quantecon - twitter_logo_url: https://assets.quantecon.org/img/qe-twitter-logo.png - og_logo_url: https://assets.quantecon.org/img/qe-og-logo.png - description: | - This website is the companion site for Economic Networks: Theory and Computation written by John Stachurski and Thomas J. Sargent - keywords: Python, QuantEcon, Quantitative Economics, Economics, Networks, Tom J. Sargent, John Stachurski - # google_analytics_id: UA-54984338-10 - mathjax3_config: - tex: - macros: - "Exp" : ["\\operatorname{Exp}"] - "Binomial" : ["\\operatorname{Binomial}"] - "Poisson" : ["\\operatorname{Poisson}"] - "BB" : ["\\mathbb{B}"] - "EE" : ["\\mathbb{E}"] - "PP" : ["\\mathbb{P}"] - "RR" : ["\\mathbb{R}"] - "NN" : ["\\mathbb{N}"] - "ZZ" : ["\\mathbb{Z}"] - "dD" : ["\\mathcal{D}"] - "fF" : ["\\mathcal{F}"] - "lL" : ["\\mathcal{L}"] - "linop" : ["\\mathcal{L}(\\mathbb{B})"] - "linopell" : ["\\mathcal{L}(\\ell_1)"] - -latex: - latex_engine: "xelatex" - latex_documents: - targetname: book.tex \ No newline at end of file diff --git a/code_book/_toc.yml b/code_book/_toc.yml deleted file mode 100644 index 1405efd..0000000 --- a/code_book/_toc.yml +++ /dev/null @@ -1,14 +0,0 @@ -format: jb-book -root: intro -chapters: -- url: https://networks.quantecon.org - title: Home -- file: ch_intro -- file: ch_production -- file: ch_opt -- file: ch_opt_julia -- file: ch_mcs -- file: ch_fpms -- file: appendix -- file: pkg_funcs -- file: status diff --git a/code_book/appendix.md b/code_book/appendix.md deleted file mode 100644 index 9e4909d..0000000 --- a/code_book/appendix.md +++ /dev/null @@ -1,547 +0,0 @@ ---- -jupytext: - cell_metadata_filter: -all - formats: md:myst - text_representation: - extension: .md - format_name: myst - format_version: 0.13 - jupytext_version: 1.10.3 -kernelspec: - display_name: Python 3 - language: python - name: python3 ---- - -# Appendix Code - -We begin with some imports - -```{code-cell} -import numpy as np -import matplotlib.pyplot as plt -import quantecon_book_networks -from mpl_toolkits.mplot3d.axes3d import Axes3D, proj3d -from matplotlib import cm -export_figures = False -quantecon_book_networks.config("matplotlib") -``` - -## Functions - -### One-to-one and onto functions on $(0,1)$ - -We start by defining the domain and our one-to-one and onto function examples. - - -```{code-cell} -x = np.linspace(0, 1, 100) -titles = 'one-to-one', 'onto' -labels = '$f(x)=1/2 + x/2$', '$f(x)=4x(1-x)$' -funcs = lambda x: 1/2 + x/2, lambda x: 4 * x * (1-x) -``` - -The figure can now be produced as follows. - -```{code-cell} -fig, axes = plt.subplots(1, 2, figsize=(10, 4)) -for f, ax, lb, ti in zip(funcs, axes, labels, titles): - ax.set_xlim(0, 1) - ax.set_ylim(0, 1.01) - ax.plot(x, f(x), label=lb) - ax.set_title(ti, fontsize=12) - ax.legend(loc='lower center', fontsize=12, frameon=False) - ax.set_xticks((0, 1)) - ax.set_yticks((0, 1)) - -if export_figures: - plt.savefig("figures/func_types_1.pdf") -plt.show() -``` - -### Some functions are bijections - -This figure can be produced in a similar manner to 6.1. - -```{code-cell} -x = np.linspace(0, 1, 100) -titles = 'constant', 'bijection' -labels = '$f(x)=1/2$', '$f(x)=1-x$' -funcs = lambda x: 1/2 + 0 * x, lambda x: 1-x - -fig, axes = plt.subplots(1, 2, figsize=(10, 4)) -for f, ax, lb, ti in zip(funcs, axes, labels, titles): - ax.set_xlim(0, 1) - ax.set_ylim(0, 1.01) - ax.plot(x, f(x), label=lb) - ax.set_title(ti, fontsize=12) - ax.legend(loc='lower center', fontsize=12, frameon=False) - ax.set_xticks((0, 1)) - ax.set_yticks((0, 1)) - -if export_figures: - plt.savefig("figures/func_types_2.pdf") -plt.show() -``` - -## Fixed Points - -### Graph and fixed points of $G \colon x \mapsto 2.125/(1 + x^{-4})$ - -We begin by defining the domain and the function. - -```{code-cell} -xmin, xmax = 0.0000001, 2 -xgrid = np.linspace(xmin, xmax, 200) -g = lambda x: 2.125 / (1 + x**(-4)) -``` - -Next we define our fixed points. - -```{code-cell} -fps_labels = ('$x_\ell$', '$x_m$', '$x_h$' ) -fps = (0.01, 0.94, 1.98) -coords = ((40, 80), (40, -40), (-40, -80)) -``` - -Finally we can produce the figure. - -```{code-cell} -fig, ax = plt.subplots(figsize=(6.5, 6)) - -ax.set_xlim(xmin, xmax) -ax.set_ylim(xmin, xmax) - -ax.plot(xgrid, g(xgrid), 'b-', lw=2, alpha=0.6, label='$G$') -ax.plot(xgrid, xgrid, 'k-', lw=1, alpha=0.7, label='$45^o$') - -ax.legend(fontsize=14) - -ax.plot(fps, fps, 'ro', ms=8, alpha=0.6) - -for (fp, lb, coord) in zip(fps, fps_labels, coords): - ax.annotate(lb, - xy=(fp, fp), - xycoords='data', - xytext=coord, - textcoords='offset points', - fontsize=16, - arrowprops=dict(arrowstyle="->")) - -if export_figures: - plt.savefig("figures/three_fixed_points.pdf") -plt.show() -``` - -## Complex Numbers - -### The complex number $(a, b) = r e^{i \phi}$ - -We start by abbreviating some useful values and functions. - -```{code-cell} -π = np.pi -zeros = np.zeros -ones = np.ones -fs = 18 -``` - -Next we set our parameters. - -```{code-cell} -r = 2 -φ = π/3 -x = r * np.cos(φ) -x_range = np.linspace(0, x, 1000) -φ_range = np.linspace(0, φ, 1000) -``` - -Finally we produce the plot. - -```{code-cell} -fig = plt.figure(figsize=(7, 7)) -ax = plt.subplot(111, projection='polar') - -ax.plot((0, φ), (0, r), marker='o', color='b', alpha=0.5) # Plot r -ax.plot(zeros(x_range.shape), x_range, color='b', alpha=0.5) # Plot x -ax.plot(φ_range, x / np.cos(φ_range), color='b', alpha=0.5) # Plot y -ax.plot(φ_range, ones(φ_range.shape) * 0.15, color='green') # Plot φ - -ax.margins(0) # Let the plot starts at origin - -ax.set_rmax(2) -ax.set_rticks((1, 2)) # Less radial ticks -ax.set_rlabel_position(-88.5) # Get radial labels away from plotted line - -ax.text(φ, r+0.04 , r'$(a, b) = (1, \sqrt{3})$', fontsize=fs) # Label z -ax.text(φ+0.4, 1 , '$r = 2$', fontsize=fs) # Label r -ax.text(0-0.4, 0.5, '$1$', fontsize=fs) # Label x -ax.text(0.5, 1.2, '$\sqrt{3}$', fontsize=fs) # Label y -ax.text(0.3, 0.25, '$\\varphi = \\pi/3$', fontsize=fs) # Label θ - -xT=plt.xticks()[0] -xL=['0', - r'$\frac{\pi}{4}$', - r'$\frac{\pi}{2}$', - r'$\frac{3\pi}{4}$', - r'$\pi$', - r'$\frac{5\pi}{4}$', - r'$\frac{3\pi}{2}$', - r'$\frac{7\pi}{4}$'] - -plt.xticks(xT, xL, fontsize=fs+2) -ax.grid(True) - -if export_figures: - plt.savefig("figures/complex_number.pdf") -plt.show() -``` - -## Convergence - -### Convergence of a sequence to the origin in $\mathbb{R}^3$ - -We define our transformation matrix, initial point, and number of iterations. - -```{code-cell} -θ = 0.1 -A = ((np.cos(θ), - np.sin(θ), 0.0001), - (np.sin(θ), np.cos(θ), 0.001), - (np.sin(θ), np.cos(θ), 1)) - -A = 0.98 * np.array(A) -p = np.array((1, 1, 1)) -n = 200 -``` - -Now we can produce the plot by repeatedly transforming our point with the transformation matrix and plotting each resulting point. - -```{code-cell} -fig = plt.figure(figsize=(6, 6)) -ax = fig.add_subplot(111, projection='3d') -ax.view_init(elev=20, azim=-40) - -ax.set_xlim((-1.5, 1.5)) -ax.set_ylim((-1.5, 1.5)) -ax.set_xticks((-1,0,1)) -ax.set_yticks((-1,0,1)) - -for i in range(n): - x, y, z = p - ax.plot([x], [y], [z], 'o', ms=4, color=cm.jet_r(i / n)) - p = A @ p - -if export_figures: - plt.savefig("figures/euclidean_convergence_1.pdf") -plt.show() -``` - -## Linear Algebra - -### The span of vectors $u$, $v$, $w$ in $\mathbb{R}$ - -We begin by importing the `FancyArrowPatch` class and extending it. - -```{code-cell} -from matplotlib.patches import FancyArrowPatch - -class Arrow3D(FancyArrowPatch): - def __init__(self, xs, ys, zs, *args, **kwargs): - FancyArrowPatch.__init__(self, (0,0), (0,0), *args, **kwargs) - self._verts3d = xs, ys, zs - - def do_3d_projection(self, renderer=None): - xs3d, ys3d, zs3d = self._verts3d - xs, ys, zs = proj3d.proj_transform(xs3d, ys3d, zs3d, self.axes.M) - self.set_positions((xs[0],ys[0]),(xs[1],ys[1])) - - return np.min(zs) - - def draw(self, renderer): - xs3d, ys3d, zs3d = self._verts3d - xs, ys, zs = proj3d.proj_transform(xs3d, ys3d, zs3d, self.axes.M) - self.set_positions((xs[0],ys[0]),(xs[1],ys[1])) - FancyArrowPatch.draw(self, renderer) -``` - -Next we generate our vectors $u$, $v$, $w$, ensuring linear dependence. - -```{code-cell} -α, β = 0.2, 0.1 -def f(x, y): - return α * x + β * y - -# Vector locations, by coordinate -x_coords = np.array((3, 3, -3.5)) -y_coords = np.array((4, -4, 3.0)) -z_coords = f(x_coords, y_coords) - -vecs = [np.array((x, y, z)) for x, y, z in zip(x_coords, y_coords, z_coords)] -``` - -Next we define the spanning plane. - -```{code-cell} -x_min, x_max = -5, 5 -y_min, y_max = -5, 5 - -grid_size = 20 -xr2 = np.linspace(x_min, x_max, grid_size) -yr2 = np.linspace(y_min, y_max, grid_size) -x2, y2 = np.meshgrid(xr2, yr2) -z2 = f(x2, y2) -``` - -Finally we generate the plot. - -```{code-cell} -fig = plt.figure(figsize=(12, 7)) -ax = plt.axes(projection ='3d') -ax.view_init(elev=10., azim=-80) - -ax.set(xlim=(x_min, x_max), - ylim=(x_min, x_max), - zlim=(x_min, x_max), - xticks=(0,), yticks=(0,), zticks=(0,)) - -# Draw the axes -gs = 3 -z = np.linspace(x_min, x_max, gs) -x = np.zeros(gs) -y = np.zeros(gs) -ax.plot(x, y, z, 'k-', lw=2, alpha=0.5) -ax.plot(z, x, y, 'k-', lw=2, alpha=0.5) -ax.plot(y, z, x, 'k-', lw=2, alpha=0.5) - -# Draw the vectors -for v in vecs: - a = Arrow3D([0, v[0]], - [0, v[1]], - [0, v[2]], - mutation_scale=20, - lw=1, - arrowstyle="-|>", - color="b") - ax.add_artist(a) - - -for v, label in zip(vecs, ('u', 'v', 'w')): - v = v * 1.1 - ax.text(v[0], v[1], v[2], - f'${label}$', - fontsize=14) - -# Draw the plane -grid_size = 20 -xr2 = np.linspace(x_min, x_max, grid_size) -yr2 = np.linspace(y_min, y_max, grid_size) -x2, y2 = np.meshgrid(xr2, yr2) -z2 = f(x2, y2) - -ax.plot_surface(x2, y2, z2, rstride=1, cstride=1, cmap=cm.jet, - linewidth=0, antialiased=True, alpha=0.2) - - -if export_figures: - plt.savefig("figures/span1.pdf") -plt.show() -``` - -## Linear Maps Are Matrices - -### Equivalence of the onto and one-to-one properties (for linear maps) - -This plot is produced similarly to Figures 6.1 and 6.2. - -```{code-cell} -fig, axes = plt.subplots(1, 2, figsize=(10, 4)) - -x = np.linspace(-2, 2, 10) - -titles = 'non-bijection', 'bijection' -labels = '$f(x)=0 x$', '$f(x)=0.5 x$' -funcs = lambda x: 0*x, lambda x: 0.5 * x - -for ax, f, lb, ti in zip(axes, funcs, labels, titles): - - # Set the axes through the origin - for spine in ['left', 'bottom']: - ax.spines[spine].set_position('zero') - for spine in ['right', 'top']: - ax.spines[spine].set_color('none') - ax.set_yticks((-1, 1)) - ax.set_ylim((-1, 1)) - ax.set_xlim((-1, 1)) - ax.set_xticks((-1, 1)) - y = f(x) - ax.plot(x, y, '-', linewidth=4, label=lb, alpha=0.6) - ax.text(-0.8, 0.5, ti, fontsize=14) - ax.legend(loc='lower right', fontsize=12) - -if export_figures: - plt.savefig("figures/func_types_3.pdf") -plt.show() -``` - -## Convexity and Polyhedra - -### A polyhedron $P$ represented as intersecting halfspaces - -Inequalities are of the form - -$$ a x + b y \leq c $$ - -To plot the halfspace we plot the line - -$$ y = c/b - a/b x $$ - -and then fill in the halfspace using `fill_between` on points $x, y, \hat y$, -where $\hat y$ is either `y_min` or `y_max`. - -```{code-cell} -fig, ax = plt.subplots() -plt.axis('off') - -x = np.linspace(-10, 14, 200) -y_min, y_max = -2, 3 - -a1, b1, c1 = 1.0, 8.0, -5.0 -y = c1 / b1 - (a1 / b1) * x -ax.plot(x, y, label='$a_1 x_1 + b_1 x_2 = c_1$') -ax.fill_between(x, y, y_max, alpha=0.2) - - -a2, b2, c2 = 0.5, 0.75, 1.5 -y = c2 / b2 - (a2 / b2) * x -ax.plot(x, y, label='$a_2 x_1 + b_2 x_2 = c_2$') -ax.fill_between(x, y, y_min, alpha=0.2) - - -a3, b3, c3 = -1.0, 1.0, 1.5 -y = c3 / b3 - (a3 / b3) * x -ax.plot(x, y, label='$a_3 x_1 + b_3 x_2 = c_3$') -ax.fill_between(x, y, y_min, alpha=0.2) - -ax.plot((0.23,), (1.82,), 'ko') -ax.plot((-1.95,), (-0.4,), 'ko') -ax.plot((4.8,), (-1.2,), 'ko') - - -ax.annotate('$P$', xy=(0, 0), fontsize=12) - -ax.set_ylim(y_min, y_max) -if export_figures: - plt.savefig("figures/polyhedron1.pdf") -plt.show() -``` - -## Saddle Points and Duality - -```{code-cell} -fig = plt.figure(figsize=(8.5, 6)) - -## Top left Plot - -ax = fig.add_subplot(221, projection='3d') - -plot_args = {'rstride': 1, 'cstride': 1, 'cmap':"viridis", - 'linewidth': 0.4, 'antialiased': True, "alpha":0.75, - 'vmin': -1, 'vmax': 1} - -x, y = np.mgrid[-1:1:31j, -1:1:31j] -z = x**2 - y**2 - -ax.plot_surface(x, y, z, **plot_args) - -ax.view_init(azim=245, elev=20) - -ax.set_xlim(-1, 1) -ax.set_ylim(-1, 1) -ax.set_zlim(-1, 1) - -ax.set_xticks([0]) -ax.set_xticklabels([r"$x^*$"], fontsize=16) -ax.set_yticks([0]) -ax.set_yticklabels([r"$\theta^*$"], fontsize=16) - - -ax.set_zticks([]) -ax.set_zticklabels([]) - -ax.w_xaxis.set_pane_color((1.0, 1.0, 1.0, 0.0)) -ax.w_yaxis.set_pane_color((1.0, 1.0, 1.0, 0.0)) -ax.w_zaxis.set_pane_color((1.0, 1.0, 1.0, 0.0)) - -ax.set_zlabel("$L(x,\\theta)$", fontsize=14) - -## Top Right Plot - -ax = fig.add_subplot(222) - - -plot_args = {'cmap':"viridis", 'antialiased': True, "alpha":0.6, - 'vmin': -1, 'vmax': 1} - -x, y = np.mgrid[-1:1:31j, -1:1:31j] -z = x**2 - y**2 - -ax.contourf(x, y, z, **plot_args) - -ax.plot([0], [0], 'ko') - -ax.set_xlim(-1, 1) -ax.set_ylim(-1, 1) - -plt.xticks([ 0], - [r"$x^*$"], fontsize=16) - -plt.yticks([0], - [r"$\theta^*$"], fontsize=16) - -ax.hlines(0, -1, 1, color='k', ls='-', lw=1) -ax.vlines(0, -1, 1, color='k', ls='-', lw=1) - -coords=(-35, 30) -ax.annotate(r'$L(x, \theta^*)$', - xy=(-0.5, 0), - xycoords="data", - xytext=coords, - textcoords="offset points", - fontsize=12, - arrowprops={"arrowstyle" : "->"}) - -coords=(35, 30) -ax.annotate(r'$L(x^*, \theta)$', - xy=(0, 0.25), - xycoords="data", - xytext=coords, - textcoords="offset points", - fontsize=12, - arrowprops={"arrowstyle" : "->"}) - -## Bottom Left Plot - -ax = fig.add_subplot(223) - -x = np.linspace(-1, 1, 100) -ax.plot(x, -x**2, label='$\\theta \mapsto L(x^*, \\theta)$') -ax.set_ylim((-1, 1)) -ax.legend(fontsize=14) -ax.set_xticks([]) -ax.set_yticks([]) - -## Bottom Right Plot - -ax = fig.add_subplot(224) - -x = np.linspace(-1, 1, 100) -ax.plot(x, x**2, label='$x \mapsto L(x, \\theta^*)$') -ax.set_ylim((-1, 1)) -ax.legend(fontsize=14, loc='lower right') -ax.set_xticks([]) -ax.set_yticks([]) - -if export_figures: - plt.savefig("figures/saddle_1.pdf") -plt.show() -``` diff --git a/code_book/ch_fpms.md b/code_book/ch_fpms.md deleted file mode 100644 index d033658..0000000 --- a/code_book/ch_fpms.md +++ /dev/null @@ -1,178 +0,0 @@ ---- -jupytext: - cell_metadata_filter: -all - formats: md:myst - text_representation: - extension: .md - format_name: myst -kernelspec: - display_name: Python 3 - language: python - name: python3 ---- - -# Chapter 5 - Nonlinear Interactions (Python Code) - -```{code-cell} -:tags: [hide-output] - -! pip install --upgrade quantecon_book_networks -``` - -We begin with some imports - -```{code-cell} -import quantecon as qe -import quantecon_book_networks -import quantecon_book_networks.input_output as qbn_io -import quantecon_book_networks.plotting as qbn_plt -import quantecon_book_networks.data as qbn_data -export_figures = False -``` - -```{code-cell} -import numpy as np -import networkx as nx -import matplotlib.pyplot as plt -from matplotlib import cm -quantecon_book_networks.config("matplotlib") -``` - -## Financial Networks - -### Equity-Cross Holdings - -Here we define a class for modelling a financial network where firms are linked by share cross-holdings, -and there are failure costs as described by [Elliott et al. (2014)](https://www.aeaweb.org/articles?id=10.1257/aer.104.10.3115). - -```{code-cell} -class FinNet: - - def __init__(self, n=100, c=0.72, d=1, θ=0.5, β=1.0, seed=1234): - - self.n, self.c, self.d, self.θ, self.β = n, c, d, θ, β - np.random.seed(seed) - - self.e = np.ones(n) - self.C, self.C_hat = self.generate_primitives() - self.A = self.C_hat @ np.linalg.inv(np.identity(n) - self.C) - self.v_bar = self.A @ self.e - self.t = np.full(n, θ) - - def generate_primitives(self): - - n, c, d = self.n, self.c, self.d - B = np.zeros((n, n)) - C = np.zeros_like(B) - - for i in range(n): - for j in range(n): - if i != j and np.random.rand() < d/(n-1): - B[i,j] = 1 - - for i in range(n): - for j in range(n): - k = np.sum(B[:,j]) - if k > 0: - C[i,j] = c * B[i,j] / k - - C_hat = np.identity(n) * (1 - c) - - return C, C_hat - - def T(self, v): - Tv = self.A @ (self.e - self.β * np.where(v < self.t, 1, 0)) - return Tv - - def compute_equilibrium(self): - i = 0 - v = self.v_bar - error = 1 - while error > 1e-10: - print(f"number of failing firms is ", np.sum(v < self.θ)) - new_v = self.T(v) - error = np.max(np.abs(new_v - v)) - v = new_v - i = i+1 - - print(f"Terminated after {i} iterations") - return v - - def map_values_to_colors(self, v, j): - cols = cm.plasma(qbn_io.to_zero_one(v)) - if j != 0: - for i in range(len(v)): - if v[i] < self.t[i]: - cols[i] = 0.0 - return cols -``` - -Now we create a financial network. - -```{code-cell} -fn = FinNet(n=100, c=0.72, d=1, θ=0.3, β=1.0) -``` - -And compute its equilibrium. - -```{code-cell} -fn.compute_equilibrium() -``` - -### Waves of bankruptcies in a financial network - -Now we visualise the network after different numbers of iterations. - -For convenience we will first define a function to plot the graphs of the financial network. - -```{code-cell} -def plot_fin_graph(G, ax, node_color_list): - - n = G.number_of_nodes() - - node_pos_dict = nx.spring_layout(G, k=1.1) - edge_colors = [] - - for i in range(n): - for j in range(n): - edge_colors.append(node_color_list[i]) - - - nx.draw_networkx_nodes(G, - node_pos_dict, - node_color=node_color_list, - edgecolors='grey', - node_size=100, - linewidths=2, - alpha=0.8, - ax=ax) - - nx.draw_networkx_edges(G, - node_pos_dict, - edge_color=edge_colors, - alpha=0.4, - ax=ax) -``` - -Now we will iterate by applying the operator $T$ to the vector of firm values $v$ and produce the plots. - -```{code-cell} -G = nx.from_numpy_array(np.array(fn.C), create_using=nx.DiGraph) -v = fn.v_bar - -k = 15 -d = 3 -fig, axes = plt.subplots(int(k/d), 1, figsize=(10, 12)) - -for i in range(k): - if i % d == 0: - ax = axes[int(i/d)] - ax.set_title(f"iteration {i}") - - plot_fin_graph(G, ax, fn.map_values_to_colors(v, i)) - v = fn.T(v) -if export_figures: - plt.savefig("figures/fin_network_sims_1.pdf") -plt.show() -``` - diff --git a/code_book/ch_intro.md b/code_book/ch_intro.md deleted file mode 100644 index 775b781..0000000 --- a/code_book/ch_intro.md +++ /dev/null @@ -1,788 +0,0 @@ ---- -jupytext: - cell_metadata_filter: -all - formats: md:myst - text_representation: - extension: .md - format_name: myst - format_version: 0.13 - jupytext_version: 1.15.1 -kernelspec: - display_name: Python 3 (ipykernel) - language: python - name: python3 ---- - -# Chapter 1 - Introduction (Python Code) - -```{code-cell} ipython3 -:tags: [hide-output] -! pip install --upgrade quantecon_book_networks kaleido -``` - -We begin by importing the `quantecon` package as well as some functions and data that have been packaged for release with this text. - -```{code-cell} ipython3 -import quantecon as qe -import quantecon_book_networks -import quantecon_book_networks.input_output as qbn_io -import quantecon_book_networks.data as qbn_data -import quantecon_book_networks.plotting as qbn_plot -ch1_data = qbn_data.introduction() -default_figsize = (6, 4) -export_figures = False -``` - -Next we import some common python libraries. - -```{code-cell} ipython3 -import numpy as np -import pandas as pd -import networkx as nx -import statsmodels.api as sm -import matplotlib.pyplot as plt -import matplotlib.cm as cm -import matplotlib.ticker as ticker -from mpl_toolkits.mplot3d.art3d import Poly3DCollection -import matplotlib.patches as mpatches -import plotly.graph_objects as go -quantecon_book_networks.config("matplotlib") -``` - -## Motivation - -### International trade in crude oil 2021 - -We begin by loading a `NetworkX` directed graph object that represents international trade in crude oil. - -```{code-cell} ipython3 -DG = ch1_data["crude_oil"] -``` - -Next we transform the data to prepare it for display as a Sankey diagram. - -```{code-cell} ipython3 -nodeid = {} -for ix,nd in enumerate(DG.nodes()): - nodeid[nd] = ix - -# Links -source = [] -target = [] -value = [] -for src,tgt in DG.edges(): - source.append(nodeid[src]) - target.append(nodeid[tgt]) - value.append(DG[src][tgt]['weight']) -``` - -Finally we produce our plot. - -```{code-cell} ipython3 -fig = go.Figure(data=[go.Sankey( - node = dict( - pad = 15, - thickness = 20, - line = dict(color = "black", width = 0.5), - label = list(nodeid.keys()), - color = "blue" - ), - link = dict( - source = source, - target = target, - value = value - ))]) - - -fig.update_layout(title_text="Crude Oil", font_size=10, width=600, height=800) -if export_figures: - fig.write_image("figures/crude_oil_2021.pdf") -fig.show(renderer='svg') -``` - -### International trade in commercial aircraft during 2019 - -For this plot we will use a cleaned dataset from -[Harvard, CID Dataverse](https://dataverse.harvard.edu/dataverse/atlas). - -```{code-cell} ipython3 -DG = ch1_data['aircraft_network'] -pos = ch1_data['aircraft_network_pos'] -``` - -We begin by calculating some features of our graph using the `NetworkX` and -the `quantecon_book_networks` packages. - -```{code-cell} ipython3 -centrality = nx.eigenvector_centrality(DG) -node_total_exports = qbn_io.node_total_exports(DG) -edge_weights = qbn_io.edge_weights(DG) -``` - -Now we convert our graph features to plot features. - -```{code-cell} ipython3 -node_pos_dict = pos - -node_sizes = qbn_io.normalise_weights(node_total_exports,10000) -edge_widths = qbn_io.normalise_weights(edge_weights,10) - -node_colors = qbn_io.colorise_weights(list(centrality.values()),color_palette=cm.viridis) -node_to_color = dict(zip(DG.nodes,node_colors)) -edge_colors = [] -for src,_ in DG.edges: - edge_colors.append(node_to_color[src]) -``` - -Finally we produce the plot. - -```{code-cell} ipython3 -fig, ax = plt.subplots(figsize=(10, 10)) -ax.axis('off') - -nx.draw_networkx_nodes(DG, - node_pos_dict, - node_color=node_colors, - node_size=node_sizes, - linewidths=2, - alpha=0.6, - ax=ax) - -nx.draw_networkx_labels(DG, - node_pos_dict, - ax=ax) - -nx.draw_networkx_edges(DG, - node_pos_dict, - edge_color=edge_colors, - width=edge_widths, - arrows=True, - arrowsize=20, - ax=ax, - node_size=node_sizes, - connectionstyle='arc3,rad=0.15') - -if export_figures: - plt.savefig("figures/commercial_aircraft_2019_1.pdf") -plt.show() -``` - -## Spectral Theory - -### Spectral Radii - -Here we provide code for computing the spectral radius of a matrix. - -```{code-cell} ipython3 -def spec_rad(M): - """ - Compute the spectral radius of M. - """ - return np.max(np.abs(np.linalg.eigvals(M))) -``` - -```{code-cell} ipython3 -M = np.array([[1,2],[2,1]]) -spec_rad(M) -``` - -This function is available in the `quantecon_book_networks` package, along with -several other functions for used repeatedly in the text. Source code for -these functions can be seen [here](pkg_funcs). - -```{code-cell} ipython3 -qbn_io.spec_rad(M) -``` - -## Probability - -### The unit simplex in $\mathbb{R}^3$ - -Here we define a function for plotting the unit simplex. - -```{code-cell} ipython3 -def unit_simplex(angle): - - fig = plt.figure(figsize=(10, 8)) - ax = fig.add_subplot(111, projection='3d') - - vtx = [[0, 0, 1], - [0, 1, 0], - [1, 0, 0]] - - tri = Poly3DCollection([vtx], color='darkblue', alpha=0.3) - tri.set_facecolor([0.5, 0.5, 1]) - ax.add_collection3d(tri) - - ax.set(xlim=(0, 1), ylim=(0, 1), zlim=(0, 1), - xticks=(1,), yticks=(1,), zticks=(1,)) - - ax.set_xticklabels(['$(1, 0, 0)$'], fontsize=16) - ax.set_yticklabels([f'$(0, 1, 0)$'], fontsize=16) - ax.set_zticklabels([f'$(0, 0, 1)$'], fontsize=16) - - ax.xaxis.majorTicks[0].set_pad(15) - ax.yaxis.majorTicks[0].set_pad(15) - ax.zaxis.majorTicks[0].set_pad(35) - - ax.view_init(30, angle) - - # Move axis to origin - ax.xaxis._axinfo['juggled'] = (0, 0, 0) - ax.yaxis._axinfo['juggled'] = (1, 1, 1) - ax.zaxis._axinfo['juggled'] = (2, 2, 0) - - ax.grid(False) - - return ax -``` - -We can now produce the plot. - -```{code-cell} ipython3 -unit_simplex(50) -if export_figures: - plt.savefig("figures/simplex_1.pdf") -plt.show() -``` - -### Independent draws from Student’s t and Normal distributions - -Here we illustrate the occurrence of "extreme" events in heavy tailed distributions. -We start by generating 1,000 samples from a normal distribution and a Student's t distribution. - -```{code-cell} ipython3 -from scipy.stats import t -n = 1000 -np.random.seed(123) - -s = 2 -n_data = np.random.randn(n) * s - -t_dist = t(df=1.5) -t_data = t_dist.rvs(n) -``` - -When we plot our samples, we see the Student's t distribution frequently -generates samples many standard deviations from the mean. - -```{code-cell} ipython3 -fig, axes = plt.subplots(1, 2, figsize=(8, 3.4)) - -for ax in axes: - ax.set_ylim((-50, 50)) - ax.plot((0, n), (0, 0), 'k-', lw=0.3) - -ax = axes[0] -ax.plot(list(range(n)), t_data, linestyle='', marker='o', alpha=0.5, ms=4) -ax.vlines(list(range(n)), 0, t_data, 'k', lw=0.2) -ax.get_xaxis().set_major_formatter( - ticker.FuncFormatter(lambda x, p: format(int(x), ','))) -ax.set_title(f"Student t draws", fontsize=11) - -ax = axes[1] -ax.plot(list(range(n)), n_data, linestyle='', marker='o', alpha=0.5, ms=4) -ax.vlines(list(range(n)), 0, n_data, lw=0.2) -ax.get_xaxis().set_major_formatter( - ticker.FuncFormatter(lambda x, p: format(int(x), ','))) -ax.set_title(f"$N(0, \sigma^2)$ with $\sigma = {s}$", fontsize=11) - -plt.tight_layout() -if export_figures: - plt.savefig("figures/heavy_tailed_draws.pdf") -plt.show() -``` - -### CCDF plots for the Pareto and Exponential distributions - -When the Pareto tail property holds, the CCDF is eventually log linear. Here -we illustrates this using a Pareto distribution. For comparison, an exponential -distribution is also shown. First we define our domain and the Pareto and -Exponential distributions. - -```{code-cell} ipython3 -x = np.linspace(1, 10, 500) -``` - -```{code-cell} ipython3 -α = 1.5 -def Gp(x): - return x**(-α) -``` - -```{code-cell} ipython3 -λ = 1.0 -def Ge(x): - return np.exp(-λ * x) -``` - -We then plot our distribution on a log-log scale. - -```{code-cell} ipython3 -fig, ax = plt.subplots(figsize=default_figsize) - -ax.plot(np.log(x), np.log(Gp(x)), label="Pareto") -ax.plot(np.log(x), np.log(Ge(x)), label="Exponential") - -ax.legend(fontsize=12, frameon=False, loc="lower left") -ax.set_xlabel("$\ln x$", fontsize=12) -ax.set_ylabel("$\ln G(x)$", fontsize=12) - -if export_figures: - plt.savefig("figures/ccdf_comparison_1.pdf") -plt.show() -``` - -### Empirical CCDF plots for largest firms (Forbes) - -Here we show that the distribution of firm sizes has a Pareto tail. We start -by loading the `forbes_global_2000` dataset. - -```{code-cell} ipython3 -dfff = ch1_data['forbes_global_2000'] -``` - -We calculate values of the empirical CCDF. - -```{code-cell} ipython3 -data = np.asarray(dfff['Market Value'])[0:500] -y_vals = np.empty_like(data, dtype='float64') -n = len(data) -for i, d in enumerate(data): - # record fraction of sample above d - y_vals[i] = np.sum(data >= d) / n -``` - -Now we fit a linear trend line (on the log-log scale). - -```{code-cell} ipython3 -x, y = np.log(data), np.log(y_vals) -results = sm.OLS(y, sm.add_constant(x)).fit() -b, a = results.params -``` - -Finally we produce our plot. - -```{code-cell} ipython3 -fig, ax = plt.subplots(figsize=(7.3, 4)) - -ax.scatter(x, y, alpha=0.3, label="firm size (market value)") -ax.plot(x, x * a + b, 'k-', alpha=0.6, label=f"slope = ${a: 1.2f}$") - -ax.set_xlabel('log value', fontsize=12) -ax.set_ylabel("log prob.", fontsize=12) -ax.legend(loc='lower left', fontsize=12) - -if export_figures: - plt.savefig("figures/empirical_powerlaw_plots_firms_forbes.pdf") -plt.show() -``` - -## Graph Theory - -### Zeta and Pareto distributions - -We begin by defining the Zeta and Pareto distributions. - -```{code-cell} ipython3 -γ = 2.0 -α = γ - 1 -``` - -```{code-cell} ipython3 -def z(k, c=2.0): - return c * k**(-γ) - -k_grid = np.arange(1, 10+1) -``` - -```{code-cell} ipython3 -def p(x, c=2.0): - return c * x**(-γ) - -x_grid = np.linspace(1, 10, 200) -``` - -Then we can produce our plot - -```{code-cell} ipython3 -fig, ax = plt.subplots(figsize=default_figsize) -ax.plot(k_grid, z(k_grid), '-o', label='zeta distribution with $\gamma=2$') -ax.plot(x_grid, p(x_grid), label='density of Pareto with tail index $\\alpha$') -ax.legend(fontsize=12) -ax.set_yticks((0, 1, 2)) -if export_figures: - plt.savefig("figures/zeta_1.pdf") -plt.show() -``` - -### NetworkX digraph plot - -We start by creating a graph object and populating it with edges. - -```{code-cell} ipython3 -G_p = nx.DiGraph() - -edge_list = [ - ('p', 'p'), - ('m', 'p'), ('m', 'm'), ('m', 'r'), - ('r', 'p'), ('r', 'm'), ('r', 'r') -] - -for e in edge_list: - u, v = e - G_p.add_edge(u, v) -``` - -Now we can plot our graph. - -```{code-cell} ipython3 -fig, ax = plt.subplots() -nx.spring_layout(G_p, seed=4) -nx.draw_spring(G_p, ax=ax, node_size=500, with_labels=True, - font_weight='bold', arrows=True, alpha=0.8, - connectionstyle='arc3,rad=0.25', arrowsize=20) -if export_figures: - plt.savefig("figures/networkx_basics_1.pdf") -plt.show() -``` - -The `DiGraph` object has methods that calculate in-degree and out-degree of vertices. - -```{code-cell} ipython3 -G_p.in_degree('p') -``` - -```{code-cell} ipython3 -G_p.out_degree('p') -``` - -Additionally, the `NetworkX` package supplies functions for testing -communication and strong connectedness, as well as to compute strongly -connected components. - -```{code-cell} ipython3 -G = nx.DiGraph() -G.add_edge(1, 1) -G.add_edge(2, 1) -G.add_edge(2, 3) -G.add_edge(3, 2) -list(nx.strongly_connected_components(G)) -``` - -Like `NetworkX`, the Python library `quantecon` -provides access to some graph-theoretic algorithms. - -In the case of QuantEcon's `DiGraph` object, an instance is created via the adjacency matrix. - -```{code-cell} ipython3 -A = ((1, 0, 0), - (1, 1, 1), - (1, 1, 1)) -A = np.array(A) # Convert to NumPy array -G = qe.DiGraph(A) - -G.strongly_connected_components -``` - -### International private credit flows by country - -We begin by loading an adjacency matrix of international private credit flows -(in the form of a NumPy array and a list of country labels). - -```{code-cell} ipython3 -Z = ch1_data["adjacency_matrix"]["Z"] -Z_visual= ch1_data["adjacency_matrix"]["Z_visual"] -countries = ch1_data["adjacency_matrix"]["countries"] -``` - -To calculate our graph's properties, we use hub-based eigenvector -centrality as our centrality measure for this plot. - -```{code-cell} ipython3 -centrality = qbn_io.eigenvector_centrality(Z_visual, authority=False) -``` - -Now we convert our graph features to plot features. - -```{code-cell} ipython3 -node_colors = cm.plasma(qbn_io.to_zero_one_beta(centrality)) -``` - -Finally we produce the plot. - -```{code-cell} ipython3 -X = qbn_io.to_zero_one_beta(Z.sum(axis=1)) - -fig, ax = plt.subplots(figsize=(8, 10)) -plt.axis("off") - -qbn_plot.plot_graph(Z_visual, X, ax, countries, - layout_type='spring', - layout_seed=1234, - node_size_multiple=3000, - edge_size_multiple=0.000006, - tol=0.0, - node_color_list=node_colors) - -if export_figures: - plt.savefig("figures/financial_network_analysis_visualization.pdf") -plt.show() -``` - -### Centrality measures for the credit network - -This figure looks at six different centrality measures. - -We begin by defining a function for calculating eigenvector centrality. - -Hub-based centrality is calculated by default, although authority-based centrality -can be calculated by setting `authority=True`. - -```{code-cell} ipython3 -def eigenvector_centrality(A, k=40, authority=False): - """ - Computes the dominant eigenvector of A. Assumes A is - primitive and uses the power method. - - """ - A_temp = A.T if authority else A - n = len(A_temp) - r = spec_rad(A_temp) - e = r**(-k) * (np.linalg.matrix_power(A_temp, k) @ np.ones(n)) - return e / np.sum(e) -``` - -Here a similar function is defined for calculating Katz centrality. - -```{code-cell} ipython3 -def katz_centrality(A, b=1, authority=False): - """ - Computes the Katz centrality of A, defined as the x solving - - x = 1 + b A x (1 = vector of ones) - - Assumes that A is square. - - If authority=True, then A is replaced by its transpose. - """ - n = len(A) - I = np.identity(n) - C = I - b * A.T if authority else I - b * A - return np.linalg.solve(C, np.ones(n)) -``` - -Now we generate an unweighted version of our matrix to help calculate in-degree and out-degree. - -```{code-cell} ipython3 -D = qbn_io.build_unweighted_matrix(Z) -``` - -We now use the above to calculate the six centrality measures. - -```{code-cell} ipython3 -outdegree = D.sum(axis=1) -ecentral_hub = eigenvector_centrality(Z, authority=False) -kcentral_hub = katz_centrality(Z, b=1/1_700_000) - -indegree = D.sum(axis=0) -ecentral_authority = eigenvector_centrality(Z, authority=True) -kcentral_authority = katz_centrality(Z, b=1/1_700_000, authority=True) -``` - -Here we provide a helper function that returns a DataFrame for each measure. -The DataFrame is ordered by that measure and contains color information. - -```{code-cell} ipython3 -def centrality_plot_data(countries, centrality_measures): - df = pd.DataFrame({'code': countries, - 'centrality':centrality_measures, - 'color': qbn_io.colorise_weights(centrality_measures).tolist() - }) - return df.sort_values('centrality') -``` - -Finally, we plot the various centrality measures. - -```{code-cell} ipython3 -centrality_measures = [outdegree, indegree, - ecentral_hub, ecentral_authority, - kcentral_hub, kcentral_authority] - -ylabels = ['out degree', 'in degree', - 'eigenvector hub','eigenvector authority', - 'Katz hub', 'Katz authority'] - -ylims = [(0, 20), (0, 20), - None, None, - None, None] - - -fig, axes = plt.subplots(3, 2, figsize=(10, 12)) - -axes = axes.flatten() - -for i, ax in enumerate(axes): - df = centrality_plot_data(countries, centrality_measures[i]) - - ax.bar('code', 'centrality', data=df, color=df["color"], alpha=0.6) - - patch = mpatches.Patch(color=None, label=ylabels[i], visible=False) - ax.legend(handles=[patch], fontsize=12, loc="upper left", handlelength=0, frameon=False) - - if ylims[i] is not None: - ax.set_ylim(ylims[i]) - -if export_figures: - plt.savefig("figures/financial_network_analysis_centrality.pdf") -plt.show() -``` - -### Computing in and out degree distributions - -The in-degree distribution evaluated at $k$ is the fraction of nodes in a -network that have in-degree $k$. The in-degree distribution of a `NetworkX` -DiGraph can be calculated using the code below. - -```{code-cell} ipython3 -def in_degree_dist(G): - n = G.number_of_nodes() - iG = np.array([G.in_degree(v) for v in G.nodes()]) - d = [np.mean(iG == k) for k in range(n+1)] - return d -``` - -The out-degree distribution is defined analogously. - -```{code-cell} ipython3 -def out_degree_dist(G): - n = G.number_of_nodes() - oG = np.array([G.out_degree(v) for v in G.nodes()]) - d = [np.mean(oG == k) for k in range(n+1)] - return d -``` - -### Degree distribution for international aircraft trade - -Here we illustrate that the commercial aircraft international trade network is -approximately scale-free by plotting the degree distribution alongside -$f(x)=cx-\gamma$ with $c=0.2$ and $\gamma=1.1$. - -In this calculation of the degree distribution, performed by the NetworkX -function `degree_histogram`, directions are ignored and the network is treated -as an undirected graph. - -```{code-cell} ipython3 -def plot_degree_dist(G, ax, loglog=True, label=None): - "Plot the degree distribution of a graph G on axis ax." - dd = [x for x in nx.degree_histogram(G) if x > 0] - dd = np.array(dd) / np.sum(dd) # normalize - if loglog: - ax.loglog(dd, '-o', lw=0.5, label=label) - else: - ax.plot(dd, '-o', lw=0.5, label=label) -``` - -```{code-cell} ipython3 -fig, ax = plt.subplots(figsize=default_figsize) - -plot_degree_dist(DG, ax, loglog=False, label='degree distribution') - -xg = np.linspace(0.5, 25, 250) -ax.plot(xg, 0.2 * xg**(-1.1), label='power law') -ax.set_xlim(0.9, 22) -ax.set_ylim(0, 0.25) -ax.legend() -if export_figures: - plt.savefig("figures/commercial_aircraft_2019_2.pdf") -plt.show() -``` - -### Random graphs - -The code to produce the Erdos-Renyi random graph, used below, applies the -combinations function from the `itertools` library. The function -`combinations(A, k)` returns a list of all subsets of $A$ of size $k$. For -example: - -```{code-cell} ipython3 -import itertools -letters = 'a', 'b', 'c' -list(itertools.combinations(letters, 2)) -``` - -Below we generate random graphs using the Erdos-Renyi and Barabasi-Albert -algorithms. Here, for convenience, we will define a function to plot these -graphs. - -```{code-cell} ipython3 -def plot_random_graph(RG,ax): - node_pos_dict = nx.spring_layout(RG, k=1.1) - - centrality = nx.degree_centrality(RG) - node_color_list = qbn_io.colorise_weights(list(centrality.values())) - - edge_color_list = [] - for i in range(n): - for j in range(n): - edge_color_list.append(node_color_list[i]) - - nx.draw_networkx_nodes(RG, - node_pos_dict, - node_color=node_color_list, - edgecolors='grey', - node_size=100, - linewidths=2, - alpha=0.8, - ax=ax) - - nx.draw_networkx_edges(RG, - node_pos_dict, - edge_color=edge_colors, - alpha=0.4, - ax=ax) -``` - -### An instance of an Erdos–Renyi random graph - -```{code-cell} ipython3 -n = 100 -p = 0.05 -G_er = qbn_io.erdos_renyi_graph(n, p, seed=1234) -``` - -```{code-cell} ipython3 -fig, axes = plt.subplots(1, 2, figsize=(10, 3.2)) - -axes[0].set_title("Graph visualization") -plot_random_graph(G_er,axes[0]) - -axes[1].set_title("Degree distribution") -plot_degree_dist(G_er, axes[1], loglog=False) - -if export_figures: - plt.savefig("figures/rand_graph_experiments_1.pdf") -plt.show() -``` - -### An instance of a preferential attachment random graph - -```{code-cell} ipython3 -n = 100 -m = 5 -G_ba = nx.generators.random_graphs.barabasi_albert_graph(n, m, seed=123) -``` - -```{code-cell} ipython3 -fig, axes = plt.subplots(1, 2, figsize=(10, 3.2)) - -axes[0].set_title("Graph visualization") -plot_random_graph(G_ba, axes[0]) - -axes[1].set_title("Degree distribution") -plot_degree_dist(G_ba, axes[1], loglog=False) - -if export_figures: - plt.savefig("figures/rand_graph_experiments_2.pdf") -plt.show() -``` diff --git a/code_book/ch_mcs.md b/code_book/ch_mcs.md deleted file mode 100644 index 198d542..0000000 --- a/code_book/ch_mcs.md +++ /dev/null @@ -1,476 +0,0 @@ ---- -jupytext: - cell_metadata_filter: -all - formats: md:myst - text_representation: - extension: .md - format_name: myst - format_version: 0.13 - jupytext_version: 1.10.3 -kernelspec: - display_name: Python 3 - language: python - name: python3 ---- - -# Chapter 4 - Markov Chains and Networks (Python Code) - -```{code-cell} ---- -tags: [hide-output] ---- -! pip install --upgrade quantecon_book_networks -``` - -We begin with some imports. - -```{code-cell} -import quantecon as qe -import quantecon_book_networks -import quantecon_book_networks.input_output as qbn_io -import quantecon_book_networks.plotting as qbn_plt -import quantecon_book_networks.data as qbn_data -ch4_data = qbn_data.markov_chains_and_networks() -export_figures = False -``` - -```{code-cell} ipython3 -import numpy as np -import pandas as pd -import networkx as nx -import matplotlib.pyplot as plt -from matplotlib import cm -from mpl_toolkits.mplot3d import Axes3D -from mpl_toolkits.mplot3d.art3d import Poly3DCollection -quantecon_book_networks.config("matplotlib") -``` - -## Example transition matrices - -In this chapter two transition matrices are used. - -First, a Markov model is estimated in the international growth dynamics study -of [Quah (1993)](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.142.5504&rep=rep1&type=pdf). - -The state is real GDP per capita in a given country relative to the world -average. - -Quah discretizes the possible values to 0–1/4, 1/4–1/2, 1/2–1, 1–2 -and 2–inf, calling these states 1 to 5 respectively. The transitions are over -a one year period. - -```{code-cell} -P_Q = [ - [0.97, 0.03, 0, 0, 0 ], - [0.05, 0.92, 0.03, 0, 0 ], - [0, 0.04, 0.92, 0.04, 0 ], - [0, 0, 0.04, 0.94, 0.02], - [0, 0, 0, 0.01, 0.99] -] -P_Q = np.array(P_Q) -codes_Q = ('1', '2', '3', '4', '5') -``` - -Second, [Benhabib et al. (2015)](https://www.economicdynamics.org/meetpapers/2015/paper_364.pdf) estimate the following transition matrix for intergenerational social mobility. - -The states are percentiles of the wealth distribution. - -In particular, states 1, 2,..., 8, correspond to the percentiles 0-20%, 20-40%, 40-60%, 60-80%, 80-90%, 90-95%, 95-99%, 99-100%. - -```{code-cell} -P_B = [ - [0.222, 0.222, 0.215, 0.187, 0.081, 0.038, 0.029, 0.006], - [0.221, 0.22, 0.215, 0.188, 0.082, 0.039, 0.029, 0.006], - [0.207, 0.209, 0.21, 0.194, 0.09, 0.046, 0.036, 0.008], - [0.198, 0.201, 0.207, 0.198, 0.095, 0.052, 0.04, 0.009], - [0.175, 0.178, 0.197, 0.207, 0.11, 0.067, 0.054, 0.012], - [0.182, 0.184, 0.2, 0.205, 0.106, 0.062, 0.05, 0.011], - [0.123, 0.125, 0.166, 0.216, 0.141, 0.114, 0.094, 0.021], - [0.084, 0.084, 0.142, 0.228, 0.17, 0.143, 0.121, 0.028] - ] - -P_B = np.array(P_B) -codes_B = ('1', '2', '3', '4', '5', '6', '7', '8') -``` - - -## Markov Chains as Digraphs - -### Contour plot of a transition matrix - -Here we define a function for producing contour plots of matrices. - -```{code-cell} -def plot_matrices(matrix, - codes, - ax, - font_size=12, - alpha=0.6, - colormap=cm.viridis, - color45d=None, - xlabel='sector $j$', - ylabel='sector $i$'): - - ticks = range(len(matrix)) - - levels = np.sqrt(np.linspace(0, 0.75, 100)) - - - if color45d != None: - co = ax.contourf(ticks, - ticks, - matrix, - alpha=alpha, cmap=colormap) - ax.plot(ticks, ticks, color=color45d) - else: - co = ax.contourf(ticks, - ticks, - matrix, - levels, - alpha=alpha, cmap=colormap) - - ax.set_xlabel(xlabel, fontsize=font_size) - ax.set_ylabel(ylabel, fontsize=font_size) - ax.set_yticks(ticks) - ax.set_yticklabels(codes_B) - ax.set_xticks(ticks) - ax.set_xticklabels(codes_B) - -``` - -Now we use our function to produce a plot of the transition matrix for -intergenerational social mobility, $P_B$. - -```{code-cell} -fig, ax = plt.subplots(figsize=(6,6)) -plot_matrices(P_B.transpose(), codes_B, ax, alpha=0.75, - colormap=cm.viridis, color45d='black', - xlabel='state at time $t$', ylabel='state at time $t+1$') - -if export_figures: - plt.savefig("figures/markov_matrix_visualization.pdf") -plt.show() -``` - - -### Wealth percentile over time - -Here we compare the mixing of the transition matrix for intergenerational -social mobility $P_B$ and the transition matrix for international growth -dynamics $P_Q$. - -We begin by creating `quantecon` `MarkovChain` objects with each of our transition -matrices. - -```{code-cell} -mc_B = qe.MarkovChain(P_B, state_values=range(1, 9)) -mc_Q = qe.MarkovChain(P_Q, state_values=range(1, 6)) -``` - -Next we define a function to plot simulations of Markov chains. - -Two simulations will be run for each `MarkovChain`, one starting at the -minimum initial value and one at the maximum. - -```{code-cell} -def sim_fig(ax, mc, T=100, seed=14, title=None): - X1 = mc.simulate(T, init=1, random_state=seed) - X2 = mc.simulate(T, init=max(mc.state_values), random_state=seed+1) - ax.plot(X1) - ax.plot(X2) - ax.set_xlabel("time") - ax.set_ylabel("state") - ax.set_title(title, fontsize=12) -``` - -Finally, we produce the figure. - -```{code-cell} -fig, axes = plt.subplots(2, 1, figsize=(6, 4)) -ax = axes[0] -sim_fig(axes[0], mc_B, title="$P_B$") -sim_fig(axes[1], mc_Q, title="$P_Q$") -axes[1].set_yticks((1, 2, 3, 4, 5)) - -plt.tight_layout() -if export_figures: - plt.savefig("figures/benhabib_mobility_mixing.pdf") -plt.show() -``` - - -### Predicted vs realized cross-country income distributions for 2019 - -Here we load a `pandas` `DataFrame` of GDP per capita data for countries compared to the global average. - -```{code-cell} -gdppc_df = ch4_data['gdppc_df'] -gdppc_df.head() -``` - -Now we assign countries bins, as per Quah (1993). - -```{code-cell} -q = [0, 0.25, 0.5, 1.0, 2.0, np.inf] -l = [0, 1, 2, 3, 4] - -x = pd.cut(gdppc_df.gdppc_r, bins=q, labels=l) -gdppc_df['interval'] = x - -gdppc_df = gdppc_df.reset_index() -gdppc_df['interval'] = gdppc_df['interval'].astype(float) -gdppc_df['year'] = gdppc_df['year'].astype(float) -``` - -Here we define a function for calculating the cross-country income -distributions for a given date range. - -```{code-cell} -def gdp_dist_estimate(df, l, yr=(1960, 2019)): - Y = np.zeros(len(l)) - for i in l: - Y[i] = df[ - (df['interval'] == i) & - (df['year'] <= yr[1]) & - (df['year'] >= yr[0]) - ].count()[0] - - return Y / Y.sum() -``` - -We calculate the true distribution for 1985. - -```{code-cell} -ψ_1985 = gdp_dist_estimate(gdppc_df,l,yr=(1985, 1985)) -``` - -Now we use the transition matrix to update the 1985 distribution $t = 2019 - 1985 = 34$ -times to get our predicted 2019 distribution. - -```{code-cell} -ψ_2019_predicted = ψ_1985 @ np.linalg.matrix_power(P_Q, 2019-1985) -``` - -Now, calculate the true 2019 distribution. - -```{code-cell} -ψ_2019 = gdp_dist_estimate(gdppc_df,l,yr=(2019, 2019)) -``` - -Finally we produce the plot. - -```{code-cell} -states = np.arange(0, 5) -ticks = range(5) -codes_S = ('1', '2', '3', '4', '5') - -fig, ax = plt.subplots(figsize=(6, 4)) -width = 0.4 -ax.plot(states, ψ_2019_predicted, '-o', alpha=0.7, label='predicted') -ax.plot(states, ψ_2019, '-o', alpha=0.7, label='realized') -ax.set_xlabel("state") -ax.set_ylabel("probability") -ax.set_yticks((0.15, 0.2, 0.25, 0.3)) -ax.set_xticks(ticks) -ax.set_xticklabels(codes_S) - -ax.legend(loc='upper center', fontsize=12) -if export_figures: - plt.savefig("figures/quah_gdppc_prediction.pdf") -plt.show() - -``` - -### Distribution dynamics - -Here we define a function for plotting the convergence of marginal -distributions $\psi$ under a transition matrix $P$ on the unit simplex. - -```{code-cell} -def convergence_plot(ψ, P, n=14, angle=50): - - ax = qbn_plt.unit_simplex(angle) - - # Convergence plot - - P = np.array(P) - - ψ = ψ # Initial condition - - x_vals, y_vals, z_vals = [], [], [] - for t in range(n): - x_vals.append(ψ[0]) - y_vals.append(ψ[1]) - z_vals.append(ψ[2]) - ψ = ψ @ P - - ax.scatter(x_vals, y_vals, z_vals, c='darkred', s=80, alpha=0.7, depthshade=False) - - mc = qe.MarkovChain(P) - ψ_star = mc.stationary_distributions[0] - ax.scatter(ψ_star[0], ψ_star[1], ψ_star[2], c='k', s=80) - - return ψ - -``` - -Now we define P. - -```{code-cell} -P = ( - (0.9, 0.1, 0.0), - (0.4, 0.4, 0.2), - (0.1, 0.1, 0.8) - ) -``` - -#### A trajectory from $\psi_0 = (0, 0, 1)$ - -Here we see the sequence of marginals appears to converge. - -```{code-cell} -ψ_0 = (0, 0, 1) -ψ = convergence_plot(ψ_0, P) -if export_figures: - plt.savefig("figures/simplex_2.pdf") -plt.show() -``` - -#### A trajectory from $\psi_0 = (0, 1/2, 1/2)$ - -Here we see again that the sequence of marginals appears to converge, and the -limit appears not to depend on the initial distribution. - -```{code-cell} -ψ_0 = (0, 1/2, 1/2) -ψ = convergence_plot(ψ_0, P, n=12) -if export_figures: - plt.savefig("figures/simplex_3.pdf") -plt.show() -``` - - -### Distribution projections from $P_B$ - -Here we define a function for plotting $\psi$ after $n$ iterations of the -transition matrix $P$. The distribution $\psi_0$ is taken as the uniform -distribution over the -state space. - -```{code-cell} -def transition(P, n, ax=None): - - P = np.array(P) - nstates = P.shape[1] - s0 = np.ones(8) * 1/nstates - s = s0 - - for i in range(n): - s = s @ P - - if ax is None: - fig, ax = plt.subplots() - - ax.plot(range(1, nstates+1), s, '-o', alpha=0.6) - ax.set(ylim=(0, 0.25), - xticks=((1, nstates))) - ax.set_title(f"$t = {n}$") - - return ax -``` - -We now generate the marginal distributions after 0, 1, 2, and 100 iterations for $P_B$. - -```{code-cell} -ns = (0, 1, 2, 100) -fig, axes = plt.subplots(1, len(ns), figsize=(6, 4)) - -for n, ax in zip(ns, axes): - ax = transition(P_B, n, ax=ax) - ax.set_xlabel("Quantile") - -plt.tight_layout() -if export_figures: - plt.savefig("figures/benhabib_mobility_dists.pdf") -plt.show() -``` - - - -## Asymptotics - -### Convergence of the empirical distribution to $\psi^*$ - -We begin by creating a `MarkovChain` object, taking $P_B$ as the transition matrix. - -```{code-cell} -mc = qe.MarkovChain(P_B) -``` - -Next we use the `quantecon` package to calculate the true stationary distribution. - -```{code-cell} -stationary = mc.stationary_distributions[0] -n = len(mc.P) -``` - -Now we define a function to simulate the Markov chain. - -```{code-cell} -def simulate_distribution(mc, T=100): - # Simulate path - n = len(mc.P) - path = mc.simulate_indices(ts_length=T, random_state=1) - distribution = np.empty(n) - for i in range(n): - distribution[i] = np.mean(path==i) - return distribution - -``` - -We run simulations of length 10, 100, 1,000 and 10,000. - -```{code-cell} -lengths = [10, 100, 1_000, 10_000] -dists = [] - -for t in lengths: - dists.append(simulate_distribution(mc, t)) -``` - -Now we produce the plots. - -We see that the simulated distribution starts to approach the true stationary distribution. - -```{code-cell} -fig, axes = plt.subplots(2, 2, figsize=(9, 6), sharex='all')#, sharey='all') - -axes = axes.flatten() - -for dist, ax, t in zip(dists, axes, lengths): - - ax.plot(np.arange(n)+1 + .25, - stationary, - '-o', - #width = 0.25, - label='$\\psi^*$', - alpha=0.75) - - ax.plot(np.arange(n)+1, - dist, - '-o', - #width = 0.25, - label=f'$\\hat \\psi_k$ with $k={t}$', - alpha=0.75) - - - ax.set_xlabel("state", fontsize=12) - ax.set_ylabel("prob.", fontsize=12) - ax.set_xticks(np.arange(n)+1) - ax.legend(loc='upper right', fontsize=12, frameon=False) - ax.set_ylim(0, 0.5) - -if export_figures: - plt.savefig("figures/benhabib_ergodicity_1.pdf") -plt.show() -``` diff --git a/code_book/ch_opt.md b/code_book/ch_opt.md deleted file mode 100644 index 2b0dce1..0000000 --- a/code_book/ch_opt.md +++ /dev/null @@ -1,416 +0,0 @@ ---- -jupytext: - cell_metadata_filter: -all - formats: md:myst - text_representation: - extension: .md - format_name: myst - format_version: 0.13 - jupytext_version: 1.10.3 -kernelspec: - display_name: Python 3 - language: python - name: python3 ---- - -# Chapter 3 - Optimal Flows (Python Code) - -```{code-cell} ---- -tags: [hide-output] ---- -! pip install --upgrade quantecon_book_networks -``` - -We begin with some imports - -```{code-cell} -import quantecon as qe -import quantecon_book_networks -import quantecon_book_networks.input_output as qbn_io -import quantecon_book_networks.plotting as qbn_plt -import quantecon_book_networks.data as qbn_data -ch3_data = qbn_data.optimal_flows() -export_figures = False -``` - -```{code-cell} -import numpy as np -from scipy.optimize import linprog -import networkx as nx -import ot -import matplotlib.pyplot as plt -from matplotlib import cm -from matplotlib.patches import Polygon -from matplotlib.artist import Artist -quantecon_book_networks.config("matplotlib") -``` - -## Linear Programming and Duality - -### Betweenness centrality (by color and node size) for the Florentine families - -We load the Florentine Families data from the NetworkX package. -```{code-cell} -G = nx.florentine_families_graph() -``` - -Next we calculate betweenness centrality. - -```{code-cell} -bc_dict = nx.betweenness_centrality(G) -``` - -And we produce the plot. - -```{code-cell} -fig, ax = plt.subplots(figsize=(9.4, 9.4)) - -plt.axis("off") -nx.draw_networkx( - G, - ax=ax, - pos=nx.spring_layout(G, seed=1234), - with_labels=True, - alpha=.8, - arrowsize=15, - connectionstyle="arc3,rad=0.1", - node_size=[10_000*(size+0.1) for size in bc_dict.values()], - node_color=[cm.plasma(bc+0.4) for bc in bc_dict.values()], -) -if export_figures: - plt.savefig("figures/betweenness_centrality_1.pdf") -``` - -### Revenue maximizing quantities and a Python implementation of linear programming - -First we specify our linear program. - -```{code-cell} -A = ((2, 5), - (4, 2)) -b = (30, 20) -c = (-3, -4) # minus in order to minimize -``` - -And now we use SciPy's linear programing module to solve our linear program. - -```{code-cell} -from scipy.optimize import linprog -result = linprog(c, A_ub=A, b_ub=b) -print(result.x) -``` - -Here we produce a visualization of what is being done. - -```{code-cell} -fig, ax = plt.subplots(figsize=(8, 4.5)) -plt.rcParams['font.size'] = '14' - -# Draw constraint lines - -ax.plot(np.linspace(-1, 17.5, 100), 6-0.4*np.linspace(-1, 17.5, 100)) -ax.plot(np.linspace(-1, 5.5, 100), 10-2*np.linspace(-1, 5.5, 100)) -ax.text(10, 2.5, "$2q_1 + 5q_2 \leq 30$") -ax.text(1.5, 8, "$4q_1 + 2q_2 \leq 20$") -ax.text(-2, 2, "$q_2 \geq 0$") -ax.text(2.5, -0.7, "$q_1 \geq 0$") - -# Draw the feasible region -feasible_set = Polygon(np.array([[0, 0], - [0, 6], - [2.5, 5], - [5, 0]])) -ax.add_artist(feasible_set) -Artist.set_alpha(feasible_set, 0.2) - -# Draw the objective function -ax.plot(np.linspace(-1, 5.5, 100), 3.875-0.75*np.linspace(-1, 5.5, 100), 'g-') -ax.plot(np.linspace(-1, 5.5, 100), 5.375-0.75*np.linspace(-1, 5.5, 100), 'g-') -ax.plot(np.linspace(-1, 5.5, 100), 6.875-0.75*np.linspace(-1, 5.5, 100), 'g-') -ax.text(5.8, 1, "revenue $ = 3q_1 + 4q_2$") - -# Draw the optimal solution -ax.plot(2.5, 5, "o", color="black") -ax.text(2.7, 5.2, "optimal solution") - -for spine in ['right', 'top']: - ax.spines[spine].set_color('none') - -ax.set_xticks(()) -ax.set_yticks(()) - -for spine in ['left', 'bottom']: - ax.spines[spine].set_position('zero') - -ax.set_ylim(-1, 8) - -if export_figures: - plt.savefig("figures/linear_programming_1.pdf") -plt.show() -``` - -## Optimal Transport - - -### Transforming one distribution into another - -Below we provide code to produce a visualization of transforming one -distribution into another in one dimension. - -```{code-cell} -σ = 0.1 - -def ϕ(z): - return (1 / np.sqrt(2 * σ**2 * np.pi)) * np.exp(-z**2 / (2 * σ**2)) - -def v(x, a=0.4, b=0.6, s=1.0, t=1.4): - return a * ϕ(x - s) + b * ϕ(x - t) - -fig, ax = plt.subplots(figsize=(10, 4)) - -x = np.linspace(0.2, 4, 1000) -ax.plot(x, v(x), label="$\\phi$") -ax.plot(x, v(x, s=3.0, t=3.3, a=0.6), label="$\\psi$") - -ax.legend(loc='upper left', fontsize=12, frameon=False) - -ax.arrow(1.8, 1.6, 0.8, 0.0, width=0.01, head_width=0.08) -ax.annotate('transform', xy=(1.9, 1.9), fontsize=12) -if export_figures: - plt.savefig("figures/ot_figs_1.pdf") -plt.show() -``` - -### Function to solve a transport problem via linear programming - -Here we define a function to solve optimal transport problems using linear programming. - -```{code-cell} -def ot_solver(phi, psi, c, method='highs-ipm'): - """ - Solve the OT problem associated with distributions phi, psi - and cost matrix c. - Parameters - ---------- - phi : 1-D array - Distribution over the source locations. - psi : 1-D array - Distribution over the target locations. - c : 2-D array - Cost matrix. - """ - n, m = len(phi), len(psi) - - # vectorize c - c_vec = c.reshape((m * n, 1), order='F') - - # Construct A and b - A1 = np.kron(np.ones((1, m)), np.identity(n)) - A2 = np.kron(np.identity(m), np.ones((1, n))) - A = np.vstack((A1, A2)) - b = np.hstack((phi, psi)) - - # Call sover - res = linprog(c_vec, A_eq=A, b_eq=b, method=method) - - # Invert the vec operation to get the solution as a matrix - pi = res.x.reshape((n, m), order='F') - return pi -``` - -Now we can set up a simple optimal transport problem. - -```{code-cell} -phi = np.array((0.5, 0.5)) -psi = np.array((1, 0)) -c = np.ones((2, 2)) -``` - -Next we solve using the above function. - -```{code-cell} -ot_solver(phi, psi, c) -``` - -We see we get the same result as when using the Python optimal transport -package. - -```{code-cell} -ot.emd(phi, psi, c) -``` - -### An optimal transport problem solved by linear programming - -Here we demonstrate a more detailed optimal transport problem. We begin by defining a node class. - -```{code-cell} -class Node: - - def __init__(self, x, y, mass, group, name): - - self.x, self.y = x, y - self.mass, self.group = mass, group - self.name = name -``` - -Now we define a function for randomly generating nodes. - -```{code-cell} -from scipy.stats import betabinom - -def build_nodes_of_one_type(group='phi', n=100, seed=123): - - nodes = [] - np.random.seed(seed) - - for i in range(n): - - if group == 'phi': - m = 1/n - x = np.random.uniform(-2, 2) - y = np.random.uniform(-2, 2) - else: - m = betabinom.pmf(i, n-1, 2, 2) - x = 0.6 * np.random.uniform(-1.5, 1.5) - y = 0.6 * np.random.uniform(-1.5, 1.5) - - name = group + str(i) - nodes.append(Node(x, y, m, group, name)) - - return nodes -``` - -We now generate our source and target nodes. -```{code-cell} -n_phi = 32 -n_psi = 32 - -phi_list = build_nodes_of_one_type(group='phi', n=n_phi) -psi_list = build_nodes_of_one_type(group='psi', n=n_psi) - -phi_probs = [phi.mass for phi in phi_list] -psi_probs = [psi.mass for psi in psi_list] -``` - -Now we define our transport costs. -```{code-cell} -c = np.empty((n_phi, n_psi)) -for i in range(n_phi): - for j in range(n_psi): - x0, y0 = phi_list[i].x, phi_list[i].y - x1, y1 = psi_list[j].x, psi_list[j].y - c[i, j] = np.sqrt((x0-x1)**2 + (y0-y1)**2) -``` - -We solve our optimal transport problem using the Python optimal transport package. - -```{code-cell} -pi = ot.emd(phi_probs, psi_probs, c) -``` - -Finally we produce a graph of our sources, targets, and optimal transport plan. - -```{code-cell} -g = nx.DiGraph() -g.add_nodes_from([phi.name for phi in phi_list]) -g.add_nodes_from([psi.name for psi in psi_list]) - -for i in range(n_phi): - for j in range(n_psi): - if pi[i, j] > 0: - g.add_edge(phi_list[i].name, psi_list[j].name, weight=pi[i, j]) - -node_pos_dict={} -for phi in phi_list: - node_pos_dict[phi.name] = (phi.x, phi.y) -for psi in psi_list: - node_pos_dict[psi.name] = (psi.x, psi.y) - -node_color_list = [] -node_size_list = [] -scale = 8_000 -for phi in phi_list: - node_color_list.append('blue') - node_size_list.append(phi.mass * scale) -for psi in psi_list: - node_color_list.append('red') - node_size_list.append(psi.mass * scale) - -fig, ax = plt.subplots(figsize=(7, 10)) -plt.axis('off') - -nx.draw_networkx_nodes(g, - node_pos_dict, - node_color=node_color_list, - node_size=node_size_list, - edgecolors='grey', - linewidths=1, - alpha=0.5, - ax=ax) - -nx.draw_networkx_edges(g, - node_pos_dict, - arrows=True, - connectionstyle='arc3,rad=0.1', - alpha=0.6) -if export_figures: - plt.savefig("figures/ot_large_scale_1.pdf") -plt.show() -``` - -### Solving linear assignment as an optimal transport problem - -Here we set up a linear assignment problem (matching $n$ workers to $n$ jobs). - -```{code-cell} -n = 4 -phi = np.ones(n) -psi = np.ones(n) -``` - -We generate our cost matrix (the cost of training the $i$th worker for the $j$th job) - -```{code-cell} -c = np.random.uniform(size=(n, n)) -``` - -Finally, we solve our linear assignment problem as a special case of optimal transport. - -```{code-cell} -ot.emd(phi, psi, c) -``` - -### Python Spatial Analysis library - -Readers interested in computational optimal transport should also consider -PySAL, the [Python Spatial Analysis library](https://pysal.org/). See, for -example, https://pysal.org/spaghetti/notebooks/transportation-problem.html. - -### The General Flow Problem - -Here we solve a simple network flow problem as a linear program. We begin by -defining the node-edge incidence matrix. - -```{code-cell} -A = ( -( 1, 1, 0, 0), -(-1, 0, 1, 0), -( 0, 0, -1, 1), -( 0, -1, 0, -1) -) -``` - -Now we define exogenous supply and transport costs. - -```{code-cell} -b = (10, 0, 0, -10) -c = (1, 4, 1, 1) -``` - -Finally we solve as a linear program. - -```{code-cell} -result = linprog(c, A_eq=A, b_eq=b, method='highs-ipm') -print(result.x) -``` diff --git a/code_book/ch_opt_julia.md b/code_book/ch_opt_julia.md deleted file mode 100644 index 6cd231f..0000000 --- a/code_book/ch_opt_julia.md +++ /dev/null @@ -1,118 +0,0 @@ ---- -jupytext: - cell_metadata_filter: -all - formats: md:myst - text_representation: - extension: .md - format_name: myst - format_version: 0.13 - jupytext_version: 1.10.3 -kernelspec: - display_name: Julia - language: Julia - name: julia-1.9 ---- - - -# Chapter 3 - Optimal Flows (Julia Code) - -## Bellman’s Method - -Here we demonstrate solving a shortest path problem using Bellman's method. - -Our first step is to set up the cost function, which we store as an array -called `c`. Note that we set `c[i, j] = Inf` when no edge exists from `i` to -`j`. - -```{code-cell} -:tags: ["remove-output"] - -c = fill(Inf, (7, 7)) -c[1, 2], c[1, 3], c[1, 4] = 1, 5, 3 -c[2, 4], c[2, 5] = 9, 6 -c[3, 6] = 2 -c[4, 6] = 4 -c[5, 7] = 4 -c[6, 7] = 1 -c[7, 7] = 0 -``` - -Next we define the Bellman operator. - -```{code-cell} -:tags: ["remove-output"] - -function T(q) - Tq = similar(q) - n = length(q) - for x in 1:n - Tq[x] = minimum(c[x, :] + q[:]) - end - return Tq -end -``` - -Now we arbitrarily set $q \equiv 0$, generate the sequence of iterates $T_q$, -$T^2_q$, $T^3_q$ and plot them. By $T^3_q$ has already converged on $q^∗$. - -```{code-cell} -using PyPlot -export_figures = false -fig, ax = plt.subplots(figsize=(6, 4)) - -n = 7 -q = zeros(n) -ax.plot(1:n, q) -ax.set_xlabel("cost-to-go") -ax.set_ylabel("nodes") - -for i in 1:3 - new_q = T(q) - ax.plot(1:n, new_q, "-o", alpha=0.7, label="iterate $i") - q = new_q -end - -ax.legend() -if export_figures == true - plt.savefig("figures/shortest_path_iter_1.pdf") -end -``` - -## Linear programming - -When solving linear programs, one option is to use a domain specific modeling -language to set out the objective and constraints in the optimization problem. -Here we demonstrate the Julia package `JuMP`. - -```{code-cell} -using JuMP -using GLPK -``` - -We create our model object and select our solver. - -```{code-cell} -m = Model() -set_optimizer(m, GLPK.Optimizer) -``` - -Now we add variables, constraints and an objective to our model. - -```{code-cell} -:tags: ["remove-output"] - -@variable(m, q1 >= 0) -@variable(m, q2 >= 0) -@constraint(m, 2q1 + 5q2 <= 30) -@constraint(m, 4q1 + 2q2 <= 20) -@objective(m, Max, 3q1 + 4q2) -``` - -Finally we solve our linear program. - -```{code-cell} -optimize!(m) - -println(value.(q1)) -println(value.(q2)) -``` diff --git a/code_book/ch_production.md b/code_book/ch_production.md deleted file mode 100644 index c6d7b4a..0000000 --- a/code_book/ch_production.md +++ /dev/null @@ -1,416 +0,0 @@ ---- -jupytext: - cell_metadata_filter: -all - formats: md:myst - text_representation: - extension: .md - format_name: myst - format_version: 0.13 - jupytext_version: 1.11.5 -kernelspec: - display_name: Python 3 - language: python - name: python3 ---- - -# Chapter 2 - Production (Python Code) - -```{code-cell} ipython3 ---- -tags: [hide-output] ---- -! pip install --upgrade quantecon_book_networks -``` - -We begin with some imports. - -```{code-cell} ipython3 -import quantecon as qe -import quantecon_book_networks -import quantecon_book_networks.input_output as qbn_io -import quantecon_book_networks.plotting as qbn_plt -import quantecon_book_networks.data as qbn_data -ch2_data = qbn_data.production() -default_figsize = (6, 4) -export_figures = False -``` - -```{code-cell} ipython3 -import numpy as np -import pandas as pd -import networkx as nx -import matplotlib.pyplot as plt -import matplotlib.cm as cm -import matplotlib.colors as plc -from matplotlib import cm -quantecon_book_networks.config("matplotlib") -``` - -## Multisector Models - -We start by loading a graph of linkages between 15 US sectors in 2021. - -Our graph comes as a list of sector codes, an adjacency matrix of sales between -the sectors, and a list the total sales of each sector. - -In particular, `Z[i,j]` is the sales from industry `i` to industry `j`, and `X[i]` is the the total sales -of each sector `i`. - -```{code-cell} ipython3 -codes = ch2_data["us_sectors_15"]["codes"] -Z = ch2_data["us_sectors_15"]["adjacency_matrix"] -X = ch2_data["us_sectors_15"]["total_industry_sales"] -``` - -Now we define a function to build coefficient matrices. - -Two coefficient matrices are returned. The backward linkage case, where sales -between sector `i` and `j` are given as a fraction of total sales of sector -`j`. The forward linkage case, where sales between sector `i` and `j` are -given as a fraction of total sales of sector `i`. - -```{code-cell} ipython3 -def build_coefficient_matrices(Z, X): - """ - Build coefficient matrices A and F from Z and X via - - A[i, j] = Z[i, j] / X[j] - F[i, j] = Z[i, j] / X[i] - - """ - A, F = np.empty_like(Z), np.empty_like(Z) - n = A.shape[0] - for i in range(n): - for j in range(n): - A[i, j] = Z[i, j] / X[j] - F[i, j] = Z[i, j] / X[i] - - return A, F - -A, F = build_coefficient_matrices(Z, X) -``` - -### Backward linkages for 15 US sectors in 2021 - -Here we calculate the hub-based eigenvector centrality of our backward linkage coefficient matrix. - -```{code-cell} ipython3 -centrality = qbn_io.eigenvector_centrality(A) -``` - -Now we use the `quantecon_book_networks` package to produce our plot. - -```{code-cell} ipython3 -fig, ax = plt.subplots(figsize=(8, 10)) -plt.axis("off") -color_list = qbn_io.colorise_weights(centrality, beta=False) -# Remove self-loops -A1 = A.copy() -for i in range(A1.shape[0]): - A1[i][i] = 0 -qbn_plt.plot_graph(A1, X, ax, codes, - layout_type='spring', - layout_seed=5432167, - tol=0.0, - node_color_list=color_list) - -if export_figures: - plt.savefig("figures/input_output_analysis_15.pdf") -plt.show() -``` - -### Eigenvector centrality of across US industrial sectors - -Now we plot a bar chart of hub-based eigenvector centrality by sector. - -```{code-cell} ipython3 -fig, ax = plt.subplots(figsize=default_figsize) -ax.bar(codes, centrality, color=color_list, alpha=0.6) -ax.set_ylabel("eigenvector centrality", fontsize=12) -if export_figures: - plt.savefig("figures/input_output_analysis_15_ec.pdf") -plt.show() -``` - -### Output multipliers across 15 US industrial sectors - -Output multipliers are equal to the authority-based Katz centrality measure of -the backward linkage coefficient matrix. - -Here we calculate authority-based -Katz centrality using the `quantecon_book_networks` package. - -```{code-cell} ipython3 -omult = qbn_io.katz_centrality(A, authority=True) -``` - -```{code-cell} ipython3 -fig, ax = plt.subplots(figsize=default_figsize) -omult_color_list = qbn_io.colorise_weights(omult,beta=False) -ax.bar(codes, omult, color=omult_color_list, alpha=0.6) -ax.set_ylabel("Output multipliers", fontsize=12) -if export_figures: - plt.savefig("figures/input_output_analysis_15_omult.pdf") -plt.show() -``` - -### Forward linkages and upstreamness over US industrial sectors - -Upstreamness is the hub-based Katz centrality of the forward linkage -coefficient matrix. - -Here we calculate hub-based Katz centrality. - -```{code-cell} ipython3 -upstreamness = qbn_io.katz_centrality(F) -``` - -Now we plot the network. - -```{code-cell} ipython3 -fig, ax = plt.subplots(figsize=(8, 10)) -plt.axis("off") -upstreamness_color_list = qbn_io.colorise_weights(upstreamness,beta=False) -# Remove self-loops -for i in range(F.shape[0]): - F[i][i] = 0 -qbn_plt.plot_graph(F, X, ax, codes, - layout_type='spring', # alternative layouts: spring, circular, random, spiral - layout_seed=5432167, - tol=0.0, - node_color_list=upstreamness_color_list) - -if export_figures: - plt.savefig("figures/input_output_analysis_15_fwd.pdf") -plt.show() -``` - -### Relative upstreamness of US industrial sectors - -Here we produce a barplot of upstreamness. - -```{code-cell} ipython3 -fig, ax = plt.subplots(figsize=default_figsize) -ax.bar(codes, upstreamness, color=upstreamness_color_list, alpha=0.6) -ax.set_ylabel("upstreamness", fontsize=12) -if export_figures: - plt.savefig("figures/input_output_analysis_15_up.pdf") -plt.show() -``` - -### Hub-based Katz centrality of across 15 US industrial sectors - -Next we plot the hub-based Katz centrality of the backward linkage coefficient matrix. - -```{code-cell} ipython3 -kcentral = qbn_io.katz_centrality(A) -``` - -```{code-cell} ipython3 -fig, ax = plt.subplots(figsize=default_figsize) -kcentral_color_list = qbn_io.colorise_weights(kcentral,beta=False) -ax.bar(codes, kcentral, color=kcentral_color_list, alpha=0.6) -ax.set_ylabel("Katz hub centrality", fontsize=12) -if export_figures: - plt.savefig("figures/input_output_analysis_15_katz.pdf") -plt.show() -``` - -### The Leontief inverse 𝐿 (hot colors are larger values) - -We construct the Leontief inverse matrix from 15 sector adjacency matrix. - -```{code-cell} ipython3 -I = np.identity(len(A)) -L = np.linalg.inv(I - A) -``` - -Now we produce the plot. - -```{code-cell} ipython3 -fig, ax = plt.subplots(figsize=(6.5, 5.5)) - -ticks = range(len(L)) - -levels = np.sqrt(np.linspace(0, 0.75, 100)) - -co = ax.contourf(ticks, - ticks, - L, - levels, - alpha=0.85, cmap=cm.plasma) - -ax.set_xlabel('sector $j$', fontsize=12) -ax.set_ylabel('sector $i$', fontsize=12) -ax.set_yticks(ticks) -ax.set_yticklabels(codes) -ax.set_xticks(ticks) -ax.set_xticklabels(codes) - -if export_figures: - plt.savefig("figures/input_output_analysis_15_leo.pdf") -plt.show() -``` - -### Propagation of demand shocks via backward linkages - -We begin by generating a demand shock vector $d$. - -```{code-cell} ipython3 -N = len(A) -np.random.seed(1234) -d = np.random.rand(N) -d[6] = 1 # positive shock to agriculture -``` - -Now we simulate the demand shock propagating through the economy. - -```{code-cell} ipython3 -sim_length = 11 -x = d -x_vecs = [] -for i in range(sim_length): - if i % 2 ==0: - x_vecs.append(x) - x = A @ x -``` - -Finally, we plot the shock propagating through the economy. - -```{code-cell} ipython3 -fig, axes = plt.subplots(3, 2, figsize=(8, 10)) -axes = axes.flatten() - -for ax, x_vec, i in zip(axes, x_vecs, range(sim_length)): - if i % 2 != 0: - pass - ax.set_title(f"round {i*2}") - x_vec_cols = qbn_io.colorise_weights(x_vec,beta=False) - # remove self-loops - for i in range(len(A)): - A[i][i] = 0 - qbn_plt.plot_graph(A, X, ax, codes, - layout_type='spring', - layout_seed=342156, - node_color_list=x_vec_cols, - node_size_multiple=0.00028, - edge_size_multiple=0.8) - -plt.tight_layout() -if export_figures: - plt.savefig("figures/input_output_analysis_15_shocks.pdf") -plt.show() -``` - -### Network for 71 US sectors in 2021 - -We start by loading a graph of linkages between 71 US sectors in 2021. - -```{code-cell} ipython3 -codes_71 = ch2_data['us_sectors_71']['codes'] -A_71 = ch2_data['us_sectors_71']['adjacency_matrix'] -X_71 = ch2_data['us_sectors_71']['total_industry_sales'] -``` - -Next we calculate our graph's properties. - -We use hub-based eigenvector centrality as our centrality measure for this plot. - -```{code-cell} ipython3 -centrality_71 = qbn_io.eigenvector_centrality(A_71) -color_list_71 = qbn_io.colorise_weights(centrality_71,beta=False) -``` - -Finally we produce the plot. - -```{code-cell} ipython3 -fig, ax = plt.subplots(figsize=(10, 12)) -plt.axis("off") -# Remove self-loops -for i in range(A_71.shape[0]): - A_71[i][i] = 0 -qbn_plt.plot_graph(A_71, X_71, ax, codes_71, - node_size_multiple=0.0005, - edge_size_multiple=4.0, - layout_type='spring', - layout_seed=5432167, - tol=0.01, - node_color_list=color_list_71) - -if export_figures: - plt.savefig("figures/input_output_analysis_71.pdf") -plt.show() -``` - -### Network for 114 Australian industry sectors in 2020 - -Next we load a graph of linkages between 114 Australian sectors in 2020. - -```{code-cell} ipython3 -codes_114 = ch2_data['au_sectors_114']['codes'] -A_114 = ch2_data['au_sectors_114']['adjacency_matrix'] -X_114 = ch2_data['au_sectors_114']['total_industry_sales'] -``` - -Next we calculate our graph's properties. - -We use hub-based eigenvector centrality as our centrality measure for this plot. - -```{code-cell} ipython3 -centrality_114 = qbn_io.eigenvector_centrality(A_114) -color_list_114 = qbn_io.colorise_weights(centrality_114,beta=False) -``` - -Finally we produce the plot. - -```{code-cell} ipython3 -fig, ax = plt.subplots(figsize=(11, 13.2)) -plt.axis("off") -# Remove self-loops -for i in range(A_114.shape[0]): - A_114[i][i] = 0 -qbn_plt.plot_graph(A_114, X_114, ax, codes_114, - node_size_multiple=0.008, - edge_size_multiple=5.0, - layout_type='spring', - layout_seed=5432167, - tol=0.03, - node_color_list=color_list_114) - -if export_figures: - plt.savefig("figures/input_output_analysis_aus_114.pdf") -plt.show() -``` - -### GDP growth rates and std. deviations (in parentheses) for 8 countries - -Here we load a `pandas` DataFrame of GDP growth rates. - -```{code-cell} ipython3 -gdp_df = ch2_data['gdp_df'] -gdp_df.head() -``` - -Now we plot the growth rates and calculate their standard deviations. - -```{code-cell} ipython3 -fig, axes = plt.subplots(5, 2, figsize=(8, 9)) -axes = axes.flatten() - -countries = gdp_df.columns -t = np.asarray(gdp_df.index.astype(float)) -series = [np.asarray(gdp_df[country].astype(float)) for country in countries] - - -for ax, country, gdp_data in zip(axes, countries, series): - - ax.plot(t, gdp_data) - ax.set_title(f'{country} 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Sargent](http://www.tomsargent.com/) - -```{tableofcontents} -``` \ No newline at end of file diff --git a/code_book/pkg_funcs.md b/code_book/pkg_funcs.md deleted file mode 100644 index 2ed0e5d..0000000 --- a/code_book/pkg_funcs.md +++ /dev/null @@ -1,359 +0,0 @@ ---- -jupytext: - cell_metadata_filter: -all - formats: md:myst - text_representation: - extension: .md - format_name: myst - format_version: 0.13 - jupytext_version: 1.10.3 -kernelspec: - display_name: Python 3 - language: python - name: python3 ---- - -# quantecon_book_networks - -## input_output - -### node_total_exports -```{code-cell} -def node_total_exports(G): - node_exports = [] - for node1 in G.nodes(): - total_export = 0 - for node2 in G[node1]: - total_export += G[node1][node2]['weight'] - node_exports.append(total_export) - return node_exports -``` - -### node_total_imports -```{code-cell} -def node_total_imports(G): - node_imports = [] - for node1 in G.nodes(): - total_import = 0 - for node2 in G[node1]: - total_import += G[node2][node1]['weight'] - node_imports.append(total_import) - return node_imports -``` - -### edge_weights -```{code-cell} -def edge_weights(G): - edge_weights = [G[u][v]['weight'] for u,v in G.edges()] - return edge_weights -``` - -### normalise_weights -```{code-cell} -def normalise_weights(weights,scalar=1): - max_value = np.max(weights) - return [scalar * (weight / max_value) for weight in weights] -``` - -### to_zero_one -```{code-cell} -def to_zero_one(x): - "Map vector x to the zero one interval." - x = np.array(x) - x_min, x_max = x.min(), x.max() - return (x - x_min)/(x_max - x_min) -``` - -### to_zero_one_beta -```{code-cell} -def to_zero_one_beta(x, - qrange=[0.25, 0.75], - beta_para=[0.5, 0.5]): - - """ - Nonlinearly map vector x to the zero one interval with beta distribution. - https://en.wikipedia.org/wiki/Beta_distribution - """ - x = np.array(x) - x_min, x_max = x.min(), x.max() - if beta_para != None: - a, b = beta_para - return beta.cdf((x - x_min) /(x_max - x_min), a, b) - else: - q1, q2 = qrange - return (x - x_min) * (q2 - q1) /(x_max - x_min) + q1 -``` - -### colorise_weights -```{code-cell} -import matplotlib.cm as cm -def colorise_weights(weights,beta=True,color_palette=cm.plasma): - if beta: - cp = color_palette(to_zero_one_beta(weights)) - else: - cp = color_palette(to_zero_one(weights)) - return cp -``` - -### spec_rad -```{code-cell} -def spec_rad(M): - """ - Compute the spectral radius of M. - """ - return np.max(np.abs(np.linalg.eigvals(M))) -``` - -### adjacency_matrix_to_graph -```{code-cell} -def adjacency_matrix_to_graph(A, - codes, - tol=0.0): # clip entries below tol - """ - Build a networkx graph object given an adjacency matrix - """ - G = nx.DiGraph() - N = len(A) - - # Add nodes - for i, code in enumerate(codes): - G.add_node(code, name=code) - - # Add the edges - for i in range(N): - for j in range(N): - a = A[i, j] - if a > tol: - G.add_edge(codes[i], codes[j], weight=a) - - return G -``` - -### eigenvector_centrality -```{code-cell} -def eigenvector_centrality(A, k=40, authority=False): - """ - Computes the dominant eigenvector of A. Assumes A is - primitive and uses the power method. - """ - A_temp = A.T if authority else A - n = len(A_temp) - r = spec_rad(A_temp) - e = r**(-k) * (np.linalg.matrix_power(A_temp, k) @ np.ones(n)) - return e / np.sum(e) -``` - -### katz_centrality -```{code-cell} -def katz_centrality(A, b=1, authority=False): - """ - Computes the Katz centrality of A, defined as the x solving - - x = 1 + b A x (1 = vector of ones) - - Assumes that A is square. - - If authority=True, then A is replaced by its transpose. - """ - n = len(A) - I = np.identity(n) - C = I - b * A.T if authority else I - b * A - return np.linalg.solve(C, np.ones(n)) -``` - -### build_unweighted_matrix -```{code-cell} -def build_unweighted_matrix(Z, tol=1e-5): - """ - return a unweighted adjacency matrix - """ - return 1*(Z>tol) -``` - -### erdos_renyi_graph -```{code-cell} -def erdos_renyi_graph(n=100, p=0.5, seed=1234): - "Returns an Erdős-Rényi random graph." - - np.random.seed(seed) - edges = itertools.combinations(range(n), 2) - G = nx.Graph() - - for e in edges: - if np.random.rand() < p: - G.add_edge(*e) - return G -``` - -### build_coefficient_matrices -```{code-cell} -def build_coefficient_matrices(Z, X): - """ - Build coefficient matrices A and F from Z and X via - - A[i, j] = Z[i, j] / X[j] - F[i, j] = Z[i, j] / X[i] - - """ - A, F = np.empty_like(Z), np.empty_like(Z) - n = A.shape[0] - for i in range(n): - for j in range(n): - A[i, j] = Z[i, j] / X[j] - F[i, j] = Z[i, j] / X[i] - - return A, F -``` - -## plotting - -### plot_graph -```{code-cell} -def plot_graph(A, - X, - ax, - codes, - node_color_list=None, - node_size_multiple=0.0005, - edge_size_multiple=14, - layout_type='circular', - layout_seed=1234, - tol=0.03): # clip entries below tol - - G = nx.DiGraph() - N = len(A) - - # Add nodes, with weights by sales of the sector - for i, w in enumerate(X): - G.add_node(codes[i], weight=w, name=codes[i]) - - node_sizes = X * node_size_multiple - - # Position the nodes - if layout_type == 'circular': - node_pos_dict = nx.circular_layout(G) - elif layout_type == 'spring': - node_pos_dict = nx.spring_layout(G, seed=layout_seed) - elif layout_type == 'random': - node_pos_dict = nx.random_layout(G, seed=layout_seed) - elif layout_type == 'spiral': - node_pos_dict = nx.spiral_layout(G) - - # Add the edges, along with their colors and widths - edge_colors = [] - edge_widths = [] - for i in range(N): - for j in range(N): - a = A[i, j] - if a > tol: - G.add_edge(codes[i], codes[j]) - edge_colors.append(node_color_list[i]) - width = a * edge_size_multiple - edge_widths.append(width) - - # Plot the networks - nx.draw_networkx_nodes(G, - node_pos_dict, - node_color=node_color_list, - node_size=node_sizes, - edgecolors='grey', - linewidths=2, - alpha=0.6, - ax=ax) - - nx.draw_networkx_labels(G, - node_pos_dict, - font_size=10, - ax=ax) - - nx.draw_networkx_edges(G, - node_pos_dict, - edge_color=edge_colors, - width=edge_widths, - arrows=True, - arrowsize=20, - alpha=0.6, - ax=ax, - arrowstyle='->', - node_size=node_sizes, - connectionstyle='arc3,rad=0.15') -``` - -### plot_matrices -```{code-cell} -def plot_matrices(matrix, - codes, - ax, - font_size=12, - alpha=0.6, - colormap=cm.viridis, - color45d=None, - xlabel='sector $j$', - ylabel='sector $i$'): - - ticks = range(len(matrix)) - - levels = np.sqrt(np.linspace(0, 0.75, 100)) - - - if color45d != None: - co = ax.contourf(ticks, - ticks, - matrix, -# levels, - alpha=alpha, cmap=colormap) - ax.plot(ticks, ticks, color=color45d) - else: - co = ax.contourf(ticks, - ticks, - matrix, - levels, - alpha=alpha, cmap=colormap) - - #plt.colorbar(co) - - ax.set_xlabel(xlabel, fontsize=font_size) - ax.set_ylabel(ylabel, fontsize=font_size) - ax.set_yticks(ticks) - ax.set_yticklabels(codes) - ax.set_xticks(ticks) - ax.set_xticklabels(codes) -``` - -### unit_simplex -```{code-cell} -def unit_simplex(angle): - - fig = plt.figure(figsize=(10, 8)) - ax = fig.add_subplot(111, projection='3d') - - vtx = [[0, 0, 1], - [0, 1, 0], - [1, 0, 0]] - - tri = Poly3DCollection([vtx], color='darkblue', alpha=0.3) - tri.set_facecolor([0.5, 0.5, 1]) - ax.add_collection3d(tri) - - ax.set(xlim=(0, 1), ylim=(0, 1), zlim=(0, 1), - xticks=(1,), yticks=(1,), zticks=(1,)) - - ax.set_xticklabels(['$(1, 0, 0)$'], fontsize=16) - ax.set_yticklabels([f'$(0, 1, 0)$'], fontsize=16) - ax.set_zticklabels([f'$(0, 0, 1)$'], fontsize=16) - - ax.xaxis.majorTicks[0].set_pad(15) - ax.yaxis.majorTicks[0].set_pad(15) - ax.zaxis.majorTicks[0].set_pad(35) - - ax.view_init(30, angle) - - # Move axis to origin - ax.xaxis._axinfo['juggled'] = (0, 0, 0) - ax.yaxis._axinfo['juggled'] = (1, 1, 1) - ax.zaxis._axinfo['juggled'] = (2, 2, 0) - - ax.grid(False) - - return ax -``` \ No newline at end of file diff --git a/code_book/status.md b/code_book/status.md deleted file mode 100644 index 56b9450..0000000 --- a/code_book/status.md +++ /dev/null @@ -1,17 +0,0 @@ ---- -jupytext: - text_representation: - extension: .md - format_name: myst -kernelspec: - display_name: Python 3 - language: python - name: python3 ---- - -# Execution Statistics - -This table contains the latest execution statistics. - -```{nb-exec-table} -``` diff --git a/ipynb/appendix.ipynb b/ipynb/appendix.ipynb new file mode 100644 index 0000000..316c944 --- /dev/null +++ b/ipynb/appendix.ipynb @@ -0,0 +1,776 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "35da8276", + "metadata": {}, + "source": [ + "# Appendix Code\n", + "\n", + "We begin with some imports" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "46328226", + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np \n", + "import matplotlib.pyplot as plt \n", + "import quantecon_book_networks \n", + "from mpl_toolkits.mplot3d.axes3d import Axes3D, proj3d \n", + "from matplotlib import cm \n", + "export_figures = False\n", + "quantecon_book_networks.config(\"matplotlib\") " + ] + }, + { + "cell_type": "markdown", + "id": "483e031e", + "metadata": {}, + "source": [ + "## Functions\n", + "\n", + "### One-to-one and onto functions on $(0,1)$\n", + "\n", + "We start by defining the domain and our one-to-one and onto function examples." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "0b617e66", + "metadata": {}, + "outputs": [], + "source": [ + "x = np.linspace(0, 1, 100)\n", + "titles = 'one-to-one', 'onto'\n", + "labels = '$f(x)=1/2 + x/2$', '$f(x)=4x(1-x)$'\n", + "funcs = lambda x: 1/2 + x/2, lambda x: 4 * x * (1-x)" + ] + }, + { + "cell_type": "markdown", + "id": "40b1bb2d", + "metadata": {}, + "source": [ + "The figure can now be produced as follows." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "12ba8e7b", + "metadata": {}, + "outputs": [], + "source": [ + "fig, axes = plt.subplots(1, 2, figsize=(10, 4))\n", + "for f, ax, lb, ti in zip(funcs, axes, labels, titles):\n", + " ax.set_xlim(0, 1)\n", + " ax.set_ylim(0, 1.01)\n", + " ax.plot(x, f(x), label=lb)\n", + " ax.set_title(ti, fontsize=12)\n", + " ax.legend(loc='lower center', fontsize=12, frameon=False)\n", + " ax.set_xticks((0, 1))\n", + " ax.set_yticks((0, 1))\n", + "\n", + "if export_figures:\n", + " plt.savefig(\"figures/func_types_1.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "fe7452ac", + "metadata": {}, + "source": [ + "### Some functions are bijections\n", + "\n", + "This figure can be produced in a similar manner to 6.1." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "5907691a", + "metadata": {}, + "outputs": [], + "source": [ + "x = np.linspace(0, 1, 100)\n", + "titles = 'constant', 'bijection'\n", + "labels = '$f(x)=1/2$', '$f(x)=1-x$'\n", + "funcs = lambda x: 1/2 + 0 * x, lambda x: 1-x\n", + "\n", + "fig, axes = plt.subplots(1, 2, figsize=(10, 4))\n", + "for f, ax, lb, ti in zip(funcs, axes, labels, titles):\n", + " ax.set_xlim(0, 1)\n", + " ax.set_ylim(0, 1.01)\n", + " ax.plot(x, f(x), label=lb)\n", + " ax.set_title(ti, fontsize=12)\n", + " ax.legend(loc='lower center', fontsize=12, frameon=False)\n", + " ax.set_xticks((0, 1))\n", + " ax.set_yticks((0, 1))\n", + " \n", + "if export_figures:\n", + " plt.savefig(\"figures/func_types_2.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "bbfde515", + "metadata": {}, + "source": [ + "## Fixed Points\n", + "\n", + "### Graph and fixed points of $G \\colon x \\mapsto 2.125/(1 + x^{-4})$\n", + "\n", + "We begin by defining the domain and the function." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "3110e5cd", + "metadata": {}, + "outputs": [], + "source": [ + "xmin, xmax = 0.0000001, 2\n", + "xgrid = np.linspace(xmin, xmax, 200)\n", + "g = lambda x: 2.125 / (1 + x**(-4))" + ] + }, + { + "cell_type": "markdown", + "id": "6532632d", + "metadata": {}, + "source": [ + "Next we define our fixed points." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "3b9ef820", + "metadata": {}, + "outputs": [], + "source": [ + "fps_labels = ('$x_\\ell$', '$x_m$', '$x_h$' )\n", + "fps = (0.01, 0.94, 1.98)\n", + "coords = ((40, 80), (40, -40), (-40, -80))" + ] + }, + { + "cell_type": "markdown", + "id": "87f4c5c2", + "metadata": {}, + "source": [ + "Finally we can produce the figure." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "f8f4567d", + "metadata": {}, + "outputs": [], + "source": [ + "fig, ax = plt.subplots(figsize=(6.5, 6))\n", + "\n", + "ax.set_xlim(xmin, xmax)\n", + "ax.set_ylim(xmin, xmax)\n", + "\n", + "ax.plot(xgrid, g(xgrid), 'b-', lw=2, alpha=0.6, label='$G$')\n", + "ax.plot(xgrid, xgrid, 'k-', lw=1, alpha=0.7, label='$45^o$')\n", + "\n", + "ax.legend(fontsize=14)\n", + "\n", + "ax.plot(fps, fps, 'ro', ms=8, alpha=0.6)\n", + "\n", + "for (fp, lb, coord) in zip(fps, fps_labels, coords):\n", + " ax.annotate(lb, \n", + " xy=(fp, fp),\n", + " xycoords='data',\n", + " xytext=coord,\n", + " textcoords='offset points',\n", + " fontsize=16,\n", + " arrowprops=dict(arrowstyle=\"->\"))\n", + "\n", + "if export_figures:\n", + " plt.savefig(\"figures/three_fixed_points.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "b6b47191", + "metadata": {}, + "source": [ + "## Complex Numbers\n", + "\n", + "### The complex number $(a, b) = r e^{i \\phi}$\n", + "\n", + "We start by abbreviating some useful values and functions." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "ee845f66", + "metadata": {}, + "outputs": [], + "source": [ + "π = np.pi\n", + "zeros = np.zeros\n", + "ones = np.ones\n", + "fs = 18" + ] + }, + { + "cell_type": "markdown", + "id": "4a40f91e", + "metadata": {}, + "source": [ + "Next we set our parameters." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "fa26afd6", + "metadata": {}, + "outputs": [], + "source": [ + "r = 2\n", + "φ = π/3\n", + "x = r * np.cos(φ)\n", + "x_range = np.linspace(0, x, 1000)\n", + "φ_range = np.linspace(0, φ, 1000)" + ] + }, + { + "cell_type": "markdown", + "id": "d55d19ec", + "metadata": {}, + "source": [ + "Finally we produce the plot." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "7036969e", + "metadata": {}, + "outputs": [], + "source": [ + "fig = plt.figure(figsize=(7, 7))\n", + "ax = plt.subplot(111, projection='polar')\n", + "\n", + "ax.plot((0, φ), (0, r), marker='o', color='b', alpha=0.5) # Plot r\n", + "ax.plot(zeros(x_range.shape), x_range, color='b', alpha=0.5) # Plot x\n", + "ax.plot(φ_range, x / np.cos(φ_range), color='b', alpha=0.5) # Plot y\n", + "ax.plot(φ_range, ones(φ_range.shape) * 0.15, color='green') # Plot φ\n", + "\n", + "ax.margins(0) # Let the plot starts at origin\n", + "\n", + "ax.set_rmax(2)\n", + "ax.set_rticks((1, 2)) # Less radial ticks\n", + "ax.set_rlabel_position(-88.5) # Get radial labels away from plotted line\n", + "\n", + "ax.text(φ, r+0.04 , r'$(a, b) = (1, \\sqrt{3})$', fontsize=fs) # Label z\n", + "ax.text(φ+0.4, 1 , '$r = 2$', fontsize=fs) # Label r\n", + "ax.text(0-0.4, 0.5, '$1$', fontsize=fs) # Label x\n", + "ax.text(0.5, 1.2, '$\\sqrt{3}$', fontsize=fs) # Label y\n", + "ax.text(0.3, 0.25, '$\\\\varphi = \\\\pi/3$', fontsize=fs) # Label θ\n", + "\n", + "xT=plt.xticks()[0]\n", + "xL=['0',\n", + " r'$\\frac{\\pi}{4}$',\n", + " r'$\\frac{\\pi}{2}$',\n", + " r'$\\frac{3\\pi}{4}$',\n", + " r'$\\pi$',\n", + " r'$\\frac{5\\pi}{4}$',\n", + " r'$\\frac{3\\pi}{2}$',\n", + " r'$\\frac{7\\pi}{4}$']\n", + "\n", + "plt.xticks(xT, xL, fontsize=fs+2)\n", + "ax.grid(True)\n", + "\n", + "if export_figures:\n", + " plt.savefig(\"figures/complex_number.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "418ac96b", + "metadata": {}, + "source": [ + "## Convergence\n", + "\n", + "### Convergence of a sequence to the origin in $\\mathbb{R}^3$\n", + "\n", + "We define our transformation matrix, initial point, and number of iterations." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "ebbf6747", + "metadata": {}, + "outputs": [], + "source": [ + "θ = 0.1\n", + "A = ((np.cos(θ), - np.sin(θ), 0.0001),\n", + " (np.sin(θ), np.cos(θ), 0.001),\n", + " (np.sin(θ), np.cos(θ), 1))\n", + "\n", + "A = 0.98 * np.array(A)\n", + "p = np.array((1, 1, 1))\n", + "n = 200" + ] + }, + { + "cell_type": "markdown", + "id": "815fed7d", + "metadata": {}, + "source": [ + "Now we can produce the plot by repeatedly transforming our point with the transformation matrix and plotting each resulting point." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "14e5a9a7", + "metadata": {}, + "outputs": [], + "source": [ + "fig = plt.figure(figsize=(6, 6))\n", + "ax = fig.add_subplot(111, projection='3d')\n", + "ax.view_init(elev=20, azim=-40)\n", + "\n", + "ax.set_xlim((-1.5, 1.5))\n", + "ax.set_ylim((-1.5, 1.5))\n", + "ax.set_xticks((-1,0,1))\n", + "ax.set_yticks((-1,0,1))\n", + "\n", + "for i in range(n):\n", + " x, y, z = p\n", + " ax.plot([x], [y], [z], 'o', ms=4, color=cm.jet_r(i / n))\n", + " p = A @ p\n", + " \n", + "if export_figures:\n", + " plt.savefig(\"figures/euclidean_convergence_1.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "8804299c", + "metadata": {}, + "source": [ + "## Linear Algebra\n", + "\n", + "### The span of vectors $u$, $v$, $w$ in $\\mathbb{R}$\n", + "\n", + "We begin by importing the `FancyArrowPatch` class and extending it." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "07b47dce", + "metadata": {}, + "outputs": [], + "source": [ + "from matplotlib.patches import FancyArrowPatch\n", + "\n", + "class Arrow3D(FancyArrowPatch):\n", + " def __init__(self, xs, ys, zs, *args, **kwargs):\n", + " FancyArrowPatch.__init__(self, (0,0), (0,0), *args, **kwargs)\n", + " self._verts3d = xs, ys, zs\n", + "\n", + " def do_3d_projection(self, renderer=None):\n", + " xs3d, ys3d, zs3d = self._verts3d\n", + " xs, ys, zs = proj3d.proj_transform(xs3d, ys3d, zs3d, self.axes.M)\n", + " self.set_positions((xs[0],ys[0]),(xs[1],ys[1]))\n", + "\n", + " return np.min(zs)\n", + "\n", + " def draw(self, renderer):\n", + " xs3d, ys3d, zs3d = self._verts3d\n", + " xs, ys, zs = proj3d.proj_transform(xs3d, ys3d, zs3d, self.axes.M)\n", + " self.set_positions((xs[0],ys[0]),(xs[1],ys[1]))\n", + " FancyArrowPatch.draw(self, renderer)" + ] + }, + { + "cell_type": "markdown", + "id": "2a087a52", + "metadata": {}, + "source": [ + "Next we generate our vectors $u$, $v$, $w$, ensuring linear dependence." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "ced11ab1", + "metadata": {}, + "outputs": [], + "source": [ + "α, β = 0.2, 0.1\n", + "def f(x, y):\n", + " return α * x + β * y\n", + "\n", + "# Vector locations, by coordinate\n", + "x_coords = np.array((3, 3, -3.5))\n", + "y_coords = np.array((4, -4, 3.0))\n", + "z_coords = f(x_coords, y_coords)\n", + "\n", + "vecs = [np.array((x, y, z)) for x, y, z in zip(x_coords, y_coords, z_coords)]" + ] + }, + { + "cell_type": "markdown", + "id": "165912ee", + "metadata": {}, + "source": [ + "Next we define the spanning plane." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "97870984", + "metadata": {}, + "outputs": [], + "source": [ + "x_min, x_max = -5, 5\n", + "y_min, y_max = -5, 5\n", + "\n", + "grid_size = 20\n", + "xr2 = np.linspace(x_min, x_max, grid_size)\n", + "yr2 = np.linspace(y_min, y_max, grid_size)\n", + "x2, y2 = np.meshgrid(xr2, yr2)\n", + "z2 = f(x2, y2)" + ] + }, + { + "cell_type": "markdown", + "id": "be62269c", + "metadata": {}, + "source": [ + "Finally we generate the plot." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "49b4eff0", + "metadata": {}, + "outputs": [], + "source": [ + "fig = plt.figure(figsize=(12, 7))\n", + "ax = plt.axes(projection ='3d')\n", + "ax.view_init(elev=10., azim=-80)\n", + "\n", + "ax.set(xlim=(x_min, x_max), \n", + " ylim=(x_min, x_max), \n", + " zlim=(x_min, x_max),\n", + " xticks=(0,), yticks=(0,), zticks=(0,))\n", + "\n", + "# Draw the axes\n", + "gs = 3\n", + "z = np.linspace(x_min, x_max, gs)\n", + "x = np.zeros(gs)\n", + "y = np.zeros(gs)\n", + "ax.plot(x, y, z, 'k-', lw=2, alpha=0.5)\n", + "ax.plot(z, x, y, 'k-', lw=2, alpha=0.5)\n", + "ax.plot(y, z, x, 'k-', lw=2, alpha=0.5)\n", + "\n", + "# Draw the vectors\n", + "for v in vecs:\n", + " a = Arrow3D([0, v[0]], \n", + " [0, v[1]], \n", + " [0, v[2]], \n", + " mutation_scale=20, \n", + " lw=1, \n", + " arrowstyle=\"-|>\", \n", + " color=\"b\")\n", + " ax.add_artist(a)\n", + "\n", + "\n", + "for v, label in zip(vecs, ('u', 'v', 'w')):\n", + " v = v * 1.1\n", + " ax.text(v[0], v[1], v[2], \n", + " f'${label}$', \n", + " fontsize=14)\n", + "\n", + "# Draw the plane\n", + "grid_size = 20\n", + "xr2 = np.linspace(x_min, x_max, grid_size)\n", + "yr2 = np.linspace(y_min, y_max, grid_size)\n", + "x2, y2 = np.meshgrid(xr2, yr2)\n", + "z2 = f(x2, y2)\n", + "\n", + "ax.plot_surface(x2, y2, z2, rstride=1, cstride=1, cmap=cm.jet,\n", + " linewidth=0, antialiased=True, alpha=0.2)\n", + "\n", + "\n", + "if export_figures:\n", + " plt.savefig(\"figures/span1.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "b87b7088", + "metadata": {}, + "source": [ + "## Linear Maps Are Matrices\n", + "\n", + "### Equivalence of the onto and one-to-one properties (for linear maps)\n", + "\n", + "This plot is produced similarly to Figures 6.1 and 6.2." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "a243b6e1", + "metadata": {}, + "outputs": [], + "source": [ + "fig, axes = plt.subplots(1, 2, figsize=(10, 4))\n", + "\n", + "x = np.linspace(-2, 2, 10)\n", + "\n", + "titles = 'non-bijection', 'bijection'\n", + "labels = '$f(x)=0 x$', '$f(x)=0.5 x$'\n", + "funcs = lambda x: 0*x, lambda x: 0.5 * x\n", + "\n", + "for ax, f, lb, ti in zip(axes, funcs, labels, titles):\n", + "\n", + " # Set the axes through the origin\n", + " for spine in ['left', 'bottom']:\n", + " ax.spines[spine].set_position('zero')\n", + " for spine in ['right', 'top']:\n", + " ax.spines[spine].set_color('none')\n", + " ax.set_yticks((-1, 1))\n", + " ax.set_ylim((-1, 1))\n", + " ax.set_xlim((-1, 1))\n", + " ax.set_xticks((-1, 1))\n", + " y = f(x)\n", + " ax.plot(x, y, '-', linewidth=4, label=lb, alpha=0.6)\n", + " ax.text(-0.8, 0.5, ti, fontsize=14)\n", + " ax.legend(loc='lower right', fontsize=12)\n", + " \n", + "if export_figures:\n", + " plt.savefig(\"figures/func_types_3.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "e3aa06c3", + "metadata": {}, + "source": [ + "## Convexity and Polyhedra\n", + "\n", + "### A polyhedron $P$ represented as intersecting halfspaces\n", + "\n", + "Inequalities are of the form\n", + "\n", + "$$ a x + b y \\leq c $$\n", + "\n", + "To plot the halfspace we plot the line\n", + "\n", + "$$ y = c/b - a/b x $$\n", + "\n", + "and then fill in the halfspace using `fill_between` on points $x, y, \\hat y$,\n", + "where $\\hat y$ is either `y_min` or `y_max`." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "ae4e8b4d", + "metadata": {}, + "outputs": [], + "source": [ + "fig, ax = plt.subplots()\n", + "plt.axis('off')\n", + "\n", + "x = np.linspace(-10, 14, 200)\n", + "y_min, y_max = -2, 3\n", + "\n", + "a1, b1, c1 = 1.0, 8.0, -5.0\n", + "y = c1 / b1 - (a1 / b1) * x\n", + "ax.plot(x, y, label='$a_1 x_1 + b_1 x_2 = c_1$')\n", + "ax.fill_between(x, y, y_max, alpha=0.2)\n", + "\n", + "\n", + "a2, b2, c2 = 0.5, 0.75, 1.5\n", + "y = c2 / b2 - (a2 / b2) * x\n", + "ax.plot(x, y, label='$a_2 x_1 + b_2 x_2 = c_2$')\n", + "ax.fill_between(x, y, y_min, alpha=0.2)\n", + "\n", + "\n", + "a3, b3, c3 = -1.0, 1.0, 1.5\n", + "y = c3 / b3 - (a3 / b3) * x\n", + "ax.plot(x, y, label='$a_3 x_1 + b_3 x_2 = c_3$')\n", + "ax.fill_between(x, y, y_min, alpha=0.2)\n", + "\n", + "ax.plot((0.23,), (1.82,), 'ko')\n", + "ax.plot((-1.95,), (-0.4,), 'ko')\n", + "ax.plot((4.8,), (-1.2,), 'ko')\n", + "\n", + "\n", + "ax.annotate('$P$', xy=(0, 0), fontsize=12)\n", + "\n", + "ax.set_ylim(y_min, y_max)\n", + "if export_figures:\n", + " plt.savefig(\"figures/polyhedron1.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "45b13eed", + "metadata": {}, + "source": [ + "## Saddle Points and Duality" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "53bdce43", + "metadata": {}, + "outputs": [], + "source": [ + "fig = plt.figure(figsize=(8.5, 6))\n", + "\n", + "## Top left Plot\n", + "\n", + "ax = fig.add_subplot(221, projection='3d')\n", + "\n", + "plot_args = {'rstride': 1, 'cstride': 1, 'cmap':\"viridis\",\n", + " 'linewidth': 0.4, 'antialiased': True, \"alpha\":0.75,\n", + " 'vmin': -1, 'vmax': 1}\n", + "\n", + "x, y = np.mgrid[-1:1:31j, -1:1:31j]\n", + "z = x**2 - y**2\n", + "\n", + "ax.plot_surface(x, y, z, **plot_args)\n", + "\n", + "ax.view_init(azim=245, elev=20)\n", + "\n", + "ax.set_xlim(-1, 1)\n", + "ax.set_ylim(-1, 1)\n", + "ax.set_zlim(-1, 1)\n", + "\n", + "ax.set_xticks([0])\n", + "ax.set_xticklabels([r\"$x^*$\"], fontsize=16)\n", + "ax.set_yticks([0])\n", + "ax.set_yticklabels([r\"$\\theta^*$\"], fontsize=16)\n", + "\n", + "\n", + "ax.set_zticks([])\n", + "ax.set_zticklabels([])\n", + "\n", + "ax.w_xaxis.set_pane_color((1.0, 1.0, 1.0, 0.0)) \n", + "ax.w_yaxis.set_pane_color((1.0, 1.0, 1.0, 0.0)) \n", + "ax.w_zaxis.set_pane_color((1.0, 1.0, 1.0, 0.0))\n", + "\n", + "ax.set_zlabel(\"$L(x,\\\\theta)$\", fontsize=14)\n", + "\n", + "## Top Right Plot\n", + "\n", + "ax = fig.add_subplot(222)\n", + "\n", + "\n", + "plot_args = {'cmap':\"viridis\", 'antialiased': True, \"alpha\":0.6,\n", + " 'vmin': -1, 'vmax': 1}\n", + "\n", + "x, y = np.mgrid[-1:1:31j, -1:1:31j]\n", + "z = x**2 - y**2\n", + "\n", + "ax.contourf(x, y, z, **plot_args)\n", + "\n", + "ax.plot([0], [0], 'ko')\n", + "\n", + "ax.set_xlim(-1, 1)\n", + "ax.set_ylim(-1, 1)\n", + "\n", + "plt.xticks([ 0],\n", + " [r\"$x^*$\"], fontsize=16)\n", + "\n", + "plt.yticks([0],\n", + " [r\"$\\theta^*$\"], fontsize=16)\n", + "\n", + "ax.hlines(0, -1, 1, color='k', ls='-', lw=1)\n", + "ax.vlines(0, -1, 1, color='k', ls='-', lw=1)\n", + "\n", + "coords=(-35, 30)\n", + "ax.annotate(r'$L(x, \\theta^*)$', \n", + " xy=(-0.5, 0), \n", + " xycoords=\"data\",\n", + " xytext=coords,\n", + " textcoords=\"offset points\",\n", + " fontsize=12,\n", + " arrowprops={\"arrowstyle\" : \"->\"})\n", + "\n", + "coords=(35, 30)\n", + "ax.annotate(r'$L(x^*, \\theta)$', \n", + " xy=(0, 0.25), \n", + " xycoords=\"data\",\n", + " xytext=coords,\n", + " textcoords=\"offset points\",\n", + " fontsize=12,\n", + " arrowprops={\"arrowstyle\" : \"->\"})\n", + "\n", + "## Bottom Left Plot\n", + "\n", + "ax = fig.add_subplot(223)\n", + "\n", + "x = np.linspace(-1, 1, 100)\n", + "ax.plot(x, -x**2, label='$\\\\theta \\mapsto L(x^*, \\\\theta)$')\n", + "ax.set_ylim((-1, 1))\n", + "ax.legend(fontsize=14)\n", + "ax.set_xticks([])\n", + "ax.set_yticks([])\n", + "\n", + "## Bottom Right Plot\n", + "\n", + "ax = fig.add_subplot(224)\n", + "\n", + "x = np.linspace(-1, 1, 100)\n", + "ax.plot(x, x**2, label='$x \\mapsto L(x, \\\\theta^*)$')\n", + "ax.set_ylim((-1, 1))\n", + "ax.legend(fontsize=14, loc='lower right')\n", + "ax.set_xticks([])\n", + "ax.set_yticks([])\n", + "\n", + "if export_figures:\n", + "\tplt.savefig(\"figures/saddle_1.pdf\")\n", + "plt.show()" + ] + } + ], + "metadata": { + "jupytext": { + "cell_metadata_filter": "-all" + }, + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/ipynb/ch_fpms.ipynb b/ipynb/ch_fpms.ipynb new file mode 100644 index 0000000..8327586 --- /dev/null +++ b/ipynb/ch_fpms.ipynb @@ -0,0 +1,273 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "ab032fde", + "metadata": {}, + "source": [ + "# Chapter 5 - Nonlinear Interactions (Python Code)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "f0b02209", + "metadata": { + "tags": [ + "hide-output" + ] + }, + "outputs": [], + "source": [ + "! pip install --upgrade quantecon_book_networks" + ] + }, + { + "cell_type": "markdown", + "id": "7d82d322", + "metadata": {}, + "source": [ + "We begin with some imports" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "be38b591", + "metadata": {}, + "outputs": [], + "source": [ + "import quantecon as qe\n", + "import quantecon_book_networks\n", + "import quantecon_book_networks.input_output as qbn_io\n", + "import quantecon_book_networks.plotting as qbn_plt\n", + "import quantecon_book_networks.data as qbn_data\n", + "export_figures = False" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "e481b4fb", + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "import networkx as nx\n", + "import matplotlib.pyplot as plt\n", + "from matplotlib import cm\n", + "quantecon_book_networks.config(\"matplotlib\")" + ] + }, + { + "cell_type": "markdown", + "id": "7f027495", + "metadata": {}, + "source": [ + "## Financial Networks\n", + "\n", + "### Equity-Cross Holdings\n", + "\n", + "Here we define a class for modelling a financial network where firms are linked by share cross-holdings,\n", + "and there are failure costs as described by [Elliott et al. (2014)](https://www.aeaweb.org/articles?id=10.1257/aer.104.10.3115)." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "58e02f09", + "metadata": {}, + "outputs": [], + "source": [ + "class FinNet:\n", + " \n", + " def __init__(self, n=100, c=0.72, d=1, θ=0.5, β=1.0, seed=1234):\n", + " \n", + " self.n, self.c, self.d, self.θ, self.β = n, c, d, θ, β\n", + " np.random.seed(seed)\n", + " \n", + " self.e = np.ones(n)\n", + " self.C, self.C_hat = self.generate_primitives()\n", + " self.A = self.C_hat @ np.linalg.inv(np.identity(n) - self.C)\n", + " self.v_bar = self.A @ self.e\n", + " self.t = np.full(n, θ)\n", + " \n", + " def generate_primitives(self):\n", + " \n", + " n, c, d = self.n, self.c, self.d\n", + " B = np.zeros((n, n))\n", + " C = np.zeros_like(B)\n", + "\n", + " for i in range(n):\n", + " for j in range(n):\n", + " if i != j and np.random.rand() < d/(n-1):\n", + " B[i,j] = 1\n", + " \n", + " for i in range(n):\n", + " for j in range(n):\n", + " k = np.sum(B[:,j])\n", + " if k > 0:\n", + " C[i,j] = c * B[i,j] / k\n", + " \n", + " C_hat = np.identity(n) * (1 - c)\n", + " \n", + " return C, C_hat\n", + " \n", + " def T(self, v):\n", + " Tv = self.A @ (self.e - self.β * np.where(v < self.t, 1, 0))\n", + " return Tv\n", + " \n", + " def compute_equilibrium(self):\n", + " i = 0\n", + " v = self.v_bar\n", + " error = 1\n", + " while error > 1e-10:\n", + " print(f\"number of failing firms is \", np.sum(v < self.θ))\n", + " new_v = self.T(v)\n", + " error = np.max(np.abs(new_v - v))\n", + " v = new_v\n", + " i = i+1\n", + " \n", + " print(f\"Terminated after {i} iterations\")\n", + " return v\n", + " \n", + " def map_values_to_colors(self, v, j):\n", + " cols = cm.plasma(qbn_io.to_zero_one(v))\n", + " if j != 0:\n", + " for i in range(len(v)):\n", + " if v[i] < self.t[i]:\n", + " cols[i] = 0.0\n", + " return cols" + ] + }, + { + "cell_type": "markdown", + "id": "dbf9b681", + "metadata": {}, + "source": [ + "Now we create a financial network." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "663e7992", + "metadata": {}, + "outputs": [], + "source": [ + "fn = FinNet(n=100, c=0.72, d=1, θ=0.3, β=1.0)" + ] + }, + { + "cell_type": "markdown", + "id": "c89db6a0", + "metadata": {}, + "source": [ + "And compute its equilibrium." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "81efad24", + "metadata": {}, + "outputs": [], + "source": [ + "fn.compute_equilibrium()" + ] + }, + { + "cell_type": "markdown", + "id": "25a95eb9", + "metadata": {}, + "source": [ + "### Waves of bankruptcies in a financial network\n", + "\n", + "Now we visualise the network after different numbers of iterations. \n", + "\n", + "For convenience we will first define a function to plot the graphs of the financial network." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "ff6c8d54", + "metadata": {}, + "outputs": [], + "source": [ + "def plot_fin_graph(G, ax, node_color_list):\n", + " \n", + " n = G.number_of_nodes()\n", + "\n", + " node_pos_dict = nx.spring_layout(G, k=1.1)\n", + " edge_colors = []\n", + "\n", + " for i in range(n):\n", + " for j in range(n):\n", + " edge_colors.append(node_color_list[i])\n", + "\n", + " \n", + " nx.draw_networkx_nodes(G, \n", + " node_pos_dict, \n", + " node_color=node_color_list, \n", + " edgecolors='grey', \n", + " node_size=100,\n", + " linewidths=2, \n", + " alpha=0.8, \n", + " ax=ax)\n", + "\n", + " nx.draw_networkx_edges(G, \n", + " node_pos_dict, \n", + " edge_color=edge_colors, \n", + " alpha=0.4, \n", + " ax=ax)" + ] + }, + { + "cell_type": "markdown", + "id": "bd11cd19", + "metadata": {}, + "source": [ + "Now we will iterate by applying the operator $T$ to the vector of firm values $v$ and produce the plots." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "c4e15ff6", + "metadata": {}, + "outputs": [], + "source": [ + "G = nx.from_numpy_array(np.array(fn.C), create_using=nx.DiGraph)\n", + "v = fn.v_bar\n", + "\n", + "k = 15\n", + "d = 3\n", + "fig, axes = plt.subplots(int(k/d), 1, figsize=(10, 12))\n", + "\n", + "for i in range(k):\n", + " if i % d == 0:\n", + " ax = axes[int(i/d)]\n", + " ax.set_title(f\"iteration {i}\")\n", + "\n", + " plot_fin_graph(G, ax, fn.map_values_to_colors(v, i))\n", + " v = fn.T(v)\n", + "if export_figures:\n", + " plt.savefig(\"figures/fin_network_sims_1.pdf\")\n", + "plt.show()" + ] + } + ], + "metadata": { + "jupytext": { + "cell_metadata_filter": "-all" + }, + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/ipynb/ch_intro.ipynb b/ipynb/ch_intro.ipynb new file mode 100644 index 0000000..a796306 --- /dev/null +++ b/ipynb/ch_intro.ipynb @@ -0,0 +1,1399 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "5175177c", + "metadata": {}, + "source": [ + "# Chapter 1 - Introduction (Python Code)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "0e0179da", + "metadata": { + "tags": [ + "hide-output" + ] + }, + "outputs": [], + "source": [ + "! pip install --upgrade quantecon_book_networks kaleido" + ] + }, + { + "cell_type": "markdown", + "id": "36c17bdf", + "metadata": {}, + "source": [ + "We begin by importing the `quantecon` package as well as some functions and data that have been packaged for release with this text." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "cf584c62", + "metadata": {}, + "outputs": [], + "source": [ + "import quantecon as qe\n", + "import quantecon_book_networks\n", + "import quantecon_book_networks.input_output as qbn_io\n", + "import quantecon_book_networks.data as qbn_data\n", + "import quantecon_book_networks.plotting as qbn_plot\n", + "ch1_data = qbn_data.introduction()\n", + "default_figsize = (6, 4)\n", + "export_figures = False" + ] + }, + { + "cell_type": "markdown", + "id": "858f6c0d", + "metadata": {}, + "source": [ + "Next we import some common python libraries." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "622fff13", + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "import pandas as pd\n", + "import networkx as nx\n", + "import statsmodels.api as sm\n", + "import matplotlib.pyplot as plt\n", + "import matplotlib.cm as cm\n", + "import matplotlib.ticker as ticker\n", + "from mpl_toolkits.mplot3d.art3d import Poly3DCollection\n", + "import matplotlib.patches as mpatches\n", + "import plotly.graph_objects as go\n", + "quantecon_book_networks.config(\"matplotlib\")" + ] + }, + { + "cell_type": "markdown", + "id": "0ee79c32", + "metadata": {}, + "source": [ + "## Motivation\n", + "\n", + "### International trade in crude oil 2021\n", + "\n", + "We begin by loading a `NetworkX` directed graph object that represents international trade in crude oil." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "7e495966", + "metadata": {}, + "outputs": [], + "source": [ + "DG = ch1_data[\"crude_oil\"]" + ] + }, + { + "cell_type": "markdown", + "id": "284d4d87", + "metadata": {}, + "source": [ + "Next we transform the data to prepare it for display as a Sankey diagram." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "0c944e03", + "metadata": {}, + "outputs": [], + "source": [ + "nodeid = {}\n", + "for ix,nd in enumerate(DG.nodes()):\n", + " nodeid[nd] = ix\n", + "\n", + "# Links\n", + "source = []\n", + "target = []\n", + "value = []\n", + "for src,tgt in DG.edges():\n", + " source.append(nodeid[src])\n", + " target.append(nodeid[tgt])\n", + " value.append(DG[src][tgt]['weight'])" + ] + }, + { + "cell_type": "markdown", + "id": "2cd4537f", + "metadata": {}, + "source": [ + "Finally we produce our plot." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "f6a9a856", + "metadata": {}, + "outputs": [], + "source": [ + "fig = go.Figure(data=[go.Sankey(\n", + " node = dict(\n", + " pad = 15,\n", + " thickness = 20,\n", + " line = dict(color = \"black\", width = 0.5),\n", + " label = list(nodeid.keys()),\n", + " color = \"blue\"\n", + " ),\n", + " link = dict(\n", + " source = source,\n", + " target = target,\n", + " value = value\n", + " ))])\n", + "\n", + "\n", + "fig.update_layout(title_text=\"Crude Oil\", font_size=10, width=600, height=800)\n", + "if export_figures:\n", + " fig.write_image(\"figures/crude_oil_2021.pdf\")\n", + "fig.show(renderer='svg')" + ] + }, + { + "cell_type": "markdown", + "id": "663c1374", + "metadata": {}, + "source": [ + "### International trade in commercial aircraft during 2019\n", + "\n", + "For this plot we will use a cleaned dataset from \n", + "[Harvard, CID Dataverse](https://dataverse.harvard.edu/dataverse/atlas)." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "31cf9ca7", + "metadata": {}, + "outputs": [], + "source": [ + "DG = ch1_data['aircraft_network']\n", + "pos = ch1_data['aircraft_network_pos']" + ] + }, + { + "cell_type": "markdown", + "id": "990418cb", + "metadata": {}, + "source": [ + "We begin by calculating some features of our graph using the `NetworkX` and\n", + "the `quantecon_book_networks` packages." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "4da4ce46", + "metadata": {}, + "outputs": [], + "source": [ + "centrality = nx.eigenvector_centrality(DG)\n", + "node_total_exports = qbn_io.node_total_exports(DG)\n", + "edge_weights = qbn_io.edge_weights(DG)" + ] + }, + { + "cell_type": "markdown", + "id": "ead51019", + "metadata": {}, + "source": [ + "Now we convert our graph features to plot features." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "f9be3605", + "metadata": {}, + "outputs": [], + "source": [ + "node_pos_dict = pos\n", + "\n", + "node_sizes = qbn_io.normalise_weights(node_total_exports,10000)\n", + "edge_widths = qbn_io.normalise_weights(edge_weights,10)\n", + "\n", + "node_colors = qbn_io.colorise_weights(list(centrality.values()),color_palette=cm.viridis)\n", + "node_to_color = dict(zip(DG.nodes,node_colors))\n", + "edge_colors = []\n", + "for src,_ in DG.edges:\n", + " edge_colors.append(node_to_color[src])" + ] + }, + { + "cell_type": "markdown", + "id": "b44e9249", + "metadata": {}, + "source": [ + "Finally we produce the plot." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "5dfccd9c", + "metadata": {}, + "outputs": [], + "source": [ + "fig, ax = plt.subplots(figsize=(10, 10))\n", + "ax.axis('off')\n", + "\n", + "nx.draw_networkx_nodes(DG, \n", + " node_pos_dict, \n", + " node_color=node_colors, \n", + " node_size=node_sizes, \n", + " linewidths=2, \n", + " alpha=0.6, \n", + " ax=ax)\n", + "\n", + "nx.draw_networkx_labels(DG, \n", + " node_pos_dict, \n", + " ax=ax)\n", + "\n", + "nx.draw_networkx_edges(DG, \n", + " node_pos_dict, \n", + " edge_color=edge_colors, \n", + " width=edge_widths, \n", + " arrows=True, \n", + " arrowsize=20, \n", + " ax=ax,\n", + " node_size=node_sizes, \n", + " connectionstyle='arc3,rad=0.15')\n", + "\n", + "if export_figures:\n", + " plt.savefig(\"figures/commercial_aircraft_2019_1.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "70e419f0", + "metadata": {}, + "source": [ + "## Spectral Theory\n", + "\n", + "### Spectral Radii\n", + "\n", + "Here we provide code for computing the spectral radius of a matrix." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "d0ac0f17", + "metadata": {}, + "outputs": [], + "source": [ + "def spec_rad(M):\n", + " \"\"\"\n", + " Compute the spectral radius of M.\n", + " \"\"\"\n", + " return np.max(np.abs(np.linalg.eigvals(M)))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "0bdca499", + "metadata": {}, + "outputs": [], + "source": [ + "M = np.array([[1,2],[2,1]])\n", + "spec_rad(M)" + ] + }, + { + "cell_type": "markdown", + "id": "f4fa90a5", + "metadata": {}, + "source": [ + "This function is available in the `quantecon_book_networks` package, along with \n", + "several other functions for used repeatedly in the text. Source code for\n", + "these functions can be seen [here](pkg_funcs)." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "2a1e4cba", + "metadata": {}, + "outputs": [], + "source": [ + "qbn_io.spec_rad(M)" + ] + }, + { + "cell_type": "markdown", + "id": "9ac85796", + "metadata": {}, + "source": [ + "## Probability\n", + "\n", + "### The unit simplex in $\\mathbb{R}^3$\n", + "\n", + "Here we define a function for plotting the unit simplex." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "a661d596", + "metadata": {}, + "outputs": [], + "source": [ + "def unit_simplex(angle):\n", + " \n", + " fig = plt.figure(figsize=(10, 8))\n", + " ax = fig.add_subplot(111, projection='3d')\n", + "\n", + " vtx = [[0, 0, 1],\n", + " [0, 1, 0], \n", + " [1, 0, 0]]\n", + " \n", + " tri = Poly3DCollection([vtx], color='darkblue', alpha=0.3)\n", + " tri.set_facecolor([0.5, 0.5, 1])\n", + " ax.add_collection3d(tri)\n", + "\n", + " ax.set(xlim=(0, 1), ylim=(0, 1), zlim=(0, 1), \n", + " xticks=(1,), yticks=(1,), zticks=(1,))\n", + "\n", + " ax.set_xticklabels(['$(1, 0, 0)$'], fontsize=16)\n", + " ax.set_yticklabels([f'$(0, 1, 0)$'], fontsize=16)\n", + " ax.set_zticklabels([f'$(0, 0, 1)$'], fontsize=16)\n", + "\n", + " ax.xaxis.majorTicks[0].set_pad(15)\n", + " ax.yaxis.majorTicks[0].set_pad(15)\n", + " ax.zaxis.majorTicks[0].set_pad(35)\n", + "\n", + " ax.view_init(30, angle)\n", + "\n", + " # Move axis to origin\n", + " ax.xaxis._axinfo['juggled'] = (0, 0, 0)\n", + " ax.yaxis._axinfo['juggled'] = (1, 1, 1)\n", + " ax.zaxis._axinfo['juggled'] = (2, 2, 0)\n", + " \n", + " ax.grid(False) \n", + " \n", + " return ax" + ] + }, + { + "cell_type": "markdown", + "id": "33c4089d", + "metadata": {}, + "source": [ + "We can now produce the plot." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "337d3792", + "metadata": {}, + "outputs": [], + "source": [ + "unit_simplex(50)\n", + "if export_figures:\n", + " plt.savefig(\"figures/simplex_1.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "8086d97f", + "metadata": {}, + "source": [ + "### Independent draws from Student’s t and Normal distributions\n", + "\n", + "Here we illustrate the occurrence of \"extreme\" events in heavy tailed distributions. \n", + "We start by generating 1,000 samples from a normal distribution and a Student's t distribution." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "7be0e011", + "metadata": {}, + "outputs": [], + "source": [ + "from scipy.stats import t\n", + "n = 1000\n", + "np.random.seed(123)\n", + "\n", + "s = 2\n", + "n_data = np.random.randn(n) * s\n", + "\n", + "t_dist = t(df=1.5)\n", + "t_data = t_dist.rvs(n)" + ] + }, + { + "cell_type": "markdown", + "id": "37b4b6cb", + "metadata": {}, + "source": [ + "When we plot our samples, we see the Student's t distribution frequently\n", + "generates samples many standard deviations from the mean." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "90cd4891", + "metadata": {}, + "outputs": [], + "source": [ + "fig, axes = plt.subplots(1, 2, figsize=(8, 3.4))\n", + "\n", + "for ax in axes:\n", + " ax.set_ylim((-50, 50))\n", + " ax.plot((0, n), (0, 0), 'k-', lw=0.3)\n", + "\n", + "ax = axes[0]\n", + "ax.plot(list(range(n)), t_data, linestyle='', marker='o', alpha=0.5, ms=4)\n", + "ax.vlines(list(range(n)), 0, t_data, 'k', lw=0.2)\n", + "ax.get_xaxis().set_major_formatter(\n", + " ticker.FuncFormatter(lambda x, p: format(int(x), ',')))\n", + "ax.set_title(f\"Student t draws\", fontsize=11)\n", + "\n", + "ax = axes[1]\n", + "ax.plot(list(range(n)), n_data, linestyle='', marker='o', alpha=0.5, ms=4)\n", + "ax.vlines(list(range(n)), 0, n_data, lw=0.2)\n", + "ax.get_xaxis().set_major_formatter(\n", + " ticker.FuncFormatter(lambda x, p: format(int(x), ',')))\n", + "ax.set_title(f\"$N(0, \\sigma^2)$ with $\\sigma = {s}$\", fontsize=11)\n", + "\n", + "plt.tight_layout()\n", + "if export_figures:\n", + " plt.savefig(\"figures/heavy_tailed_draws.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "ca8d8028", + "metadata": {}, + "source": [ + "### CCDF plots for the Pareto and Exponential distributions\n", + "\n", + "When the Pareto tail property holds, the CCDF is eventually log linear. Here\n", + "we illustrates this using a Pareto distribution. For comparison, an exponential\n", + "distribution is also shown. First we define our domain and the Pareto and\n", + "Exponential distributions." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "8a79084d", + "metadata": {}, + "outputs": [], + "source": [ + "x = np.linspace(1, 10, 500)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "89d0baa5", + "metadata": {}, + "outputs": [], + "source": [ + "α = 1.5\n", + "def Gp(x):\n", + " return x**(-α)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "fb412bf5", + "metadata": {}, + "outputs": [], + "source": [ + "λ = 1.0\n", + "def Ge(x):\n", + " return np.exp(-λ * x)" + ] + }, + { + "cell_type": "markdown", + "id": "90785fac", + "metadata": {}, + "source": [ + "We then plot our distribution on a log-log scale." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "a6b2f844", + "metadata": {}, + "outputs": [], + "source": [ + "fig, ax = plt.subplots(figsize=default_figsize)\n", + "\n", + "ax.plot(np.log(x), np.log(Gp(x)), label=\"Pareto\")\n", + "ax.plot(np.log(x), np.log(Ge(x)), label=\"Exponential\")\n", + "\n", + "ax.legend(fontsize=12, frameon=False, loc=\"lower left\")\n", + "ax.set_xlabel(\"$\\ln x$\", fontsize=12)\n", + "ax.set_ylabel(\"$\\ln G(x)$\", fontsize=12)\n", + "\n", + "if export_figures:\n", + " plt.savefig(\"figures/ccdf_comparison_1.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "63906e68", + "metadata": {}, + "source": [ + "### Empirical CCDF plots for largest firms (Forbes)\n", + "\n", + "Here we show that the distribution of firm sizes has a Pareto tail. We start\n", + "by loading the `forbes_global_2000` dataset." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "8b9e33a0", + "metadata": {}, + "outputs": [], + "source": [ + "dfff = ch1_data['forbes_global_2000']" + ] + }, + { + "cell_type": "markdown", + "id": "9266e970", + "metadata": {}, + "source": [ + "We calculate values of the empirical CCDF." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "2a342d5a", + "metadata": {}, + "outputs": [], + "source": [ + "data = np.asarray(dfff['Market Value'])[0:500]\n", + "y_vals = np.empty_like(data, dtype='float64')\n", + "n = len(data)\n", + "for i, d in enumerate(data):\n", + " # record fraction of sample above d\n", + " y_vals[i] = np.sum(data >= d) / n" + ] + }, + { + "cell_type": "markdown", + "id": "690fa754", + "metadata": {}, + "source": [ + "Now we fit a linear trend line (on the log-log scale)." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "956c019d", + "metadata": {}, + "outputs": [], + "source": [ + "x, y = np.log(data), np.log(y_vals)\n", + "results = sm.OLS(y, sm.add_constant(x)).fit()\n", + "b, a = results.params" + ] + }, + { + "cell_type": "markdown", + "id": "c90d836b", + "metadata": {}, + "source": [ + "Finally we produce our plot." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "a13d4b05", + "metadata": {}, + "outputs": [], + "source": [ + "fig, ax = plt.subplots(figsize=(7.3, 4))\n", + "\n", + "ax.scatter(x, y, alpha=0.3, label=\"firm size (market value)\")\n", + "ax.plot(x, x * a + b, 'k-', alpha=0.6, label=f\"slope = ${a: 1.2f}$\")\n", + "\n", + "ax.set_xlabel('log value', fontsize=12)\n", + "ax.set_ylabel(\"log prob.\", fontsize=12)\n", + "ax.legend(loc='lower left', fontsize=12)\n", + " \n", + "if export_figures:\n", + " plt.savefig(\"figures/empirical_powerlaw_plots_firms_forbes.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "d80c5a0a", + "metadata": {}, + "source": [ + "## Graph Theory\n", + "\n", + "### Zeta and Pareto distributions\n", + "\n", + "We begin by defining the Zeta and Pareto distributions." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "19032324", + "metadata": {}, + "outputs": [], + "source": [ + "γ = 2.0\n", + "α = γ - 1" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "abccb3b0", + "metadata": {}, + "outputs": [], + "source": [ + "def z(k, c=2.0):\n", + " return c * k**(-γ)\n", + "\n", + "k_grid = np.arange(1, 10+1)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "fb68c8bc", + "metadata": {}, + "outputs": [], + "source": [ + "def p(x, c=2.0):\n", + " return c * x**(-γ)\n", + "\n", + "x_grid = np.linspace(1, 10, 200)" + ] + }, + { + "cell_type": "markdown", + "id": "f605c4dc", + "metadata": {}, + "source": [ + "Then we can produce our plot" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "d3661175", + "metadata": {}, + "outputs": [], + "source": [ + "fig, ax = plt.subplots(figsize=default_figsize)\n", + "ax.plot(k_grid, z(k_grid), '-o', label='zeta distribution with $\\gamma=2$')\n", + "ax.plot(x_grid, p(x_grid), label='density of Pareto with tail index $\\\\alpha$')\n", + "ax.legend(fontsize=12)\n", + "ax.set_yticks((0, 1, 2))\n", + "if export_figures:\n", + " plt.savefig(\"figures/zeta_1.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "35f01cce", + "metadata": {}, + "source": [ + "### NetworkX digraph plot\n", + "\n", + "We start by creating a graph object and populating it with edges." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "cd8fae62", + "metadata": {}, + "outputs": [], + "source": [ + "G_p = nx.DiGraph()\n", + "\n", + "edge_list = [\n", + " ('p', 'p'),\n", + " ('m', 'p'), ('m', 'm'), ('m', 'r'),\n", + " ('r', 'p'), ('r', 'm'), ('r', 'r')\n", + "]\n", + "\n", + "for e in edge_list:\n", + " u, v = e\n", + " G_p.add_edge(u, v)" + ] + }, + { + "cell_type": "markdown", + "id": "77e32349", + "metadata": {}, + "source": [ + "Now we can plot our graph." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "892bc151", + "metadata": {}, + "outputs": [], + "source": [ + "fig, ax = plt.subplots()\n", + "nx.spring_layout(G_p, seed=4)\n", + "nx.draw_spring(G_p, ax=ax, node_size=500, with_labels=True, \n", + " font_weight='bold', arrows=True, alpha=0.8,\n", + " connectionstyle='arc3,rad=0.25', arrowsize=20)\n", + "if export_figures:\n", + " plt.savefig(\"figures/networkx_basics_1.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "f089c248", + "metadata": {}, + "source": [ + "The `DiGraph` object has methods that calculate in-degree and out-degree of vertices." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "c3e1adc6", + "metadata": {}, + "outputs": [], + "source": [ + "G_p.in_degree('p')" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "a502c57a", + "metadata": {}, + "outputs": [], + "source": [ + "G_p.out_degree('p')" + ] + }, + { + "cell_type": "markdown", + "id": "84636840", + "metadata": {}, + "source": [ + "Additionally, the `NetworkX` package supplies functions for testing\n", + "communication and strong connectedness, as well as to compute strongly\n", + "connected components." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "9b41f9be", + "metadata": {}, + "outputs": [], + "source": [ + "G = nx.DiGraph()\n", + "G.add_edge(1, 1)\n", + "G.add_edge(2, 1)\n", + "G.add_edge(2, 3)\n", + "G.add_edge(3, 2)\n", + "list(nx.strongly_connected_components(G))" + ] + }, + { + "cell_type": "markdown", + "id": "79d4b29c", + "metadata": {}, + "source": [ + "Like `NetworkX`, the Python library `quantecon` \n", + "provides access to some graph-theoretic algorithms. \n", + "\n", + "In the case of QuantEcon's `DiGraph` object, an instance is created via the adjacency matrix." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "cbd43d1a", + "metadata": {}, + "outputs": [], + "source": [ + "A = ((1, 0, 0),\n", + " (1, 1, 1),\n", + " (1, 1, 1))\n", + "A = np.array(A) # Convert to NumPy array\n", + "G = qe.DiGraph(A)\n", + "\n", + "G.strongly_connected_components" + ] + }, + { + "cell_type": "markdown", + "id": "0c670e09", + "metadata": {}, + "source": [ + "### International private credit flows by country\n", + "\n", + "We begin by loading an adjacency matrix of international private credit flows\n", + "(in the form of a NumPy array and a list of country labels)." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "cd6f046d", + "metadata": {}, + "outputs": [], + "source": [ + "Z = ch1_data[\"adjacency_matrix\"][\"Z\"]\n", + "Z_visual= ch1_data[\"adjacency_matrix\"][\"Z_visual\"]\n", + "countries = ch1_data[\"adjacency_matrix\"][\"countries\"]" + ] + }, + { + "cell_type": "markdown", + "id": "ade862b8", + "metadata": {}, + "source": [ + "To calculate our graph's properties, we use hub-based eigenvector\n", + "centrality as our centrality measure for this plot." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "4a89769e", + "metadata": {}, + "outputs": [], + "source": [ + "centrality = qbn_io.eigenvector_centrality(Z_visual, authority=False)" + ] + }, + { + "cell_type": "markdown", + "id": "7be043ae", + "metadata": {}, + "source": [ + "Now we convert our graph features to plot features." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "ce2696fd", + "metadata": {}, + "outputs": [], + "source": [ + "node_colors = cm.plasma(qbn_io.to_zero_one_beta(centrality))" + ] + }, + { + "cell_type": "markdown", + "id": "8ba53d05", + "metadata": {}, + "source": [ + "Finally we produce the plot." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "1f802d18", + "metadata": {}, + "outputs": [], + "source": [ + "X = qbn_io.to_zero_one_beta(Z.sum(axis=1))\n", + "\n", + "fig, ax = plt.subplots(figsize=(8, 10))\n", + "plt.axis(\"off\")\n", + "\n", + "qbn_plot.plot_graph(Z_visual, X, ax, countries,\n", + " layout_type='spring',\n", + " layout_seed=1234,\n", + " node_size_multiple=3000,\n", + " edge_size_multiple=0.000006,\n", + " tol=0.0,\n", + " node_color_list=node_colors) \n", + "\n", + "if export_figures:\n", + " plt.savefig(\"figures/financial_network_analysis_visualization.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "88e9c96e", + "metadata": {}, + "source": [ + "### Centrality measures for the credit network\n", + "\n", + "This figure looks at six different centrality measures.\n", + "\n", + "We begin by defining a function for calculating eigenvector centrality.\n", + "\n", + "Hub-based centrality is calculated by default, although authority-based centrality\n", + "can be calculated by setting `authority=True`." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "577372e6", + "metadata": {}, + "outputs": [], + "source": [ + "def eigenvector_centrality(A, k=40, authority=False):\n", + " \"\"\"\n", + " Computes the dominant eigenvector of A. Assumes A is \n", + " primitive and uses the power method. \n", + " \n", + " \"\"\"\n", + " A_temp = A.T if authority else A\n", + " n = len(A_temp)\n", + " r = spec_rad(A_temp)\n", + " e = r**(-k) * (np.linalg.matrix_power(A_temp, k) @ np.ones(n))\n", + " return e / np.sum(e)" + ] + }, + { + "cell_type": "markdown", + "id": "c16682dc", + "metadata": {}, + "source": [ + "Here a similar function is defined for calculating Katz centrality." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "9d7fbee4", + "metadata": {}, + "outputs": [], + "source": [ + "def katz_centrality(A, b=1, authority=False):\n", + " \"\"\"\n", + " Computes the Katz centrality of A, defined as the x solving\n", + "\n", + " x = 1 + b A x (1 = vector of ones)\n", + "\n", + " Assumes that A is square.\n", + "\n", + " If authority=True, then A is replaced by its transpose.\n", + " \"\"\"\n", + " n = len(A)\n", + " I = np.identity(n)\n", + " C = I - b * A.T if authority else I - b * A\n", + " return np.linalg.solve(C, np.ones(n))" + ] + }, + { + "cell_type": "markdown", + "id": "c0812961", + "metadata": {}, + "source": [ + "Now we generate an unweighted version of our matrix to help calculate in-degree and out-degree." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "2d665a8f", + "metadata": {}, + "outputs": [], + "source": [ + "D = qbn_io.build_unweighted_matrix(Z)" + ] + }, + { + "cell_type": "markdown", + "id": "947d8b01", + "metadata": {}, + "source": [ + "We now use the above to calculate the six centrality measures." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "413f7029", + "metadata": {}, + "outputs": [], + "source": [ + "outdegree = D.sum(axis=1)\n", + "ecentral_hub = eigenvector_centrality(Z, authority=False)\n", + "kcentral_hub = katz_centrality(Z, b=1/1_700_000)\n", + "\n", + "indegree = D.sum(axis=0)\n", + "ecentral_authority = eigenvector_centrality(Z, authority=True)\n", + "kcentral_authority = katz_centrality(Z, b=1/1_700_000, authority=True)" + ] + }, + { + "cell_type": "markdown", + "id": "a14bf02e", + "metadata": {}, + "source": [ + "Here we provide a helper function that returns a DataFrame for each measure.\n", + "The DataFrame is ordered by that measure and contains color information." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "2bcb86db", + "metadata": {}, + "outputs": [], + "source": [ + "def centrality_plot_data(countries, centrality_measures):\n", + " df = pd.DataFrame({'code': countries,\n", + " 'centrality':centrality_measures, \n", + " 'color': qbn_io.colorise_weights(centrality_measures).tolist()\n", + " })\n", + " return df.sort_values('centrality')" + ] + }, + { + "cell_type": "markdown", + "id": "c5777a24", + "metadata": {}, + "source": [ + "Finally, we plot the various centrality measures." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "47130789", + "metadata": {}, + "outputs": [], + "source": [ + "centrality_measures = [outdegree, indegree, \n", + " ecentral_hub, ecentral_authority, \n", + " kcentral_hub, kcentral_authority]\n", + "\n", + "ylabels = ['out degree', 'in degree',\n", + " 'eigenvector hub','eigenvector authority', \n", + " 'Katz hub', 'Katz authority']\n", + "\n", + "ylims = [(0, 20), (0, 20), \n", + " None, None, \n", + " None, None]\n", + "\n", + "\n", + "fig, axes = plt.subplots(3, 2, figsize=(10, 12))\n", + "\n", + "axes = axes.flatten()\n", + "\n", + "for i, ax in enumerate(axes):\n", + " df = centrality_plot_data(countries, centrality_measures[i])\n", + " \n", + " ax.bar('code', 'centrality', data=df, color=df[\"color\"], alpha=0.6)\n", + " \n", + " patch = mpatches.Patch(color=None, label=ylabels[i], visible=False)\n", + " ax.legend(handles=[patch], fontsize=12, loc=\"upper left\", handlelength=0, frameon=False)\n", + " \n", + " if ylims[i] is not None:\n", + " ax.set_ylim(ylims[i])\n", + "\n", + "if export_figures:\n", + " plt.savefig(\"figures/financial_network_analysis_centrality.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "11204e33", + "metadata": {}, + "source": [ + "### Computing in and out degree distributions\n", + "\n", + "The in-degree distribution evaluated at $k$ is the fraction of nodes in a\n", + "network that have in-degree $k$. The in-degree distribution of a `NetworkX`\n", + "DiGraph can be calculated using the code below." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "61512e1d", + "metadata": {}, + "outputs": [], + "source": [ + "def in_degree_dist(G):\n", + " n = G.number_of_nodes()\n", + " iG = np.array([G.in_degree(v) for v in G.nodes()])\n", + " d = [np.mean(iG == k) for k in range(n+1)]\n", + " return d" + ] + }, + { + "cell_type": "markdown", + "id": "8121ffff", + "metadata": {}, + "source": [ + "The out-degree distribution is defined analogously." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "fc3e776b", + "metadata": {}, + "outputs": [], + "source": [ + "def out_degree_dist(G):\n", + " n = G.number_of_nodes()\n", + " oG = np.array([G.out_degree(v) for v in G.nodes()])\n", + " d = [np.mean(oG == k) for k in range(n+1)]\n", + " return d" + ] + }, + { + "cell_type": "markdown", + "id": "840131af", + "metadata": {}, + "source": [ + "### Degree distribution for international aircraft trade\n", + "\n", + "Here we illustrate that the commercial aircraft international trade network is\n", + "approximately scale-free by plotting the degree distribution alongside\n", + "$f(x)=cx-\\gamma$ with $c=0.2$ and $\\gamma=1.1$. \n", + "\n", + "In this calculation of the degree distribution, performed by the NetworkX\n", + "function `degree_histogram`, directions are ignored and the network is treated\n", + "as an undirected graph." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "ecc6928e", + "metadata": {}, + "outputs": [], + "source": [ + "def plot_degree_dist(G, ax, loglog=True, label=None):\n", + " \"Plot the degree distribution of a graph G on axis ax.\"\n", + " dd = [x for x in nx.degree_histogram(G) if x > 0]\n", + " dd = np.array(dd) / np.sum(dd) # normalize\n", + " if loglog:\n", + " ax.loglog(dd, '-o', lw=0.5, label=label)\n", + " else:\n", + " ax.plot(dd, '-o', lw=0.5, label=label)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "f5ec0060", + "metadata": {}, + "outputs": [], + "source": [ + "fig, ax = plt.subplots(figsize=default_figsize)\n", + "\n", + "plot_degree_dist(DG, ax, loglog=False, label='degree distribution')\n", + "\n", + "xg = np.linspace(0.5, 25, 250)\n", + "ax.plot(xg, 0.2 * xg**(-1.1), label='power law')\n", + "ax.set_xlim(0.9, 22)\n", + "ax.set_ylim(0, 0.25)\n", + "ax.legend()\n", + "if export_figures:\n", + " plt.savefig(\"figures/commercial_aircraft_2019_2.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "6c8cb9ba", + "metadata": {}, + "source": [ + "### Random graphs\n", + "\n", + "The code to produce the Erdos-Renyi random graph, used below, applies the\n", + "combinations function from the `itertools` library. The function\n", + "`combinations(A, k)` returns a list of all subsets of $A$ of size $k$. For\n", + "example:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "875a6ae1", + "metadata": {}, + "outputs": [], + "source": [ + "import itertools\n", + "letters = 'a', 'b', 'c'\n", + "list(itertools.combinations(letters, 2))" + ] + }, + { + "cell_type": "markdown", + "id": "fe83b981", + "metadata": {}, + "source": [ + "Below we generate random graphs using the Erdos-Renyi and Barabasi-Albert\n", + "algorithms. Here, for convenience, we will define a function to plot these\n", + "graphs." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "bc9858c8", + "metadata": {}, + "outputs": [], + "source": [ + "def plot_random_graph(RG,ax):\n", + " node_pos_dict = nx.spring_layout(RG, k=1.1)\n", + "\n", + " centrality = nx.degree_centrality(RG)\n", + " node_color_list = qbn_io.colorise_weights(list(centrality.values()))\n", + "\n", + " edge_color_list = []\n", + " for i in range(n):\n", + " for j in range(n):\n", + " edge_color_list.append(node_color_list[i])\n", + "\n", + " nx.draw_networkx_nodes(RG, \n", + " node_pos_dict, \n", + " node_color=node_color_list, \n", + " edgecolors='grey', \n", + " node_size=100,\n", + " linewidths=2, \n", + " alpha=0.8, \n", + " ax=ax)\n", + "\n", + " nx.draw_networkx_edges(RG, \n", + " node_pos_dict, \n", + " edge_color=edge_colors, \n", + " alpha=0.4, \n", + " ax=ax)" + ] + }, + { + "cell_type": "markdown", + "id": "968c2f98", + "metadata": {}, + "source": [ + "### An instance of an Erdos–Renyi random graph" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "027cd2dc", + "metadata": {}, + "outputs": [], + "source": [ + "n = 100\n", + "p = 0.05\n", + "G_er = qbn_io.erdos_renyi_graph(n, p, seed=1234)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "86c89612", + "metadata": {}, + "outputs": [], + "source": [ + "fig, axes = plt.subplots(1, 2, figsize=(10, 3.2))\n", + "\n", + "axes[0].set_title(\"Graph visualization\")\n", + "plot_random_graph(G_er,axes[0])\n", + "\n", + "axes[1].set_title(\"Degree distribution\")\n", + "plot_degree_dist(G_er, axes[1], loglog=False)\n", + "\n", + "if export_figures:\n", + " plt.savefig(\"figures/rand_graph_experiments_1.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "1c2f0411", + "metadata": {}, + "source": [ + "### An instance of a preferential attachment random graph" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "1fcae0d0", + "metadata": {}, + "outputs": [], + "source": [ + "n = 100\n", + "m = 5\n", + "G_ba = nx.generators.random_graphs.barabasi_albert_graph(n, m, seed=123)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "5b513178", + "metadata": {}, + "outputs": [], + "source": [ + "fig, axes = plt.subplots(1, 2, figsize=(10, 3.2))\n", + "\n", + "axes[0].set_title(\"Graph visualization\")\n", + "plot_random_graph(G_ba, axes[0])\n", + "\n", + "axes[1].set_title(\"Degree distribution\")\n", + "plot_degree_dist(G_ba, axes[1], loglog=False)\n", + "\n", + "if export_figures:\n", + " plt.savefig(\"figures/rand_graph_experiments_2.pdf\")\n", + "plt.show()" + ] + } + ], + "metadata": { + "jupytext": { + "cell_metadata_filter": "-all" + }, + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/ipynb/ch_mcs.ipynb b/ipynb/ch_mcs.ipynb new file mode 100644 index 0000000..f8690e2 --- /dev/null +++ b/ipynb/ch_mcs.ipynb @@ -0,0 +1,799 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "9eba78b3", + "metadata": {}, + "source": [ + "# Chapter 4 - Markov Chains and Networks (Python Code)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "0c5585b0", + "metadata": { + "tags": [ + "hide-output" + ] + }, + "outputs": [], + "source": [ + "! pip install --upgrade quantecon_book_networks" + ] + }, + { + "cell_type": "markdown", + "id": "03d19647", + "metadata": {}, + "source": [ + "We begin with some imports." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "48b566bb", + "metadata": {}, + "outputs": [], + "source": [ + "import quantecon as qe\n", + "import quantecon_book_networks\n", + "import quantecon_book_networks.input_output as qbn_io\n", + "import quantecon_book_networks.plotting as qbn_plt\n", + "import quantecon_book_networks.data as qbn_data\n", + "ch4_data = qbn_data.markov_chains_and_networks()\n", + "export_figures = False" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "d76606ee", + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "import pandas as pd\n", + "import networkx as nx\n", + "import matplotlib.pyplot as plt\n", + "from matplotlib import cm\n", + "from mpl_toolkits.mplot3d import Axes3D\n", + "from mpl_toolkits.mplot3d.art3d import Poly3DCollection\n", + "quantecon_book_networks.config(\"matplotlib\")" + ] + }, + { + "cell_type": "markdown", + "id": "da06eb17", + "metadata": {}, + "source": [ + "## Example transition matrices\n", + "\n", + "In this chapter two transition matrices are used.\n", + "\n", + "First, a Markov model is estimated in the international growth dynamics study\n", + "of [Quah (1993)](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.142.5504&rep=rep1&type=pdf).\n", + "\n", + "The state is real GDP per capita in a given country relative to the world\n", + "average. \n", + "\n", + "Quah discretizes the possible values to 0–1/4, 1/4–1/2, 1/2–1, 1–2\n", + "and 2–inf, calling these states 1 to 5 respectively. The transitions are over\n", + "a one year period." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "b1d19891", + "metadata": {}, + "outputs": [], + "source": [ + "P_Q = [\n", + " [0.97, 0.03, 0, 0, 0 ],\n", + " [0.05, 0.92, 0.03, 0, 0 ],\n", + " [0, 0.04, 0.92, 0.04, 0 ],\n", + " [0, 0, 0.04, 0.94, 0.02],\n", + " [0, 0, 0, 0.01, 0.99]\n", + "]\n", + "P_Q = np.array(P_Q)\n", + "codes_Q = ('1', '2', '3', '4', '5')" + ] + }, + { + "cell_type": "markdown", + "id": "c9d3f18b", + "metadata": {}, + "source": [ + "Second, [Benhabib et al. (2015)](https://www.economicdynamics.org/meetpapers/2015/paper_364.pdf) estimate the following transition matrix for intergenerational social mobility.\n", + "\n", + "The states are percentiles of the wealth distribution. \n", + "\n", + "In particular, states 1, 2,..., 8, correspond to the percentiles 0-20%, 20-40%, 40-60%, 60-80%, 80-90%, 90-95%, 95-99%, 99-100%." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "8a2ff2d3", + "metadata": {}, + "outputs": [], + "source": [ + "P_B = [\n", + " [0.222, 0.222, 0.215, 0.187, 0.081, 0.038, 0.029, 0.006],\n", + " [0.221, 0.22, 0.215, 0.188, 0.082, 0.039, 0.029, 0.006],\n", + " [0.207, 0.209, 0.21, 0.194, 0.09, 0.046, 0.036, 0.008],\n", + " [0.198, 0.201, 0.207, 0.198, 0.095, 0.052, 0.04, 0.009],\n", + " [0.175, 0.178, 0.197, 0.207, 0.11, 0.067, 0.054, 0.012],\n", + " [0.182, 0.184, 0.2, 0.205, 0.106, 0.062, 0.05, 0.011],\n", + " [0.123, 0.125, 0.166, 0.216, 0.141, 0.114, 0.094, 0.021],\n", + " [0.084, 0.084, 0.142, 0.228, 0.17, 0.143, 0.121, 0.028]\n", + " ]\n", + "\n", + "P_B = np.array(P_B)\n", + "codes_B = ('1', '2', '3', '4', '5', '6', '7', '8')" + ] + }, + { + "cell_type": "markdown", + "id": "53bd61df", + "metadata": {}, + "source": [ + "## Markov Chains as Digraphs\n", + "\n", + "### Contour plot of a transition matrix \n", + "\n", + "Here we define a function for producing contour plots of matrices." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "a95d7285", + "metadata": {}, + "outputs": [], + "source": [ + "def plot_matrices(matrix,\n", + " codes,\n", + " ax,\n", + " font_size=12,\n", + " alpha=0.6, \n", + " colormap=cm.viridis, \n", + " color45d=None, \n", + " xlabel='sector $j$', \n", + " ylabel='sector $i$'):\n", + " \n", + " ticks = range(len(matrix))\n", + "\n", + " levels = np.sqrt(np.linspace(0, 0.75, 100))\n", + " \n", + " \n", + " if color45d != None:\n", + " co = ax.contourf(ticks, \n", + " ticks,\n", + " matrix,\n", + " alpha=alpha, cmap=colormap)\n", + " ax.plot(ticks, ticks, color=color45d)\n", + " else:\n", + " co = ax.contourf(ticks, \n", + " ticks,\n", + " matrix,\n", + " levels,\n", + " alpha=alpha, cmap=colormap)\n", + "\n", + " ax.set_xlabel(xlabel, fontsize=font_size)\n", + " ax.set_ylabel(ylabel, fontsize=font_size)\n", + " ax.set_yticks(ticks)\n", + " ax.set_yticklabels(codes_B)\n", + " ax.set_xticks(ticks)\n", + " ax.set_xticklabels(codes_B)\n" + ] + }, + { + "cell_type": "markdown", + "id": "15a6af09", + "metadata": {}, + "source": [ + "Now we use our function to produce a plot of the transition matrix for\n", + "intergenerational social mobility, $P_B$." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "8dc01df7", + "metadata": {}, + "outputs": [], + "source": [ + "fig, ax = plt.subplots(figsize=(6,6))\n", + "plot_matrices(P_B.transpose(), codes_B, ax, alpha=0.75, \n", + " colormap=cm.viridis, color45d='black',\n", + " xlabel='state at time $t$', ylabel='state at time $t+1$')\n", + "\n", + "if export_figures:\n", + " plt.savefig(\"figures/markov_matrix_visualization.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "b2476ede", + "metadata": {}, + "source": [ + "### Wealth percentile over time\n", + "\n", + "Here we compare the mixing of the transition matrix for intergenerational\n", + "social mobility $P_B$ and the transition matrix for international growth\n", + "dynamics $P_Q$. \n", + "\n", + "We begin by creating `quantecon` `MarkovChain` objects with each of our transition\n", + "matrices." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "f13e25bb", + "metadata": {}, + "outputs": [], + "source": [ + "mc_B = qe.MarkovChain(P_B, state_values=range(1, 9))\n", + "mc_Q = qe.MarkovChain(P_Q, state_values=range(1, 6))" + ] + }, + { + "cell_type": "markdown", + "id": "df645ec9", + "metadata": {}, + "source": [ + "Next we define a function to plot simulations of Markov chains. \n", + "\n", + "Two simulations will be run for each `MarkovChain`, one starting at the\n", + "minimum initial value and one at the maximum." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "174ec3fc", + "metadata": {}, + "outputs": [], + "source": [ + "def sim_fig(ax, mc, T=100, seed=14, title=None):\n", + " X1 = mc.simulate(T, init=1, random_state=seed)\n", + " X2 = mc.simulate(T, init=max(mc.state_values), random_state=seed+1)\n", + " ax.plot(X1)\n", + " ax.plot(X2)\n", + " ax.set_xlabel(\"time\")\n", + " ax.set_ylabel(\"state\")\n", + " ax.set_title(title, fontsize=12)" + ] + }, + { + "cell_type": "markdown", + "id": "05dc505f", + "metadata": {}, + "source": [ + "Finally, we produce the figure." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "6c98817b", + "metadata": {}, + "outputs": [], + "source": [ + "fig, axes = plt.subplots(2, 1, figsize=(6, 4))\n", + "ax = axes[0]\n", + "sim_fig(axes[0], mc_B, title=\"$P_B$\")\n", + "sim_fig(axes[1], mc_Q, title=\"$P_Q$\")\n", + "axes[1].set_yticks((1, 2, 3, 4, 5))\n", + "\n", + "plt.tight_layout()\n", + "if export_figures:\n", + " plt.savefig(\"figures/benhabib_mobility_mixing.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "2bf229d0", + "metadata": {}, + "source": [ + "### Predicted vs realized cross-country income distributions for 2019\n", + "\n", + "Here we load a `pandas` `DataFrame` of GDP per capita data for countries compared to the global average." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "44dc111f", + "metadata": {}, + "outputs": [], + "source": [ + "gdppc_df = ch4_data['gdppc_df']\n", + "gdppc_df.head()" + ] + }, + { + "cell_type": "markdown", + "id": "d3a50768", + "metadata": {}, + "source": [ + "Now we assign countries bins, as per Quah (1993)." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "09775bab", + "metadata": {}, + "outputs": [], + "source": [ + "q = [0, 0.25, 0.5, 1.0, 2.0, np.inf]\n", + "l = [0, 1, 2, 3, 4]\n", + "\n", + "x = pd.cut(gdppc_df.gdppc_r, bins=q, labels=l)\n", + "gdppc_df['interval'] = x\n", + "\n", + "gdppc_df = gdppc_df.reset_index()\n", + "gdppc_df['interval'] = gdppc_df['interval'].astype(float)\n", + "gdppc_df['year'] = gdppc_df['year'].astype(float)" + ] + }, + { + "cell_type": "markdown", + "id": "8c91bb21", + "metadata": {}, + "source": [ + "Here we define a function for calculating the cross-country income\n", + "distributions for a given date range." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "e7e86382", + "metadata": {}, + "outputs": [], + "source": [ + "def gdp_dist_estimate(df, l, yr=(1960, 2019)):\n", + " Y = np.zeros(len(l))\n", + " for i in l:\n", + " Y[i] = df[\n", + " (df['interval'] == i) & \n", + " (df['year'] <= yr[1]) & \n", + " (df['year'] >= yr[0])\n", + " ].count()[0]\n", + " \n", + " return Y / Y.sum()" + ] + }, + { + "cell_type": "markdown", + "id": "ad6d54d7", + "metadata": {}, + "source": [ + "We calculate the true distribution for 1985." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "130b145a", + "metadata": {}, + "outputs": [], + "source": [ + "ψ_1985 = gdp_dist_estimate(gdppc_df,l,yr=(1985, 1985))" + ] + }, + { + "cell_type": "markdown", + "id": "b3b24523", + "metadata": {}, + "source": [ + "Now we use the transition matrix to update the 1985 distribution $t = 2019 - 1985 = 34$\n", + "times to get our predicted 2019 distribution." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "de8e360b", + "metadata": {}, + "outputs": [], + "source": [ + "ψ_2019_predicted = ψ_1985 @ np.linalg.matrix_power(P_Q, 2019-1985)" + ] + }, + { + "cell_type": "markdown", + "id": "3e8be48f", + "metadata": {}, + "source": [ + "Now, calculate the true 2019 distribution." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "48653d34", + "metadata": {}, + "outputs": [], + "source": [ + "ψ_2019 = gdp_dist_estimate(gdppc_df,l,yr=(2019, 2019))" + ] + }, + { + "cell_type": "markdown", + "id": "886af82b", + "metadata": {}, + "source": [ + "Finally we produce the plot." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "0d6c8426", + "metadata": {}, + "outputs": [], + "source": [ + "states = np.arange(0, 5)\n", + "ticks = range(5)\n", + "codes_S = ('1', '2', '3', '4', '5')\n", + "\n", + "fig, ax = plt.subplots(figsize=(6, 4))\n", + "width = 0.4\n", + "ax.plot(states, ψ_2019_predicted, '-o', alpha=0.7, label='predicted')\n", + "ax.plot(states, ψ_2019, '-o', alpha=0.7, label='realized')\n", + "ax.set_xlabel(\"state\")\n", + "ax.set_ylabel(\"probability\")\n", + "ax.set_yticks((0.15, 0.2, 0.25, 0.3))\n", + "ax.set_xticks(ticks)\n", + "ax.set_xticklabels(codes_S)\n", + "\n", + "ax.legend(loc='upper center', fontsize=12)\n", + "if export_figures:\n", + " plt.savefig(\"figures/quah_gdppc_prediction.pdf\")\n", + "plt.show()\n" + ] + }, + { + "cell_type": "markdown", + "id": "ce838e58", + "metadata": {}, + "source": [ + "### Distribution dynamics\n", + "\n", + "Here we define a function for plotting the convergence of marginal\n", + "distributions $\\psi$ under a transition matrix $P$ on the unit simplex." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "6eaa5db8", + "metadata": {}, + "outputs": [], + "source": [ + "def convergence_plot(ψ, P, n=14, angle=50):\n", + "\n", + " ax = qbn_plt.unit_simplex(angle)\n", + "\n", + " # Convergence plot\n", + " \n", + " P = np.array(P)\n", + "\n", + " ψ = ψ # Initial condition\n", + "\n", + " x_vals, y_vals, z_vals = [], [], []\n", + " for t in range(n):\n", + " x_vals.append(ψ[0])\n", + " y_vals.append(ψ[1])\n", + " z_vals.append(ψ[2])\n", + " ψ = ψ @ P\n", + "\n", + " ax.scatter(x_vals, y_vals, z_vals, c='darkred', s=80, alpha=0.7, depthshade=False)\n", + "\n", + " mc = qe.MarkovChain(P)\n", + " ψ_star = mc.stationary_distributions[0]\n", + " ax.scatter(ψ_star[0], ψ_star[1], ψ_star[2], c='k', s=80)\n", + "\n", + " return ψ\n" + ] + }, + { + "cell_type": "markdown", + "id": "5bc093e3", + "metadata": {}, + "source": [ + "Now we define P." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "64d602c8", + "metadata": {}, + "outputs": [], + "source": [ + "P = (\n", + " (0.9, 0.1, 0.0),\n", + " (0.4, 0.4, 0.2),\n", + " (0.1, 0.1, 0.8)\n", + " )" + ] + }, + { + "cell_type": "markdown", + "id": "a38c94cd", + "metadata": {}, + "source": [ + "#### A trajectory from $\\psi_0 = (0, 0, 1)$\n", + "\n", + "Here we see the sequence of marginals appears to converge." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "39d90e91", + "metadata": {}, + "outputs": [], + "source": [ + "ψ_0 = (0, 0, 1)\n", + "ψ = convergence_plot(ψ_0, P)\n", + "if export_figures:\n", + " plt.savefig(\"figures/simplex_2.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "da3337df", + "metadata": {}, + "source": [ + "#### A trajectory from $\\psi_0 = (0, 1/2, 1/2)$\n", + "\n", + "Here we see again that the sequence of marginals appears to converge, and the\n", + "limit appears not to depend on the initial distribution." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "c4ba2e7b", + "metadata": {}, + "outputs": [], + "source": [ + "ψ_0 = (0, 1/2, 1/2)\n", + "ψ = convergence_plot(ψ_0, P, n=12)\n", + "if export_figures:\n", + " plt.savefig(\"figures/simplex_3.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "7491cf30", + "metadata": {}, + "source": [ + "### Distribution projections from $P_B$\n", + "\n", + "Here we define a function for plotting $\\psi$ after $n$ iterations of the\n", + "transition matrix $P$. The distribution $\\psi_0$ is taken as the uniform \n", + "distribution over the\n", + "state space." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "6dd25c98", + "metadata": {}, + "outputs": [], + "source": [ + "def transition(P, n, ax=None):\n", + " \n", + " P = np.array(P)\n", + " nstates = P.shape[1]\n", + " s0 = np.ones(8) * 1/nstates\n", + " s = s0\n", + " \n", + " for i in range(n):\n", + " s = s @ P\n", + " \n", + " if ax is None:\n", + " fig, ax = plt.subplots()\n", + " \n", + " ax.plot(range(1, nstates+1), s, '-o', alpha=0.6)\n", + " ax.set(ylim=(0, 0.25), \n", + " xticks=((1, nstates)))\n", + " ax.set_title(f\"$t = {n}$\")\n", + " \n", + " return ax" + ] + }, + { + "cell_type": "markdown", + "id": "659480d1", + "metadata": {}, + "source": [ + "We now generate the marginal distributions after 0, 1, 2, and 100 iterations for $P_B$." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "a1bbb0cd", + "metadata": {}, + "outputs": [], + "source": [ + "ns = (0, 1, 2, 100)\n", + "fig, axes = plt.subplots(1, len(ns), figsize=(6, 4))\n", + "\n", + "for n, ax in zip(ns, axes):\n", + " ax = transition(P_B, n, ax=ax)\n", + " ax.set_xlabel(\"Quantile\")\n", + "\n", + "plt.tight_layout()\n", + "if export_figures:\n", + " plt.savefig(\"figures/benhabib_mobility_dists.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "e7ee0c57", + "metadata": {}, + "source": [ + "## Asymptotics\n", + "\n", + "### Convergence of the empirical distribution to $\\psi^*$\n", + "\n", + "We begin by creating a `MarkovChain` object, taking $P_B$ as the transition matrix." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "2f911e34", + "metadata": {}, + "outputs": [], + "source": [ + "mc = qe.MarkovChain(P_B)" + ] + }, + { + "cell_type": "markdown", + "id": "7ca9220f", + "metadata": {}, + "source": [ + "Next we use the `quantecon` package to calculate the true stationary distribution." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "50947355", + "metadata": {}, + "outputs": [], + "source": [ + "stationary = mc.stationary_distributions[0]\n", + "n = len(mc.P)" + ] + }, + { + "cell_type": "markdown", + "id": "0ea1a5bf", + "metadata": {}, + "source": [ + "Now we define a function to simulate the Markov chain." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "d98416ff", + "metadata": {}, + "outputs": [], + "source": [ + "def simulate_distribution(mc, T=100):\n", + " # Simulate path \n", + " n = len(mc.P)\n", + " path = mc.simulate_indices(ts_length=T, random_state=1)\n", + " distribution = np.empty(n)\n", + " for i in range(n):\n", + " distribution[i] = np.mean(path==i)\n", + " return distribution\n" + ] + }, + { + "cell_type": "markdown", + "id": "7bd6bdaa", + "metadata": {}, + "source": [ + "We run simulations of length 10, 100, 1,000 and 10,000." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "bef22ec7", + "metadata": {}, + "outputs": [], + "source": [ + "lengths = [10, 100, 1_000, 10_000]\n", + "dists = []\n", + "\n", + "for t in lengths:\n", + " dists.append(simulate_distribution(mc, t))" + ] + }, + { + "cell_type": "markdown", + "id": "a185b18b", + "metadata": {}, + "source": [ + "Now we produce the plots. \n", + "\n", + "We see that the simulated distribution starts to approach the true stationary distribution." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "6ccd5bdf", + "metadata": {}, + "outputs": [], + "source": [ + "fig, axes = plt.subplots(2, 2, figsize=(9, 6), sharex='all')#, sharey='all')\n", + "\n", + "axes = axes.flatten()\n", + "\n", + "for dist, ax, t in zip(dists, axes, lengths):\n", + " \n", + " ax.plot(np.arange(n)+1 + .25, \n", + " stationary, \n", + " '-o',\n", + " #width = 0.25, \n", + " label='$\\\\psi^*$', \n", + " alpha=0.75)\n", + " \n", + " ax.plot(np.arange(n)+1, \n", + " dist, \n", + " '-o',\n", + " #width = 0.25, \n", + " label=f'$\\\\hat \\\\psi_k$ with $k={t}$', \n", + " alpha=0.75)\n", + "\n", + "\n", + " ax.set_xlabel(\"state\", fontsize=12)\n", + " ax.set_ylabel(\"prob.\", fontsize=12)\n", + " ax.set_xticks(np.arange(n)+1)\n", + " ax.legend(loc='upper right', fontsize=12, frameon=False)\n", + " ax.set_ylim(0, 0.5)\n", + " \n", + "if export_figures:\n", + " plt.savefig(\"figures/benhabib_ergodicity_1.pdf\")\n", + "plt.show()" + ] + } + ], + "metadata": { + "jupytext": { + "cell_metadata_filter": "-all" + }, + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/ipynb/ch_opt.ipynb b/ipynb/ch_opt.ipynb new file mode 100644 index 0000000..f18ddc2 --- /dev/null +++ b/ipynb/ch_opt.ipynb @@ -0,0 +1,728 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "461d8d77", + "metadata": {}, + "source": [ + "# Chapter 3 - Optimal Flows (Python Code)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "e85fdcd6", + "metadata": { + "tags": [ + "hide-output" + ] + }, + "outputs": [], + "source": [ + "! pip install --upgrade quantecon_book_networks" + ] + }, + { + "cell_type": "markdown", + "id": "d3d5fcb1", + "metadata": {}, + "source": [ + "We begin with some imports" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "2a00ff7f", + "metadata": {}, + "outputs": [], + "source": [ + "import quantecon as qe\n", + "import quantecon_book_networks\n", + "import quantecon_book_networks.input_output as qbn_io\n", + "import quantecon_book_networks.plotting as qbn_plt\n", + "import quantecon_book_networks.data as qbn_data\n", + "ch3_data = qbn_data.optimal_flows()\n", + "export_figures = False" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "2ed711ad", + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "from scipy.optimize import linprog\n", + "import networkx as nx\n", + "import ot\n", + "import matplotlib.pyplot as plt\n", + "from matplotlib import cm\n", + "from matplotlib.patches import Polygon\n", + "from matplotlib.artist import Artist \n", + "quantecon_book_networks.config(\"matplotlib\")" + ] + }, + { + "cell_type": "markdown", + "id": "7a5e9cd1", + "metadata": {}, + "source": [ + "## Linear Programming and Duality\n", + "\n", + "### Betweenness centrality (by color and node size) for the Florentine families\n", + "\n", + "We load the Florentine Families data from the NetworkX package." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "4e14e416", + "metadata": {}, + "outputs": [], + "source": [ + "G = nx.florentine_families_graph()" + ] + }, + { + "cell_type": "markdown", + "id": "5199cc62", + "metadata": {}, + "source": [ + "Next we calculate betweenness centrality." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "dab1b1d9", + "metadata": {}, + "outputs": [], + "source": [ + "bc_dict = nx.betweenness_centrality(G)" + ] + }, + { + "cell_type": "markdown", + "id": "700349de", + "metadata": {}, + "source": [ + "And we produce the plot." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "33557ff6", + "metadata": {}, + "outputs": [], + "source": [ + "fig, ax = plt.subplots(figsize=(9.4, 9.4))\n", + "\n", + "plt.axis(\"off\")\n", + "nx.draw_networkx(\n", + " G, \n", + " ax=ax,\n", + " pos=nx.spring_layout(G, seed=1234), \n", + " with_labels=True,\n", + " alpha=.8,\n", + " arrowsize=15,\n", + " connectionstyle=\"arc3,rad=0.1\",\n", + " node_size=[10_000*(size+0.1) for size in bc_dict.values()], \n", + " node_color=[cm.plasma(bc+0.4) for bc in bc_dict.values()],\n", + ")\n", + "if export_figures:\n", + " plt.savefig(\"figures/betweenness_centrality_1.pdf\")" + ] + }, + { + "cell_type": "markdown", + "id": "0fa794a1", + "metadata": {}, + "source": [ + "### Revenue maximizing quantities and a Python implementation of linear programming\n", + "\n", + "First we specify our linear program." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "59dd01db", + "metadata": {}, + "outputs": [], + "source": [ + "A = ((2, 5),\n", + " (4, 2))\n", + "b = (30, 20)\n", + "c = (-3, -4) # minus in order to minimize" + ] + }, + { + "cell_type": "markdown", + "id": "f64d9118", + "metadata": {}, + "source": [ + "And now we use SciPy's linear programing module to solve our linear program." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "e0bf9ba9", + "metadata": {}, + "outputs": [], + "source": [ + "from scipy.optimize import linprog\n", + "result = linprog(c, A_ub=A, b_ub=b)\n", + "print(result.x)" + ] + }, + { + "cell_type": "markdown", + "id": "b1c7bfbb", + "metadata": {}, + "source": [ + "Here we produce a visualization of what is being done." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "41077bbd", + "metadata": {}, + "outputs": [], + "source": [ + "fig, ax = plt.subplots(figsize=(8, 4.5))\n", + "plt.rcParams['font.size'] = '14'\n", + "\n", + "# Draw constraint lines\n", + "\n", + "ax.plot(np.linspace(-1, 17.5, 100), 6-0.4*np.linspace(-1, 17.5, 100))\n", + "ax.plot(np.linspace(-1, 5.5, 100), 10-2*np.linspace(-1, 5.5, 100))\n", + "ax.text(10, 2.5, \"$2q_1 + 5q_2 \\leq 30$\")\n", + "ax.text(1.5, 8, \"$4q_1 + 2q_2 \\leq 20$\")\n", + "ax.text(-2, 2, \"$q_2 \\geq 0$\")\n", + "ax.text(2.5, -0.7, \"$q_1 \\geq 0$\")\n", + "\n", + "# Draw the feasible region\n", + "feasible_set = Polygon(np.array([[0, 0], \n", + " [0, 6], \n", + " [2.5, 5], \n", + " [5, 0]]))\n", + "ax.add_artist(feasible_set)\n", + "Artist.set_alpha(feasible_set, 0.2) \n", + "\n", + "# Draw the objective function\n", + "ax.plot(np.linspace(-1, 5.5, 100), 3.875-0.75*np.linspace(-1, 5.5, 100), 'g-')\n", + "ax.plot(np.linspace(-1, 5.5, 100), 5.375-0.75*np.linspace(-1, 5.5, 100), 'g-')\n", + "ax.plot(np.linspace(-1, 5.5, 100), 6.875-0.75*np.linspace(-1, 5.5, 100), 'g-')\n", + "ax.text(5.8, 1, \"revenue $ = 3q_1 + 4q_2$\")\n", + "\n", + "# Draw the optimal solution\n", + "ax.plot(2.5, 5, \"o\", color=\"black\")\n", + "ax.text(2.7, 5.2, \"optimal solution\")\n", + "\n", + "for spine in ['right', 'top']:\n", + " ax.spines[spine].set_color('none')\n", + " \n", + "ax.set_xticks(())\n", + "ax.set_yticks(())\n", + "\n", + "for spine in ['left', 'bottom']:\n", + " ax.spines[spine].set_position('zero')\n", + " \n", + "ax.set_ylim(-1, 8)\n", + "\n", + "if export_figures:\n", + " plt.savefig(\"figures/linear_programming_1.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "b1c4f25b", + "metadata": {}, + "source": [ + "## Optimal Transport\n", + " \n", + "\n", + "### Transforming one distribution into another\n", + "\n", + "Below we provide code to produce a visualization of transforming one\n", + "distribution into another in one dimension." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "92b7150b", + "metadata": {}, + "outputs": [], + "source": [ + "σ = 0.1\n", + "\n", + "def ϕ(z):\n", + " return (1 / np.sqrt(2 * σ**2 * np.pi)) * np.exp(-z**2 / (2 * σ**2))\n", + "\n", + "def v(x, a=0.4, b=0.6, s=1.0, t=1.4):\n", + " return a * ϕ(x - s) + b * ϕ(x - t)\n", + "\n", + "fig, ax = plt.subplots(figsize=(10, 4))\n", + "\n", + "x = np.linspace(0.2, 4, 1000)\n", + "ax.plot(x, v(x), label=\"$\\\\phi$\")\n", + "ax.plot(x, v(x, s=3.0, t=3.3, a=0.6), label=\"$\\\\psi$\")\n", + "\n", + "ax.legend(loc='upper left', fontsize=12, frameon=False)\n", + "\n", + "ax.arrow(1.8, 1.6, 0.8, 0.0, width=0.01, head_width=0.08)\n", + "ax.annotate('transform', xy=(1.9, 1.9), fontsize=12)\n", + "if export_figures:\n", + " plt.savefig(\"figures/ot_figs_1.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "7464f721", + "metadata": {}, + "source": [ + "### Function to solve a transport problem via linear programming\n", + "\n", + "Here we define a function to solve optimal transport problems using linear programming." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "d94e547a", + "metadata": {}, + "outputs": [], + "source": [ + "def ot_solver(phi, psi, c, method='highs-ipm'):\n", + " \"\"\"\n", + " Solve the OT problem associated with distributions phi, psi\n", + " and cost matrix c.\n", + " Parameters\n", + " ----------\n", + " phi : 1-D array\n", + " Distribution over the source locations.\n", + " psi : 1-D array\n", + " Distribution over the target locations.\n", + " c : 2-D array\n", + " Cost matrix.\n", + " \"\"\"\n", + " n, m = len(phi), len(psi)\n", + "\n", + " # vectorize c\n", + " c_vec = c.reshape((m * n, 1), order='F')\n", + "\n", + " # Construct A and b\n", + " A1 = np.kron(np.ones((1, m)), np.identity(n))\n", + " A2 = np.kron(np.identity(m), np.ones((1, n)))\n", + " A = np.vstack((A1, A2))\n", + " b = np.hstack((phi, psi))\n", + "\n", + " # Call sover\n", + " res = linprog(c_vec, A_eq=A, b_eq=b, method=method)\n", + "\n", + " # Invert the vec operation to get the solution as a matrix\n", + " pi = res.x.reshape((n, m), order='F')\n", + " return pi" + ] + }, + { + "cell_type": "markdown", + "id": "dfdeb358", + "metadata": {}, + "source": [ + "Now we can set up a simple optimal transport problem." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "54e87d72", + "metadata": {}, + "outputs": [], + "source": [ + "phi = np.array((0.5, 0.5))\n", + "psi = np.array((1, 0))\n", + "c = np.ones((2, 2))" + ] + }, + { + "cell_type": "markdown", + "id": "b81830ff", + "metadata": {}, + "source": [ + "Next we solve using the above function." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "3e044c8c", + "metadata": {}, + "outputs": [], + "source": [ + "ot_solver(phi, psi, c)" + ] + }, + { + "cell_type": "markdown", + "id": "f44e6ab3", + "metadata": {}, + "source": [ + "We see we get the same result as when using the Python optimal transport\n", + "package." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "82d43df5", + "metadata": {}, + "outputs": [], + "source": [ + "ot.emd(phi, psi, c) " + ] + }, + { + "cell_type": "markdown", + "id": "642f85ba", + "metadata": {}, + "source": [ + "### An optimal transport problem solved by linear programming\n", + "\n", + "Here we demonstrate a more detailed optimal transport problem. We begin by defining a node class." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "9bcd7819", + "metadata": {}, + "outputs": [], + "source": [ + "class Node:\n", + "\n", + " def __init__(self, x, y, mass, group, name):\n", + "\n", + " self.x, self.y = x, y\n", + " self.mass, self.group = mass, group\n", + " self.name = name" + ] + }, + { + "cell_type": "markdown", + "id": "ad91c7e4", + "metadata": {}, + "source": [ + "Now we define a function for randomly generating nodes." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "7d1bfd36", + "metadata": {}, + "outputs": [], + "source": [ + "from scipy.stats import betabinom\n", + "\n", + "def build_nodes_of_one_type(group='phi', n=100, seed=123):\n", + "\n", + " nodes = []\n", + " np.random.seed(seed)\n", + "\n", + " for i in range(n):\n", + " \n", + " if group == 'phi':\n", + " m = 1/n\n", + " x = np.random.uniform(-2, 2)\n", + " y = np.random.uniform(-2, 2)\n", + " else:\n", + " m = betabinom.pmf(i, n-1, 2, 2)\n", + " x = 0.6 * np.random.uniform(-1.5, 1.5)\n", + " y = 0.6 * np.random.uniform(-1.5, 1.5)\n", + " \n", + " name = group + str(i)\n", + " nodes.append(Node(x, y, m, group, name))\n", + "\n", + " return nodes" + ] + }, + { + "cell_type": "markdown", + "id": "45b701ab", + "metadata": {}, + "source": [ + "We now generate our source and target nodes." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "efa68979", + "metadata": {}, + "outputs": [], + "source": [ + "n_phi = 32\n", + "n_psi = 32\n", + "\n", + "phi_list = build_nodes_of_one_type(group='phi', n=n_phi)\n", + "psi_list = build_nodes_of_one_type(group='psi', n=n_psi)\n", + "\n", + "phi_probs = [phi.mass for phi in phi_list]\n", + "psi_probs = [psi.mass for psi in psi_list]" + ] + }, + { + "cell_type": "markdown", + "id": "61f09811", + "metadata": {}, + "source": [ + "Now we define our transport costs." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "38af4ff7", + "metadata": {}, + "outputs": [], + "source": [ + "c = np.empty((n_phi, n_psi))\n", + "for i in range(n_phi):\n", + " for j in range(n_psi):\n", + " x0, y0 = phi_list[i].x, phi_list[i].y\n", + " x1, y1 = psi_list[j].x, psi_list[j].y\n", + " c[i, j] = np.sqrt((x0-x1)**2 + (y0-y1)**2)" + ] + }, + { + "cell_type": "markdown", + "id": "cbe78a44", + "metadata": {}, + "source": [ + "We solve our optimal transport problem using the Python optimal transport package." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "c8cbef04", + "metadata": {}, + "outputs": [], + "source": [ + "pi = ot.emd(phi_probs, psi_probs, c)" + ] + }, + { + "cell_type": "markdown", + "id": "3fd92511", + "metadata": {}, + "source": [ + "Finally we produce a graph of our sources, targets, and optimal transport plan." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "fa85e02e", + "metadata": {}, + "outputs": [], + "source": [ + "g = nx.DiGraph()\n", + "g.add_nodes_from([phi.name for phi in phi_list])\n", + "g.add_nodes_from([psi.name for psi in psi_list])\n", + "\n", + "for i in range(n_phi):\n", + " for j in range(n_psi):\n", + " if pi[i, j] > 0:\n", + " g.add_edge(phi_list[i].name, psi_list[j].name, weight=pi[i, j])\n", + "\n", + "node_pos_dict={}\n", + "for phi in phi_list:\n", + " node_pos_dict[phi.name] = (phi.x, phi.y)\n", + "for psi in psi_list:\n", + " node_pos_dict[psi.name] = (psi.x, psi.y)\n", + "\n", + "node_color_list = []\n", + "node_size_list = []\n", + "scale = 8_000\n", + "for phi in phi_list:\n", + " node_color_list.append('blue')\n", + " node_size_list.append(phi.mass * scale)\n", + "for psi in psi_list:\n", + " node_color_list.append('red')\n", + " node_size_list.append(psi.mass * scale)\n", + "\n", + "fig, ax = plt.subplots(figsize=(7, 10))\n", + "plt.axis('off')\n", + "\n", + "nx.draw_networkx_nodes(g, \n", + " node_pos_dict, \n", + " node_color=node_color_list,\n", + " node_size=node_size_list,\n", + " edgecolors='grey',\n", + " linewidths=1,\n", + " alpha=0.5,\n", + " ax=ax)\n", + "\n", + "nx.draw_networkx_edges(g, \n", + " node_pos_dict, \n", + " arrows=True,\n", + " connectionstyle='arc3,rad=0.1',\n", + " alpha=0.6)\n", + "if export_figures:\n", + " plt.savefig(\"figures/ot_large_scale_1.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "4bd33217", + "metadata": {}, + "source": [ + "### Solving linear assignment as an optimal transport problem\n", + "\n", + "Here we set up a linear assignment problem (matching $n$ workers to $n$ jobs)." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "4f26dc4c", + "metadata": {}, + "outputs": [], + "source": [ + "n = 4\n", + "phi = np.ones(n)\n", + "psi = np.ones(n)" + ] + }, + { + "cell_type": "markdown", + "id": "756aa046", + "metadata": {}, + "source": [ + "We generate our cost matrix (the cost of training the $i$th worker for the $j$th job)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "437274b1", + "metadata": {}, + "outputs": [], + "source": [ + "c = np.random.uniform(size=(n, n))" + ] + }, + { + "cell_type": "markdown", + "id": "b6a66222", + "metadata": {}, + "source": [ + "Finally, we solve our linear assignment problem as a special case of optimal transport." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "69565581", + "metadata": {}, + "outputs": [], + "source": [ + "ot.emd(phi, psi, c)" + ] + }, + { + "cell_type": "markdown", + "id": "ef9fc68a", + "metadata": {}, + "source": [ + "### Python Spatial Analysis library\n", + "\n", + "Readers interested in computational optimal transport should also consider\n", + "PySAL, the [Python Spatial Analysis library](https://pysal.org/). See, for\n", + "example, https://pysal.org/spaghetti/notebooks/transportation-problem.html.\n", + "\n", + "### The General Flow Problem\n", + "\n", + "Here we solve a simple network flow problem as a linear program. We begin by\n", + "defining the node-edge incidence matrix." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "1e7fc46b", + "metadata": {}, + "outputs": [], + "source": [ + "A = (\n", + "( 1, 1, 0, 0),\n", + "(-1, 0, 1, 0),\n", + "( 0, 0, -1, 1),\n", + "( 0, -1, 0, -1)\n", + ")" + ] + }, + { + "cell_type": "markdown", + "id": "23fac21d", + "metadata": {}, + "source": [ + "Now we define exogenous supply and transport costs." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "f8c5644d", + "metadata": {}, + "outputs": [], + "source": [ + "b = (10, 0, 0, -10)\n", + "c = (1, 4, 1, 1)" + ] + }, + { + "cell_type": "markdown", + "id": "3fe351ee", + "metadata": {}, + "source": [ + "Finally we solve as a linear program." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "a257b5e1", + "metadata": {}, + "outputs": [], + "source": [ + "result = linprog(c, A_eq=A, b_eq=b, method='highs-ipm')\n", + "print(result.x)" + ] + } + ], + "metadata": { + "jupytext": { + "cell_metadata_filter": "-all" + }, + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/ipynb/ch_opt_julia.ipynb b/ipynb/ch_opt_julia.ipynb new file mode 100644 index 0000000..b88726d --- /dev/null +++ b/ipynb/ch_opt_julia.ipynb @@ -0,0 +1,209 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "736e5076", + "metadata": {}, + "source": [ + "# Chapter 3 - Optimal Flows (Julia Code)\n", + "\n", + "## Bellman’s Method\n", + "\n", + "Here we demonstrate solving a shortest path problem using Bellman's method.\n", + "\n", + "Our first step is to set up the cost function, which we store as an array\n", + "called `c`. Note that we set `c[i, j] = Inf` when no edge exists from `i` to\n", + "`j`." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "c36f47c4", + "metadata": { + "tags": [ + "remove-output" + ] + }, + "outputs": [], + "source": [ + "c = fill(Inf, (7, 7))\n", + "c[1, 2], c[1, 3], c[1, 4] = 1, 5, 3\n", + "c[2, 4], c[2, 5] = 9, 6\n", + "c[3, 6] = 2\n", + "c[4, 6] = 4\n", + "c[5, 7] = 4\n", + "c[6, 7] = 1\n", + "c[7, 7] = 0" + ] + }, + { + "cell_type": "markdown", + "id": "2b8fa3ad", + "metadata": {}, + "source": [ + "Next we define the Bellman operator." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "c2bc51e6", + "metadata": { + "tags": [ + "remove-output" + ] + }, + "outputs": [], + "source": [ + "function T(q)\n", + " Tq = similar(q)\n", + " n = length(q)\n", + " for x in 1:n\n", + " Tq[x] = minimum(c[x, :] + q[:])\n", + " end\n", + " return Tq\n", + "end" + ] + }, + { + "cell_type": "markdown", + "id": "9857bf8a", + "metadata": {}, + "source": [ + "Now we arbitrarily set $q \\equiv 0$, generate the sequence of iterates $T_q$,\n", + "$T^2_q$, $T^3_q$ and plot them. By $T^3_q$ has already converged on $q^∗$." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "757cadc8", + "metadata": {}, + "outputs": [], + "source": [ + "using PyPlot\n", + "export_figures = false\n", + "fig, ax = plt.subplots(figsize=(6, 4))\n", + "\n", + "n = 7\n", + "q = zeros(n)\n", + "ax.plot(1:n, q)\n", + "ax.set_xlabel(\"cost-to-go\")\n", + "ax.set_ylabel(\"nodes\")\n", + "\n", + "for i in 1:3\n", + " new_q = T(q)\n", + " ax.plot(1:n, new_q, \"-o\", alpha=0.7, label=\"iterate $i\")\n", + " q = new_q\n", + "end\n", + "\n", + "ax.legend()\n", + "if export_figures == true\n", + " plt.savefig(\"figures/shortest_path_iter_1.pdf\")\n", + "end" + ] + }, + { + "cell_type": "markdown", + "id": "d31abe43", + "metadata": {}, + "source": [ + "## Linear programming\n", + "\n", + "When solving linear programs, one option is to use a domain specific modeling\n", + "language to set out the objective and constraints in the optimization problem.\n", + "Here we demonstrate the Julia package `JuMP`." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "67361f35", + "metadata": {}, + "outputs": [], + "source": [ + "using JuMP\n", + "using GLPK" + ] + }, + { + "cell_type": "markdown", + "id": "9bd4aef4", + "metadata": {}, + "source": [ + "We create our model object and select our solver." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "856ef0ee", + "metadata": {}, + "outputs": [], + "source": [ + "m = Model()\n", + "set_optimizer(m, GLPK.Optimizer)" + ] + }, + { + "cell_type": "markdown", + "id": "fe817ce7", + "metadata": {}, + "source": [ + "Now we add variables, constraints and an objective to our model." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "07228cb0", + "metadata": { + "tags": [ + "remove-output" + ] + }, + "outputs": [], + "source": [ + "@variable(m, q1 >= 0)\n", + "@variable(m, q2 >= 0)\n", + "@constraint(m, 2q1 + 5q2 <= 30)\n", + "@constraint(m, 4q1 + 2q2 <= 20)\n", + "@objective(m, Max, 3q1 + 4q2)" + ] + }, + { + "cell_type": "markdown", + "id": "842781b0", + "metadata": {}, + "source": [ + "Finally we solve our linear program." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "83c0d928", + "metadata": {}, + "outputs": [], + "source": [ + "optimize!(m)\n", + "\n", + "println(value.(q1)) \n", + "println(value.(q2))" + ] + } + ], + "metadata": { + "jupytext": { + "cell_metadata_filter": "-all" + }, + "kernelspec": { + "display_name": "Julia", + "language": "Julia", + "name": "julia-1.9" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/ipynb/ch_production.ipynb b/ipynb/ch_production.ipynb new file mode 100644 index 0000000..8f51490 --- /dev/null +++ b/ipynb/ch_production.ipynb @@ -0,0 +1,737 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "7e970772", + "metadata": {}, + "source": [ + "# Chapter 2 - Production (Python Code)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "fd1d49bf", + "metadata": { + "tags": [ + "hide-output" + ] + }, + "outputs": [], + "source": [ + "! pip install --upgrade quantecon_book_networks" + ] + }, + { + "cell_type": "markdown", + "id": "5bb0e40f", + "metadata": {}, + "source": [ + "We begin with some imports." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "9b06f1ff", + "metadata": {}, + "outputs": [], + "source": [ + "import quantecon as qe\n", + "import quantecon_book_networks\n", + "import quantecon_book_networks.input_output as qbn_io\n", + "import quantecon_book_networks.plotting as qbn_plt\n", + "import quantecon_book_networks.data as qbn_data\n", + "ch2_data = qbn_data.production()\n", + "default_figsize = (6, 4)\n", + "export_figures = False" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "67d26617", + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "import pandas as pd\n", + "import networkx as nx\n", + "import matplotlib.pyplot as plt\n", + "import matplotlib.cm as cm\n", + "import matplotlib.colors as plc\n", + "from matplotlib import cm\n", + "quantecon_book_networks.config(\"matplotlib\")" + ] + }, + { + "cell_type": "markdown", + "id": "aa98eb15", + "metadata": {}, + "source": [ + "## Multisector Models\n", + "\n", + "We start by loading a graph of linkages between 15 US sectors in 2021. \n", + "\n", + "Our graph comes as a list of sector codes, an adjacency matrix of sales between\n", + "the sectors, and a list the total sales of each sector. \n", + "\n", + "In particular, `Z[i,j]` is the sales from industry `i` to industry `j`, and `X[i]` is the the total sales\n", + "of each sector `i`." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "bac5b180", + "metadata": {}, + "outputs": [], + "source": [ + "codes = ch2_data[\"us_sectors_15\"][\"codes\"]\n", + "Z = ch2_data[\"us_sectors_15\"][\"adjacency_matrix\"]\n", + "X = ch2_data[\"us_sectors_15\"][\"total_industry_sales\"]" + ] + }, + { + "cell_type": "markdown", + "id": "c3a07fd9", + "metadata": {}, + "source": [ + "Now we define a function to build coefficient matrices. \n", + "\n", + "Two coefficient matrices are returned. The backward linkage case, where sales\n", + "between sector `i` and `j` are given as a fraction of total sales of sector\n", + "`j`. The forward linkage case, where sales between sector `i` and `j` are\n", + "given as a fraction of total sales of sector `i`." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "fa4c9d05", + "metadata": {}, + "outputs": [], + "source": [ + "def build_coefficient_matrices(Z, X):\n", + " \"\"\"\n", + " Build coefficient matrices A and F from Z and X via \n", + " \n", + " A[i, j] = Z[i, j] / X[j] \n", + " F[i, j] = Z[i, j] / X[i]\n", + " \n", + " \"\"\"\n", + " A, F = np.empty_like(Z), np.empty_like(Z)\n", + " n = A.shape[0]\n", + " for i in range(n):\n", + " for j in range(n):\n", + " A[i, j] = Z[i, j] / X[j]\n", + " F[i, j] = Z[i, j] / X[i]\n", + "\n", + " return A, F\n", + "\n", + "A, F = build_coefficient_matrices(Z, X)" + ] + }, + { + "cell_type": "markdown", + "id": "66ecec7b", + "metadata": {}, + "source": [ + "### Backward linkages for 15 US sectors in 2021\n", + "\n", + "Here we calculate the hub-based eigenvector centrality of our backward linkage coefficient matrix." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "f1e25af6", + "metadata": {}, + "outputs": [], + "source": [ + "centrality = qbn_io.eigenvector_centrality(A)" + ] + }, + { + "cell_type": "markdown", + "id": "99a5def5", + "metadata": {}, + "source": [ + "Now we use the `quantecon_book_networks` package to produce our plot." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "5c9a1c8e", + "metadata": {}, + "outputs": [], + "source": [ + "fig, ax = plt.subplots(figsize=(8, 10))\n", + "plt.axis(\"off\")\n", + "color_list = qbn_io.colorise_weights(centrality, beta=False)\n", + "# Remove self-loops\n", + "A1 = A.copy()\n", + "for i in range(A1.shape[0]):\n", + " A1[i][i] = 0\n", + "qbn_plt.plot_graph(A1, X, ax, codes, \n", + " layout_type='spring',\n", + " layout_seed=5432167,\n", + " tol=0.0,\n", + " node_color_list=color_list) \n", + "\n", + "if export_figures:\n", + " plt.savefig(\"figures/input_output_analysis_15.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "30667357", + "metadata": {}, + "source": [ + "### Eigenvector centrality of across US industrial sectors\n", + "\n", + "Now we plot a bar chart of hub-based eigenvector centrality by sector." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "60e3a77c", + "metadata": {}, + "outputs": [], + "source": [ + "fig, ax = plt.subplots(figsize=default_figsize)\n", + "ax.bar(codes, centrality, color=color_list, alpha=0.6)\n", + "ax.set_ylabel(\"eigenvector centrality\", fontsize=12)\n", + "if export_figures:\n", + " plt.savefig(\"figures/input_output_analysis_15_ec.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "f0556d61", + "metadata": {}, + "source": [ + "### Output multipliers across 15 US industrial sectors\n", + "\n", + "Output multipliers are equal to the authority-based Katz centrality measure of\n", + "the backward linkage coefficient matrix. \n", + "\n", + "Here we calculate authority-based\n", + "Katz centrality using the `quantecon_book_networks` package." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "f1358ad2", + "metadata": {}, + "outputs": [], + "source": [ + "omult = qbn_io.katz_centrality(A, authority=True)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "d45c83f2", + "metadata": {}, + "outputs": [], + "source": [ + "fig, ax = plt.subplots(figsize=default_figsize)\n", + "omult_color_list = qbn_io.colorise_weights(omult,beta=False)\n", + "ax.bar(codes, omult, color=omult_color_list, alpha=0.6)\n", + "ax.set_ylabel(\"Output multipliers\", fontsize=12)\n", + "if export_figures:\n", + " plt.savefig(\"figures/input_output_analysis_15_omult.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "d07a87b3", + "metadata": {}, + "source": [ + "### Forward linkages and upstreamness over US industrial sectors\n", + "\n", + "Upstreamness is the hub-based Katz centrality of the forward linkage\n", + "coefficient matrix. \n", + "\n", + "Here we calculate hub-based Katz centrality." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "2df63367", + "metadata": {}, + "outputs": [], + "source": [ + "upstreamness = qbn_io.katz_centrality(F)" + ] + }, + { + "cell_type": "markdown", + "id": "cff33a99", + "metadata": {}, + "source": [ + "Now we plot the network." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "8ac45508", + "metadata": {}, + "outputs": [], + "source": [ + "fig, ax = plt.subplots(figsize=(8, 10))\n", + "plt.axis(\"off\")\n", + "upstreamness_color_list = qbn_io.colorise_weights(upstreamness,beta=False)\n", + "# Remove self-loops\n", + "for i in range(F.shape[0]):\n", + " F[i][i] = 0\n", + "qbn_plt.plot_graph(F, X, ax, codes, \n", + " layout_type='spring', # alternative layouts: spring, circular, random, spiral\n", + " layout_seed=5432167,\n", + " tol=0.0,\n", + " node_color_list=upstreamness_color_list) \n", + "\n", + "if export_figures:\n", + " plt.savefig(\"figures/input_output_analysis_15_fwd.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "e9cbc94b", + "metadata": {}, + "source": [ + "### Relative upstreamness of US industrial sectors\n", + "\n", + "Here we produce a barplot of upstreamness." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "abc67147", + "metadata": {}, + "outputs": [], + "source": [ + "fig, ax = plt.subplots(figsize=default_figsize)\n", + "ax.bar(codes, upstreamness, color=upstreamness_color_list, alpha=0.6)\n", + "ax.set_ylabel(\"upstreamness\", fontsize=12)\n", + "if export_figures:\n", + " plt.savefig(\"figures/input_output_analysis_15_up.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "f9223fd5", + "metadata": {}, + "source": [ + "### Hub-based Katz centrality of across 15 US industrial sectors\n", + "\n", + "Next we plot the hub-based Katz centrality of the backward linkage coefficient matrix." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "19bce52b", + "metadata": {}, + "outputs": [], + "source": [ + "kcentral = qbn_io.katz_centrality(A)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "d7563381", + "metadata": {}, + "outputs": [], + "source": [ + "fig, ax = plt.subplots(figsize=default_figsize)\n", + "kcentral_color_list = qbn_io.colorise_weights(kcentral,beta=False)\n", + "ax.bar(codes, kcentral, color=kcentral_color_list, alpha=0.6)\n", + "ax.set_ylabel(\"Katz hub centrality\", fontsize=12)\n", + "if export_figures:\n", + " plt.savefig(\"figures/input_output_analysis_15_katz.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "bf61ce72", + "metadata": {}, + "source": [ + "### The Leontief inverse 𝐿 (hot colors are larger values)\n", + "\n", + "We construct the Leontief inverse matrix from 15 sector adjacency matrix." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "87c25e09", + "metadata": {}, + "outputs": [], + "source": [ + "I = np.identity(len(A))\n", + "L = np.linalg.inv(I - A)" + ] + }, + { + "cell_type": "markdown", + "id": "a37cada4", + "metadata": {}, + "source": [ + "Now we produce the plot." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "d590c0e1", + "metadata": {}, + "outputs": [], + "source": [ + "fig, ax = plt.subplots(figsize=(6.5, 5.5))\n", + "\n", + "ticks = range(len(L))\n", + "\n", + "levels = np.sqrt(np.linspace(0, 0.75, 100))\n", + "\n", + "co = ax.contourf(ticks, \n", + " ticks,\n", + " L,\n", + " levels,\n", + " alpha=0.85, cmap=cm.plasma)\n", + "\n", + "ax.set_xlabel('sector $j$', fontsize=12)\n", + "ax.set_ylabel('sector $i$', fontsize=12)\n", + "ax.set_yticks(ticks)\n", + "ax.set_yticklabels(codes)\n", + "ax.set_xticks(ticks)\n", + "ax.set_xticklabels(codes)\n", + "\n", + "if export_figures:\n", + " plt.savefig(\"figures/input_output_analysis_15_leo.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "1a9a7d40", + "metadata": {}, + "source": [ + "### Propagation of demand shocks via backward linkages\n", + "\n", + "We begin by generating a demand shock vector $d$." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "0861b2f5", + "metadata": {}, + "outputs": [], + "source": [ + "N = len(A)\n", + "np.random.seed(1234)\n", + "d = np.random.rand(N) \n", + "d[6] = 1 # positive shock to agriculture" + ] + }, + { + "cell_type": "markdown", + "id": "7fa2840f", + "metadata": {}, + "source": [ + "Now we simulate the demand shock propagating through the economy." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "9089f879", + "metadata": {}, + "outputs": [], + "source": [ + "sim_length = 11\n", + "x = d\n", + "x_vecs = []\n", + "for i in range(sim_length):\n", + " if i % 2 ==0:\n", + " x_vecs.append(x)\n", + " x = A @ x" + ] + }, + { + "cell_type": "markdown", + "id": "4a51906c", + "metadata": {}, + "source": [ + "Finally, we plot the shock propagating through the economy." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "9d777b13", + "metadata": {}, + "outputs": [], + "source": [ + "fig, axes = plt.subplots(3, 2, figsize=(8, 10))\n", + "axes = axes.flatten()\n", + "\n", + "for ax, x_vec, i in zip(axes, x_vecs, range(sim_length)):\n", + " if i % 2 != 0:\n", + " pass\n", + " ax.set_title(f\"round {i*2}\")\n", + " x_vec_cols = qbn_io.colorise_weights(x_vec,beta=False)\n", + " # remove self-loops\n", + " for i in range(len(A)):\n", + " A[i][i] = 0\n", + " qbn_plt.plot_graph(A, X, ax, codes,\n", + " layout_type='spring',\n", + " layout_seed=342156,\n", + " node_color_list=x_vec_cols,\n", + " node_size_multiple=0.00028,\n", + " edge_size_multiple=0.8)\n", + "\n", + "plt.tight_layout()\n", + "if export_figures:\n", + " plt.savefig(\"figures/input_output_analysis_15_shocks.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "6ccd313f", + "metadata": {}, + "source": [ + "### Network for 71 US sectors in 2021\n", + "\n", + "We start by loading a graph of linkages between 71 US sectors in 2021." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "24d32882", + "metadata": {}, + "outputs": [], + "source": [ + "codes_71 = ch2_data['us_sectors_71']['codes']\n", + "A_71 = ch2_data['us_sectors_71']['adjacency_matrix']\n", + "X_71 = ch2_data['us_sectors_71']['total_industry_sales']" + ] + }, + { + "cell_type": "markdown", + "id": "e4efe0f9", + "metadata": {}, + "source": [ + "Next we calculate our graph's properties. \n", + "\n", + "We use hub-based eigenvector centrality as our centrality measure for this plot." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "c468f74b", + "metadata": {}, + "outputs": [], + "source": [ + "centrality_71 = qbn_io.eigenvector_centrality(A_71)\n", + "color_list_71 = qbn_io.colorise_weights(centrality_71,beta=False)" + ] + }, + { + "cell_type": "markdown", + "id": "37c499e0", + "metadata": {}, + "source": [ + "Finally we produce the plot." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "3d920aca", + "metadata": {}, + "outputs": [], + "source": [ + "fig, ax = plt.subplots(figsize=(10, 12))\n", + "plt.axis(\"off\")\n", + "# Remove self-loops\n", + "for i in range(A_71.shape[0]):\n", + " A_71[i][i] = 0\n", + "qbn_plt.plot_graph(A_71, X_71, ax, codes_71,\n", + " node_size_multiple=0.0005,\n", + " edge_size_multiple=4.0,\n", + " layout_type='spring',\n", + " layout_seed=5432167,\n", + " tol=0.01,\n", + " node_color_list=color_list_71)\n", + "\n", + "if export_figures:\n", + " plt.savefig(\"figures/input_output_analysis_71.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "419a96fa", + "metadata": {}, + "source": [ + "### Network for 114 Australian industry sectors in 2020\n", + "\n", + "Next we load a graph of linkages between 114 Australian sectors in 2020." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "52309e87", + "metadata": {}, + "outputs": [], + "source": [ + "codes_114 = ch2_data['au_sectors_114']['codes']\n", + "A_114 = ch2_data['au_sectors_114']['adjacency_matrix']\n", + "X_114 = ch2_data['au_sectors_114']['total_industry_sales']" + ] + }, + { + "cell_type": "markdown", + "id": "b5e46835", + "metadata": {}, + "source": [ + "Next we calculate our graph's properties. \n", + "\n", + "We use hub-based eigenvector centrality as our centrality measure for this plot." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "9555ffa8", + "metadata": {}, + "outputs": [], + "source": [ + "centrality_114 = qbn_io.eigenvector_centrality(A_114)\n", + "color_list_114 = qbn_io.colorise_weights(centrality_114,beta=False)" + ] + }, + { + "cell_type": "markdown", + "id": "1b6801cd", + "metadata": {}, + "source": [ + "Finally we produce the plot." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "1af80023", + "metadata": {}, + "outputs": [], + "source": [ + "fig, ax = plt.subplots(figsize=(11, 13.2))\n", + "plt.axis(\"off\")\n", + "# Remove self-loops\n", + "for i in range(A_114.shape[0]):\n", + " A_114[i][i] = 0\n", + "qbn_plt.plot_graph(A_114, X_114, ax, codes_114,\n", + " node_size_multiple=0.008,\n", + " edge_size_multiple=5.0,\n", + " layout_type='spring',\n", + " layout_seed=5432167,\n", + " tol=0.03,\n", + " node_color_list=color_list_114)\n", + "\n", + "if export_figures:\n", + " plt.savefig(\"figures/input_output_analysis_aus_114.pdf\")\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "a3095a50", + "metadata": {}, + "source": [ + "### GDP growth rates and std. deviations (in parentheses) for 8 countries\n", + "\n", + "Here we load a `pandas` DataFrame of GDP growth rates." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "e7aa80e5", + "metadata": {}, + "outputs": [], + "source": [ + "gdp_df = ch2_data['gdp_df']\n", + "gdp_df.head()" + ] + }, + { + "cell_type": "markdown", + "id": "c82df9d4", + "metadata": {}, + "source": [ + "Now we plot the growth rates and calculate their standard deviations." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "394a86ea", + "metadata": {}, + "outputs": [], + "source": [ + "fig, axes = plt.subplots(5, 2, figsize=(8, 9))\n", + "axes = axes.flatten()\n", + "\n", + "countries = gdp_df.columns\n", + "t = np.asarray(gdp_df.index.astype(float))\n", + "series = [np.asarray(gdp_df[country].astype(float)) for country in countries]\n", + "\n", + "\n", + "for ax, country, gdp_data in zip(axes, countries, series):\n", + " \n", + " ax.plot(t, gdp_data)\n", + " ax.set_title(f'{country} (${gdp_data.std():1.2f}$\\%)' )\n", + " ax.set_ylabel('\\%')\n", + " ax.set_ylim((-12, 14))\n", + "\n", + "plt.tight_layout()\n", + "if export_figures:\n", + " plt.savefig(\"figures/gdp_growth.pdf\")\n", + "plt.show()" + ] + } + ], + "metadata": { + "jupytext": { + "cell_metadata_filter": "-all" + }, + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/code_book/figures/benhabib_mobility_dists.pdf b/ipynb/figures/benhabib_mobility_dists.pdf similarity index 100% rename from code_book/figures/benhabib_mobility_dists.pdf rename to ipynb/figures/benhabib_mobility_dists.pdf diff --git a/ipynb/pkg_funcs.ipynb b/ipynb/pkg_funcs.ipynb new file mode 100644 index 0000000..886138c --- /dev/null +++ b/ipynb/pkg_funcs.ipynb @@ -0,0 +1,582 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "83f7d689", + "metadata": {}, + "source": [ + "# quantecon_book_networks\n", + "\n", + "## input_output\n", + "\n", + "### node_total_exports" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "6e1c0e01", + "metadata": {}, + "outputs": [], + "source": [ + "def node_total_exports(G):\n", + " node_exports = []\n", + " for node1 in G.nodes():\n", + " total_export = 0\n", + " for node2 in G[node1]:\n", + " total_export += G[node1][node2]['weight']\n", + " node_exports.append(total_export)\n", + " return node_exports" + ] + }, + { + "cell_type": "markdown", + "id": "d91edba5", + "metadata": {}, + "source": [ + "### node_total_imports" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "d7942abf", + "metadata": {}, + "outputs": [], + "source": [ + "def node_total_imports(G):\n", + " node_imports = []\n", + " for node1 in G.nodes():\n", + " total_import = 0\n", + " for node2 in G[node1]:\n", + " total_import += G[node2][node1]['weight']\n", + " node_imports.append(total_import)\n", + " return node_imports" + ] + }, + { + "cell_type": "markdown", + "id": "3dd8089f", + "metadata": {}, + "source": [ + "### edge_weights" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "fdf563f5", + "metadata": {}, + "outputs": [], + "source": [ + "def edge_weights(G):\n", + " edge_weights = [G[u][v]['weight'] for u,v in G.edges()]\n", + " return edge_weights" + ] + }, + { + "cell_type": "markdown", + "id": "055dcff7", + "metadata": {}, + "source": [ + "### normalise_weights" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "54211ff8", + "metadata": {}, + "outputs": [], + "source": [ + "def normalise_weights(weights,scalar=1):\n", + " max_value = np.max(weights)\n", + " return [scalar * (weight / max_value) for weight in weights]" + ] + }, + { + "cell_type": "markdown", + "id": "bfaedc3f", + "metadata": {}, + "source": [ + "### to_zero_one" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "ed795d06", + "metadata": {}, + "outputs": [], + "source": [ + "def to_zero_one(x):\n", + " \"Map vector x to the zero one interval.\"\n", + " x = np.array(x)\n", + " x_min, x_max = x.min(), x.max()\n", + " return (x - x_min)/(x_max - x_min)" + ] + }, + { + "cell_type": "markdown", + "id": "6ce4a5bd", + "metadata": {}, + "source": [ + "### to_zero_one_beta" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "f2d51900", + "metadata": {}, + "outputs": [], + "source": [ + "def to_zero_one_beta(x, \n", + " qrange=[0.25, 0.75], \n", + " beta_para=[0.5, 0.5]):\n", + " \n", + " \"\"\"\n", + " Nonlinearly map vector x to the zero one interval with beta distribution.\n", + " https://en.wikipedia.org/wiki/Beta_distribution\n", + " \"\"\"\n", + " x = np.array(x)\n", + " x_min, x_max = x.min(), x.max()\n", + " if beta_para != None:\n", + " a, b = beta_para\n", + " return beta.cdf((x - x_min) /(x_max - x_min), a, b)\n", + " else:\n", + " q1, q2 = qrange\n", + " return (x - x_min) * (q2 - q1) /(x_max - x_min) + q1" + ] + }, + { + "cell_type": "markdown", + "id": "677a3b25", + "metadata": {}, + "source": [ + "### colorise_weights" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "650554ba", + "metadata": {}, + "outputs": [], + "source": [ + "import matplotlib.cm as cm\n", + "def colorise_weights(weights,beta=True,color_palette=cm.plasma):\n", + " if beta:\n", + " cp = color_palette(to_zero_one_beta(weights))\n", + " else:\n", + " cp = color_palette(to_zero_one(weights))\n", + " return cp " + ] + }, + { + "cell_type": "markdown", + "id": "13e770e7", + "metadata": {}, + "source": [ + "### spec_rad" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "8d31e595", + "metadata": {}, + "outputs": [], + "source": [ + "def spec_rad(M):\n", + " \"\"\"\n", + " Compute the spectral radius of M.\n", + " \"\"\"\n", + " return np.max(np.abs(np.linalg.eigvals(M)))" + ] + }, + { + "cell_type": "markdown", + "id": "8b997a2b", + "metadata": {}, + "source": [ + "### adjacency_matrix_to_graph" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "1b8660e3", + "metadata": {}, + "outputs": [], + "source": [ + "def adjacency_matrix_to_graph(A, \n", + " codes,\n", + " tol=0.0): # clip entries below tol\n", + " \"\"\"\n", + " Build a networkx graph object given an adjacency matrix\n", + " \"\"\"\n", + " G = nx.DiGraph()\n", + " N = len(A)\n", + "\n", + " # Add nodes\n", + " for i, code in enumerate(codes):\n", + " G.add_node(code, name=code)\n", + "\n", + " # Add the edges\n", + " for i in range(N):\n", + " for j in range(N):\n", + " a = A[i, j]\n", + " if a > tol:\n", + " G.add_edge(codes[i], codes[j], weight=a)\n", + "\n", + " return G" + ] + }, + { + "cell_type": "markdown", + "id": "e673657f", + "metadata": {}, + "source": [ + "### eigenvector_centrality" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "90190208", + "metadata": {}, + "outputs": [], + "source": [ + "def eigenvector_centrality(A, k=40, authority=False):\n", + " \"\"\"\n", + " Computes the dominant eigenvector of A. Assumes A is \n", + " primitive and uses the power method. \n", + " \"\"\"\n", + " A_temp = A.T if authority else A\n", + " n = len(A_temp)\n", + " r = spec_rad(A_temp)\n", + " e = r**(-k) * (np.linalg.matrix_power(A_temp, k) @ np.ones(n))\n", + " return e / np.sum(e)" + ] + }, + { + "cell_type": "markdown", + "id": "78f99a4d", + "metadata": {}, + "source": [ + "### katz_centrality" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "842342f6", + "metadata": {}, + "outputs": [], + "source": [ + "def katz_centrality(A, b=1, authority=False):\n", + " \"\"\"\n", + " Computes the Katz centrality of A, defined as the x solving\n", + "\n", + " x = 1 + b A x (1 = vector of ones)\n", + "\n", + " Assumes that A is square.\n", + "\n", + " If authority=True, then A is replaced by its transpose.\n", + " \"\"\"\n", + " n = len(A)\n", + " I = np.identity(n)\n", + " C = I - b * A.T if authority else I - b * A\n", + " return np.linalg.solve(C, np.ones(n))" + ] + }, + { + "cell_type": "markdown", + "id": "76431864", + "metadata": {}, + "source": [ + "### build_unweighted_matrix" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "0e638e6a", + "metadata": {}, + "outputs": [], + "source": [ + "def build_unweighted_matrix(Z, tol=1e-5):\n", + " \"\"\"\n", + " return a unweighted adjacency matrix\n", + " \"\"\"\n", + " return 1*(Z>tol)" + ] + }, + { + "cell_type": "markdown", + "id": "db7270d9", + "metadata": {}, + "source": [ + "### erdos_renyi_graph" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "8a74e9db", + "metadata": {}, + "outputs": [], + "source": [ + "def erdos_renyi_graph(n=100, p=0.5, seed=1234):\n", + " \"Returns an Erdős-Rényi random graph.\"\n", + " \n", + " np.random.seed(seed)\n", + " edges = itertools.combinations(range(n), 2)\n", + " G = nx.Graph()\n", + " \n", + " for e in edges:\n", + " if np.random.rand() < p:\n", + " G.add_edge(*e)\n", + " return G" + ] + }, + { + "cell_type": "markdown", + "id": "02443320", + "metadata": {}, + "source": [ + "### build_coefficient_matrices" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "9316d715", + "metadata": {}, + "outputs": [], + "source": [ + "def build_coefficient_matrices(Z, X):\n", + " \"\"\"\n", + " Build coefficient matrices A and F from Z and X via \n", + " \n", + " A[i, j] = Z[i, j] / X[j] \n", + " F[i, j] = Z[i, j] / X[i]\n", + " \n", + " \"\"\"\n", + " A, F = np.empty_like(Z), np.empty_like(Z)\n", + " n = A.shape[0]\n", + " for i in range(n):\n", + " for j in range(n):\n", + " A[i, j] = Z[i, j] / X[j]\n", + " F[i, j] = Z[i, j] / X[i]\n", + "\n", + " return A, F" + ] + }, + { + "cell_type": "markdown", + "id": "54010bf3", + "metadata": {}, + "source": [ + "## plotting\n", + "\n", + "### plot_graph" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "4439b087", + "metadata": {}, + "outputs": [], + "source": [ + "def plot_graph(A, \n", + " X,\n", + " ax,\n", + " codes,\n", + " node_color_list=None,\n", + " node_size_multiple=0.0005, \n", + " edge_size_multiple=14,\n", + " layout_type='circular',\n", + " layout_seed=1234,\n", + " tol=0.03): # clip entries below tol\n", + "\n", + " G = nx.DiGraph()\n", + " N = len(A)\n", + "\n", + " # Add nodes, with weights by sales of the sector\n", + " for i, w in enumerate(X):\n", + " G.add_node(codes[i], weight=w, name=codes[i])\n", + "\n", + " node_sizes = X * node_size_multiple\n", + "\n", + " # Position the nodes\n", + " if layout_type == 'circular':\n", + " node_pos_dict = nx.circular_layout(G)\n", + " elif layout_type == 'spring':\n", + " node_pos_dict = nx.spring_layout(G, seed=layout_seed)\n", + " elif layout_type == 'random':\n", + " node_pos_dict = nx.random_layout(G, seed=layout_seed)\n", + " elif layout_type == 'spiral':\n", + " node_pos_dict = nx.spiral_layout(G)\n", + "\n", + " # Add the edges, along with their colors and widths\n", + " edge_colors = []\n", + " edge_widths = []\n", + " for i in range(N):\n", + " for j in range(N):\n", + " a = A[i, j]\n", + " if a > tol:\n", + " G.add_edge(codes[i], codes[j])\n", + " edge_colors.append(node_color_list[i])\n", + " width = a * edge_size_multiple\n", + " edge_widths.append(width)\n", + " \n", + " # Plot the networks\n", + " nx.draw_networkx_nodes(G, \n", + " node_pos_dict, \n", + " node_color=node_color_list, \n", + " node_size=node_sizes, \n", + " edgecolors='grey', \n", + " linewidths=2, \n", + " alpha=0.6, \n", + " ax=ax)\n", + "\n", + " nx.draw_networkx_labels(G, \n", + " node_pos_dict, \n", + " font_size=10, \n", + " ax=ax)\n", + "\n", + " nx.draw_networkx_edges(G, \n", + " node_pos_dict, \n", + " edge_color=edge_colors, \n", + " width=edge_widths, \n", + " arrows=True, \n", + " arrowsize=20, \n", + " alpha=0.6, \n", + " ax=ax, \n", + " arrowstyle='->', \n", + " node_size=node_sizes, \n", + " connectionstyle='arc3,rad=0.15')" + ] + }, + { + "cell_type": "markdown", + "id": "f879578e", + "metadata": {}, + "source": [ + "### plot_matrices" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "2e619a96", + "metadata": {}, + "outputs": [], + "source": [ + "def plot_matrices(matrix,\n", + " codes,\n", + " ax,\n", + " font_size=12,\n", + " alpha=0.6, \n", + " colormap=cm.viridis, \n", + " color45d=None, \n", + " xlabel='sector $j$', \n", + " ylabel='sector $i$'):\n", + " \n", + " ticks = range(len(matrix))\n", + "\n", + " levels = np.sqrt(np.linspace(0, 0.75, 100))\n", + " \n", + " \n", + " if color45d != None:\n", + " co = ax.contourf(ticks, \n", + " ticks,\n", + " matrix,\n", + "# levels,\n", + " alpha=alpha, cmap=colormap)\n", + " ax.plot(ticks, ticks, color=color45d)\n", + " else:\n", + " co = ax.contourf(ticks, \n", + " ticks,\n", + " matrix,\n", + " levels,\n", + " alpha=alpha, cmap=colormap)\n", + "\n", + " #plt.colorbar(co)\n", + "\n", + " ax.set_xlabel(xlabel, fontsize=font_size)\n", + " ax.set_ylabel(ylabel, fontsize=font_size)\n", + " ax.set_yticks(ticks)\n", + " ax.set_yticklabels(codes)\n", + " ax.set_xticks(ticks)\n", + " ax.set_xticklabels(codes)" + ] + }, + { + "cell_type": "markdown", + "id": "88631141", + "metadata": {}, + "source": [ + "### unit_simplex" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "2f4de6b0", + "metadata": {}, + "outputs": [], + "source": [ + "def unit_simplex(angle):\n", + " \n", + " fig = plt.figure(figsize=(10, 8))\n", + " ax = fig.add_subplot(111, projection='3d')\n", + "\n", + " vtx = [[0, 0, 1],\n", + " [0, 1, 0], \n", + " [1, 0, 0]]\n", + " \n", + " tri = Poly3DCollection([vtx], color='darkblue', alpha=0.3)\n", + " tri.set_facecolor([0.5, 0.5, 1])\n", + " ax.add_collection3d(tri)\n", + "\n", + " ax.set(xlim=(0, 1), ylim=(0, 1), zlim=(0, 1), \n", + " xticks=(1,), yticks=(1,), zticks=(1,))\n", + "\n", + " ax.set_xticklabels(['$(1, 0, 0)$'], fontsize=16)\n", + " ax.set_yticklabels([f'$(0, 1, 0)$'], fontsize=16)\n", + " ax.set_zticklabels([f'$(0, 0, 1)$'], fontsize=16)\n", + "\n", + " ax.xaxis.majorTicks[0].set_pad(15)\n", + " ax.yaxis.majorTicks[0].set_pad(15)\n", + " ax.zaxis.majorTicks[0].set_pad(35)\n", + "\n", + " ax.view_init(30, angle)\n", + "\n", + " # Move axis to origin\n", + " ax.xaxis._axinfo['juggled'] = (0, 0, 0)\n", + " ax.yaxis._axinfo['juggled'] = (1, 1, 1)\n", + " ax.zaxis._axinfo['juggled'] = (2, 2, 0)\n", + " \n", + " ax.grid(False)\n", + " \n", + " return ax" + ] + } + ], + "metadata": { + "jupytext": { + "cell_metadata_filter": "-all" + }, + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/frontpage/LICENSE.txt b/website/LICENSE.txt similarity index 100% rename from frontpage/LICENSE.txt rename to website/LICENSE.txt diff --git a/frontpage/README.txt b/website/README.txt similarity index 100% rename from frontpage/README.txt rename to website/README.txt diff --git a/frontpage/assets/css/fontawesome-all.min.css b/website/assets/css/fontawesome-all.min.css similarity index 100% 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Economic Networks

Theory and Computation

 -

Download the textbook here -
你可以点击 这里 下载电子版教材 +

Download the textbook here +
你可以点击 这里 下载电子版教材

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    • @@ -92,7 +92,7 @@

    Links

    • - Code Book + Notebooks
    • GitHub diff --git a/website/notebooks.html b/website/notebooks.html new file mode 100644 index 0000000..c70638d --- /dev/null +++ b/website/notebooks.html @@ -0,0 +1,85 @@ + + + + + + Notebooks + + + + + + + + + + + +
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      Design: HTML5 UP +
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      + + + + + + + + + + + + \ No newline at end of file

Thomas J. Sargent and John Stachurski