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March 7, 2015 22:32
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{ | |
"cells": [ | |
{ | |
"cell_type": "markdown", | |
"metadata": {}, | |
"source": [ | |
"# Simple Power-Law Graphs " | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"metadata": {}, | |
"source": [ | |
"We desire simple undirected, connected graphs of particular size that follow a particular power-law degree distribution. There are multiple problems as shown below related to parallel edges introduced by the configuration model and the size of the largest connected component." | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 1, | |
"metadata": { | |
"collapsed": false | |
}, | |
"outputs": [], | |
"source": [ | |
"import networkx as nx\n", | |
"import numpy as np\n", | |
"import matplotlib.pyplot as plt\n", | |
"from networkx.utils import (powerlaw_sequence, create_degree_sequence)" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 2, | |
"metadata": { | |
"collapsed": true | |
}, | |
"outputs": [], | |
"source": [ | |
"%matplotlib inline" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 3, | |
"metadata": { | |
"collapsed": false | |
}, | |
"outputs": [ | |
{ | |
"data": { | |
"text/plain": [ | |
"'1.9.1'" | |
] | |
}, | |
"execution_count": 3, | |
"metadata": {}, | |
"output_type": "execute_result" | |
} | |
], | |
"source": [ | |
"nx.__version__" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 4, | |
"metadata": { | |
"collapsed": true | |
}, | |
"outputs": [], | |
"source": [ | |
"exponents = [2.3, 2.5, 2.7, 3.0]" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 5, | |
"metadata": { | |
"collapsed": true | |
}, | |
"outputs": [], | |
"source": [ | |
"num_nodes = [2000, 3000, 4000, 5000]" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 6, | |
"metadata": { | |
"collapsed": true | |
}, | |
"outputs": [], | |
"source": [ | |
"seed = 123456789" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": 7, | |
"metadata": { | |
"collapsed": false, | |
"scrolled": true | |
}, | |
"outputs": [ | |
{ | |
"name": "stdout", | |
"output_type": "stream", | |
"text": [ | |
"number of nodes = 2000\n", | |
"--------------------\n", | |
"exponent = -2.30\n", | |
"graph has 3380 parallel edges\n", | |
"graph has 29 self-loops\n", | |
"size of largest connected component = 1856\n" | |
] | |
}, | |
{ | |
"data": { | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7fbf2e9b58d0>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
}, | |
{ | |
"name": "stdout", | |
"output_type": "stream", | |
"text": [ | |
"exponent = -2.50\n", | |
"graph has 2810 parallel edges\n", | |
"graph has 17 self-loops\n", | |
"size of largest connected component = 1787\n" | |
] | |
}, | |
{ | |
"data": { | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7fbf30b2d080>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
}, | |
{ | |
"name": "stdout", | |
"output_type": "stream", | |
"text": [ | |
"exponent = -2.70\n", | |
"graph has 2337 parallel edges\n", | |
"graph has 3 self-loops\n", | |
"size of largest connected component = 1644\n" | |
] | |
}, | |
{ | |
"data": { | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7fbf2e707f60>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
}, | |
{ | |
"name": "stdout", | |
"output_type": "stream", | |
"text": [ | |
"exponent = -3.00\n", | |
"graph has 1989 parallel edges\n", | |
"graph has 6 self-loops\n", | |
"size of largest connected component = 1392\n" | |
] | |
}, | |
{ | |
"data": { | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7fbf2e59eb38>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
}, | |
{ | |
"name": "stdout", | |
"output_type": "stream", | |
"text": [ | |
"number of nodes = 3000\n", | |
"--------------------\n", | |
"exponent = -2.30\n", | |
"graph has 5786 parallel edges\n", | |
"graph has 46 self-loops\n", | |
"size of largest connected component = 2846\n" | |
] | |
}, | |
{ | |
"data": { | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7fbf2e90eeb8>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
}, | |
{ | |
"name": "stdout", | |
"output_type": "stream", | |
"text": [ | |
"exponent = -2.50\n", | |
"graph has 3931 parallel edges\n", | |
"graph has 5 self-loops\n", | |
"size of largest connected component = 2636\n" | |
] | |
}, | |
{ | |
"data": { | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7fbf2e7768d0>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
}, | |
{ | |
"name": "stdout", | |
"output_type": "stream", | |
"text": [ | |
"exponent = -2.70\n", | |
"graph has 3454 parallel edges\n", | |
"graph has 2 self-loops\n", | |
"size of largest connected component = 2462\n" | |
] | |
}, | |
{ | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7fbf2e7698d0>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
}, | |
{ | |
"name": "stdout", | |
"output_type": "stream", | |
"text": [ | |
"exponent = -3.00\n", | |
"graph has 2824 parallel edges\n", | |
"graph has 3 self-loops\n", | |
"size of largest connected component = 1986\n" | |
] | |
}, | |
{ | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7fbf2e6e5b38>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
}, | |
{ | |
"name": "stdout", | |
"output_type": "stream", | |
"text": [ | |
"number of nodes = 4000\n", | |
"--------------------\n", | |
"exponent = -2.30\n", | |
"graph has 7835 parallel edges\n", | |
"graph has 55 self-loops\n", | |
"size of largest connected component = 3761\n" | |
] | |
}, | |
{ | |
"data": { | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7fbf2e6bc390>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
}, | |
{ | |
"name": "stdout", | |
"output_type": "stream", | |
"text": [ | |
"exponent = -2.50\n", | |
"graph has 5669 parallel edges\n", | |
"graph has 14 self-loops\n", | |
"size of largest connected component = 3577\n" | |
] | |
}, | |
{ | |
"data": { | |
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"YII=\n" | |
], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7fbf2e54b668>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
}, | |
{ | |
"name": "stdout", | |
"output_type": "stream", | |
"text": [ | |
"exponent = -2.70\n", | |
"graph has 4776 parallel edges\n", | |
"graph has 14 self-loops\n", | |
"size of largest connected component = 3329\n" | |
] | |
}, | |
{ | |
"data": { | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7fbf2e2fb470>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
}, | |
{ | |
"name": "stdout", | |
"output_type": "stream", | |
"text": [ | |
"exponent = -3.00\n", | |
"graph has 3797 parallel edges\n", | |
"graph has 1 self-loops\n", | |
"size of largest connected component = 2658\n" | |
] | |
}, | |
{ | |
"data": { | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7fbf2e9b5a20>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
}, | |
{ | |
"name": "stdout", | |
"output_type": "stream", | |
"text": [ | |
"number of nodes = 5000\n", | |
"--------------------\n", | |
"exponent = -2.30\n", | |
"graph has 10376 parallel edges\n", | |
"graph has 52 self-loops\n", | |
"size of largest connected component = 4773\n" | |
] | |
}, | |
{ | |
"data": { | |
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"RK5CYII=\n" | |
], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7fbf2e14fc18>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
}, | |
{ | |
"name": "stdout", | |
"output_type": "stream", | |
"text": [ | |
"exponent = -2.50\n", | |
"graph has 7247 parallel edges\n", | |
"graph has 19 self-loops\n", | |
"size of largest connected component = 4428\n" | |
] | |
}, | |
{ | |
"data": { | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7fbf2e455518>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
}, | |
{ | |
"name": "stdout", | |
"output_type": "stream", | |
"text": [ | |
"exponent = -2.70\n", | |
"graph has 6166 parallel edges\n", | |
"graph has 11 self-loops\n", | |
"size of largest connected component = 4240\n" | |
] | |
}, | |
{ | |
"data": { | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7fbf2e980208>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
}, | |
{ | |
"name": "stdout", | |
"output_type": "stream", | |
"text": [ | |
"exponent = -3.00\n", | |
"graph has 4693 parallel edges\n", | |
"graph has 1 self-loops\n", | |
"size of largest connected component = 3263\n" | |
] | |
}, | |
{ | |
"data": { | |
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], | |
"text/plain": [ | |
"<matplotlib.figure.Figure at 0x7fbf2e718ba8>" | |
] | |
}, | |
"metadata": {}, | |
"output_type": "display_data" | |
} | |
], | |
"source": [ | |
"for num in num_nodes:\n", | |
" print(\"number of nodes = {0:d}\".format(num))\n", | |
" print(\"\".join([\"-\"]* 20))\n", | |
" for exp in exponents:\n", | |
" print(\"exponent = -{0:.2f}\".format(exp))\n", | |
" sequence = create_degree_sequence(num, powerlaw_sequence, exponent=exp)\n", | |
" graph = nx.configuration_model(sequence, seed=seed)\n", | |
" # count parallel edges and avoid counting A-B as well as B-A\n", | |
" num_par = sum(len(graph[node][neigh]) for node in graph for neigh in graph.neighbors_iter(node)) // 2\n", | |
" print(\"graph has {0:d} parallel edges\".format(num_par))\n", | |
" loops = graph.selfloop_edges()\n", | |
" print(\"graph has {0:d} self-loops\".format(len(loops)))\n", | |
" # remove them\n", | |
" graph = nx.Graph(graph)\n", | |
" graph.remove_edges_from(loops)\n", | |
" # get largest connected component\n", | |
" # unfortunately, the iterator over the components is not guaranteed to be sorted by size\n", | |
" components = sorted(nx.connected_components(graph), key=len, reverse=True)\n", | |
" lcc = graph.subgraph(components[0])\n", | |
" print(\"size of largest connected component = {0:d}\".format(len(lcc)))\n", | |
" # new degree sequence\n", | |
" simple_seq = [deg for (node, deg) in lcc.degree_iter()]\n", | |
" # create histograms\n", | |
" counts = np.bincount(sequence)\n", | |
" mask = (counts > 0)\n", | |
" plt.plot(np.arange(len(counts))[mask], counts[mask] / counts.sum(), \"o\", label=\"MultiGraph\")\n", | |
" simple_counts = np.bincount(simple_seq)\n", | |
" mask = (simple_counts > 0)\n", | |
" # distribution is shifted for visibility\n", | |
" plt.plot(np.arange(len(simple_counts))[mask], simple_counts[mask] / simple_counts.sum() / 10.0,\n", | |
" \"o\", label=\"Simple LCC\")\n", | |
" x = np.arange(1, len(counts))\n", | |
" plt.plot(x, np.power(x, -exp))\n", | |
" plt.xlabel(r\"Degree $k$\")\n", | |
" plt.xscale(\"log\")\n", | |
" plt.ylabel(r\"Probability $P(k)$\")\n", | |
" plt.yscale(\"log\")\n", | |
" plt.title(r\"$N = {0:d}, \\quad \\lambda = {1:.2f}$\".format(num, exp))\n", | |
" plt.legend(loc=\"best\")\n", | |
" plt.show()" | |
] | |
} | |
], | |
"metadata": { | |
"kernelspec": { | |
"display_name": "Python 3", | |
"language": "python", | |
"name": "python3" | |
}, | |
"language_info": { | |
"codemirror_mode": { | |
"name": "ipython", | |
"version": 3 | |
}, | |
"file_extension": ".py", | |
"mimetype": "text/x-python", | |
"name": "python", | |
"nbconvert_exporter": "python", | |
"pygments_lexer": "ipython3", | |
"version": "3.4.2" | |
} | |
}, | |
"nbformat": 4, | |
"nbformat_minor": 0 | |
} |
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