{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## CSCS530 Winter 2015\n", "#### Complex Systems 530 - Computer Modeling of Complex Systems (Winter 2015)\n", "\n", " * Course ID: CMPLXSYS 530\n", " * Course Title: Computer Modeling of Complex Systems\n", " * Term: Winter 2015\n", " * Schedule: Wednesdays and Friday, 1:00-2:30PM ET\n", " * Location: 120 West Hall (http://www.lsa.umich.edu/cscs/research/computerlab)\n", " * Teachers: [Mike Bommarito](https://www.linkedin.com/in/bommarito) and [Sarah Cherng](https://www.linkedin.com/pub/sarah-cherng/35/1b7/316)\n", "\n", "#### [View this repository on NBViewer](http://nbviewer.ipython.org/github/mjbommar/cscs-530-w2015/tree/master/)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Basic Networks\n", "\n", " In this notebook, we'll explore one of the most popular types of environments - a network. In the cells below, we'll be creating a small-world network and simulating a disease outbreak.\n", " \n", " ### Imports\n", "\n", " In the import section below, you'll see an important new addition: __networkx__. We'll be using __networkx__ as our preferred package for creating and managing network objects; we'll include links to more info and documentation in later cells." ] }, { "cell_type": "code", "execution_count": 98, "metadata": { "collapsed": false }, "outputs": [], "source": [ "%matplotlib inline\n", "\n", "# Imports\n", "import networkx as nx\n", "import numpy\n", "import matplotlib.pyplot as plt\n", "import pandas\n", "\n", "import seaborn; seaborn.set()\n", "\n", "# Import widget methods\n", "from IPython.html.widgets import *" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Initializing a network\n", "\n", " In some problems, we can define our own rules to _\"grow\"_ or _\"generate\"_ a network. However, for many problems, we may want to re-use an existing methodology. __networkx__ comes with a suite of methods to \"sample\" random graphs, such as:\n", " * Trees, e.g., balanced trees (networkx.balanced_tree)\n", " * Erdos-Renyi graphs (networkx.erdos_renyi_graph)\n", " * Watts-Strogatz small-world graphs (networkx.watts_strogatz_graph)\n", " * Bipartite graphs\n", " \n", " For a full list, see [this page on Graph Generators in the networkx documentation](https://networkx.github.io/documentation/latest/reference/generators.html).\n", " \n", " In the sample below, we'll create a Newman-Watts-Strogatz small-world graph and print some basic info about the graph." ] }, { "cell_type": "code", "execution_count": 99, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "(30, 38)\n" ] } ], "source": [ "# Create a random graph\n", "nodes = 30\n", "edges = 2\n", "prob_out = 0.2\n", "g = nx.newman_watts_strogatz_graph(nodes, edges, prob_out)\n", "print((g.number_of_nodes(), g.number_of_edges()))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Visualizing a network\n", "\n", " As humans, we are wired for visuals; for networks, visualizations can provide an important first characterization that isn't apparent from simple statistics.\n", " \n", " In the cell below, we show how to calculate a \"layout\" for a network and visualize it using networkx.\n", " \n", " For more examples, see [the Drawing Graphs section of the NetworkX tutorial](https://networkx.github.io/documentation/latest/tutorial/tutorial.html#drawing-graphs)." ] }, { "cell_type": "code", "execution_count": 100, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAewAAAFVCAYAAAAt79zdAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3XlYlNXbwPHvIDBIIlqaW5ato+WaIrgOCrIvbki4lKam\n", "QraZay7krpWZlb80TU0xt9xQRlQQTVMRzSWzKbP0TVHTAtPRYZl5/0AmUJCdYZj7c11el8w8yznM\n", 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"output_type": "display_data" } ], "source": [ "# Draw the random graph\n", "g_layout = nx.spring_layout(g, iterations=1000)\n", "nx.draw_networkx(g, pos=g_layout, node_color='#dddddd')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Outbreak in a network\n", "\n", " Next, we'll simulate a very simple outbreak in a network. Our disease will start by infecting a random patient, then pass with probability 1 to any connected individuals. \n", " \n", " To find the \"connected\" individuals, we'll need to learn how to find the neighbors of a node. NetworkX makes this easy." ] }, { "cell_type": "code", "execution_count": 101, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# Let's pick a node at random to infect\n", "patient_zero = numpy.random.choice(g.nodes())\n", "patient_zero\n", "\n", "healthy_nodes = g.nodes()\n", "healthy_nodes.remove(patient_zero)" ] }, { "cell_type": "code", "execution_count": 102, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAewAAAFVCAYAAAAt79zdAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xlck1f68P9PAkmURVBZBDcQIe5WrQKiRgVFwBWtVut0\n", "nbXt7ztd5tXFdpyx60yf6TPt+Ht8xs50OtPWrXUDl4AKglWrdnNatU2ttloFVFQWCSRA8vyBpGwK\n", "hCUkud6vV18zJPdyboL3lfuc61xHYbVaEUIIIUTXpnR0A4QQQgjRPAnYQgghhBOQgC2EEEI4AQnY\n", "QgghhBOQgC2EEEI4AQnYQgghhBPwbMvOWq02GviTwWCY1uD1pcBvgSrgK+Bhg8Eg88eEEEIIO9n9\n", "hK3Vap8C/gFoGrzeHXgRmGowGCYBfsDstjRSCCGEcHdt6RL/DkgFFA1erwBiDQZDxc2fPYHyNpxH\n", 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"cell_type": "markdown", "metadata": {}, "source": [ "## First model step\n", "\n", " In our simple disease outbreak model, each step of the model consists of the following:\n", "\n", " * for each infected individual, find their neighbors\n", " * infect all neighbors\n", " \n", " For our first step, we start with a single infected person; in subsequent steps, we need to use a _for_ loop to handle all currently infected persons." ] }, { "cell_type": "code", "execution_count": 103, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# Find patient zero's neighbors\n", "neighbors = g.neighbors(patient_zero)\n", "neighbors\n", "\n", "# Let's infect all of his neighbors!\n", "infected_patients = [patient_zero]\n", "infected_patients.extend(neighbors)\n", "infected_patients\n", "\n", "# Remove the infected from healthy nodes\n", "healthy_nodes = [node for node in healthy_nodes if node not in infected_patients]" ] }, { "cell_type": "code", "execution_count": 104, "metadata": { 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node_color='red')\n", "\n", "nx.draw_networkx_nodes(g, g_layout,\n", " nodelist=healthy_nodes,\n", " node_color='#dddddd')\n", "\n", "nx.draw_networkx_edges(g, g_layout, \n", " width=1.0, \n", " alpha=0.5,\n", " edge_color='#111111')\n", "\n", "_ = nx.draw_networkx_labels(g, g_layout,\n", " dict(zip(g.nodes(), g.nodes())),\n", " font_size=10)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Second model step\n", "\n", " As mentioned above, we now need to use a _for_ loop over all individuals." ] }, { "cell_type": "code", "execution_count": 105, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[25, 21]" ] }, "execution_count": 105, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Now let's infect the neighbors of all infected patients!\n", "newly_infected = []\n", "for infected_patient in infected_patients:\n", " # Find patient zero's neighbors\n", " neighbors = [neighbor for neighbor in g.neighbors(infected_patient) if neighbor not in infected_patients]\n", " newly_infected.extend(neighbors)\n", "\n", "newly_infected" ] }, { "cell_type": "code", "execution_count": 106, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# Update infected and healthy\n", "infected_patients.extend(newly_infected)\n", "\n", "# Remove the infected from healthy nodes\n", "healthy_nodes = [node for node in healthy_nodes if node not in infected_patients]" ] }, { "cell_type": "code", "execution_count": 107, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAewAAAFVCAYAAAAt79zdAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3XlYU3e+wP93AkmU3QUQ3EDEuFtXQNQoUBFwpe51rO3M\n", "nelUf/ObLk+n223HzrSd6e/2tjPe6zPObaszrW1trYobbiBQ99bWtto2dSsugIrKIoEESH5/RFI2\n", "lYQlJHxez8PTJjnn5HMInk/Od/l8FRaLBSGEEEK0b0pnByCEEEKIe5OELYQQQrgASdhCCCGEC5CE\n", "LYQQQrgASdhCCCGEC5CELYQQQrgAz+bsrNVqo4C/6PX6KfWeXwT8v0AV8C3wmF6vl/ljQgghhIMc\n", 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"nx.draw_networkx_nodes(g, g_layout,\n", " nodelist=healthy_nodes,\n", " node_color='#dddddd')\n", "\n", "nx.draw_networkx_edges(g, g_layout, \n", " width=1.0, \n", " alpha=0.5,\n", " edge_color='#111111')\n", "\n", "_ = nx.draw_networkx_labels(g, g_layout,\n", " dict(zip(g.nodes(), g.nodes())),\n", " font_size=10)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Running a few more steps\n", "\n", " Let's wrap our model step method in yet another _for_ loop, allowing us to simulate multiple steps of the model at once." ] }, { "cell_type": "code", "execution_count": 108, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAewAAAFVCAYAAAAt79zdAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xd8lFW++PFPJg3SIT2UJKQMHSFCGhCCSC8aiuIqChZ0\n", "3evWl/f+dO9aVt173fW6bnHVFcSluCogVQhKNQQColLloYeWQBJIIZM++f3xJGEyGZLMpExm5vt+\n", "vXiRzDzzzHkyyXznnPM93+NUW1uLEEIIIbo2jbUbIIQQQoiWScAWQgghbIAEbCGEEMIGSMAWQggh\n", 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"SMAWQgghAoAEbCGEECIA/H+R/BtfDtz/xAAAAABJRU5ErkJggg==\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Now let's infect the neighbors of all infected patients!\n", "model_steps = 5\n", "\n", "# Iterate over steps\n", "for i in range(model_steps):\n", " # Iterate over infected\n", " newly_infected = []\n", " for infected_patient in infected_patients:\n", " # Find patient neighbors and infect\n", " neighbors = [neighbor for neighbor in g.neighbors(infected_patient) if neighbor not in infected_patients]\n", " newly_infected.extend(neighbors)\n", " \n", " # Remove the infected from healthy nodes\n", " infected_patients.extend(newly_infected)\n", " healthy_nodes = [node for node in healthy_nodes if node not in infected_patients]\n", " \n", " \n", "\n", "\n", "\n", "# Now we can visualize the infected node's position\n", "f = plt.figure()\n", "nx.draw_networkx_nodes(g, g_layout,\n", " nodelist=infected_patients,\n", " node_color='red')\n", "\n", "nx.draw_networkx_nodes(g, g_layout,\n", " nodelist=healthy_nodes,\n", " node_color='#dddddd')\n", "\n", "nx.draw_networkx_edges(g, g_layout, \n", " width=1.0, \n", " alpha=0.5,\n", " edge_color='#111111')\n", "\n", "_ = nx.draw_networkx_labels(g, g_layout,\n", " dict(zip(g.nodes(), g.nodes())),\n", " font_size=10)" ] } ], "metadata": { "kernelspec": { "display_name": "IPython (Python 2)", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.3" } }, "nbformat": 4, "nbformat_minor": 0 }