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@Midnighter
Created 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"
]
},
{
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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"
]
},
{
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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"
]
},
{
"data": {
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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"
]
},
{
"data": {
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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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