6. Network Statistical Inference (Instructor).ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "# Load the data\n",
    "import pandas as pd\n",
    "import networkx as nx\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import numpy.random as npr\n",
    "from scipy.stats import norm, ks_2samp  # no scipy - comment out\n",
    "from custom import custom_funcs as cf\n",
    "\n",
    "%matplotlib inline\n",
    "%load_ext autoreload\n",
    "%autoreload 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Introduction\n",
    "\n",
    "In this notebook, we will walk through a hacker's approach to statistical thinking, as applied to network analysis."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Statistics in a Nutshell\n",
    "\n",
    "All of statistics can be broken down into two activities:\n",
    "\n",
    "- Descriptively summarizing data. (a.k.a. **\"descriptive statistics\"**)\n",
    "- Figuring out whether something happened by random chance. (a.k.a. **\"inferential statistics\"**)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Descriptive Statistics\n",
    "\n",
    "- Centrality measures: mean, median, mode\n",
    "- Variance measures: inter-quartile range (IQR), variance and standard deviation\n",
    "\n",
    "### Inferential Statistics\n",
    "\n",
    "- Models of Randomness (see below)\n",
    "- Hypothesis Testing\n",
    "- Fitting Statistical Models"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Load Data\n",
    "\n",
    "Let's load a [protein-protein interaction network dataset](http://konect.uni-koblenz.de/networks/moreno_propro).\n",
    "\n",
    "> This undirected network contains protein interactions contained in yeast. Research showed that proteins with a high degree were more important for the surivial of the yeast than others. A node represents a protein and an edge represents a metabolic interaction between two proteins. The network contains loops."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# Read in the data.\n",
    "# Note from above that we have to skip the first two rows, and that there's no header column,and that the edges are\n",
    "# delimited by spaces in between the nodes. Hence the syntax below:\n",
    "G = cf.load_propro_network()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Exercise\n",
    "\n",
    "Compute some basic descriptive statistics about the graph, namely:\n",
    "\n",
    "- the number of nodes,\n",
    "- the number of edges,\n",
    "- the graph density,\n",
    "- the distribution of degree centralities in the graph,\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1870"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Number of nodes:\n",
    "len(G.nodes())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2277"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Number of edges:\n",
    "len(G.edges())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0013029931073553016"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Graph density:\n",
    "nx.density(G)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "scrolled": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0.002675227394328518,\n",
       " 0.003210272873194221,\n",
       " 0.010700909577314071,\n",
       " 0.001070090957731407,\n",
       " 0.002675227394328518,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.002675227394328518,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.013376136971642588,\n",
       " 0.0037453183520599247,\n",
       " 0.0005350454788657035,\n",
       " 0.011235955056179775,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.002675227394328518,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.002140181915462814,\n",
       " 0.020331728196896735,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.004815409309791332,\n",
       " 0.002675227394328518,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.004815409309791332,\n",
       " 0.009630818619582664,\n",
       " 0.006420545746388442,\n",
       " 0.0005350454788657035,\n",
       " 0.002675227394328518,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.005885500267522739,\n",
       " 0.006420545746388442,\n",
       " 0.003210272873194221,\n",
       " 0.001070090957731407,\n",
       " 0.0037453183520599247,\n",
       " 0.003210272873194221,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.002675227394328518,\n",
       " 0.016051364365971106,\n",
       " 0.008025682182985553,\n",
       " 0.015516318887105402,\n",
       " 0.0005350454788657035,\n",
       " 0.0037453183520599247,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.005350454788657036,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.008025682182985553,\n",
       " 0.002675227394328518,\n",
       " 0.015516318887105402,\n",
       " 0.011235955056179775,\n",
       " 0.004815409309791332,\n",
       " 0.001070090957731407,\n",
       " 0.004815409309791332,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.002675227394328518,\n",
       " 0.0037453183520599247,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.005350454788657036,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.003210272873194221,\n",
       " 0.002140181915462814,\n",
       " 0.0037453183520599247,\n",
       " 0.002675227394328518,\n",
       " 0.002140181915462814,\n",
       " 0.002675227394328518,\n",
       " 0.002675227394328518,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.003210272873194221,\n",
       " 0.0016051364365971105,\n",
       " 0.002675227394328518,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.005885500267522739,\n",
       " 0.0037453183520599247,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.002140181915462814,\n",
       " 0.0016051364365971105,\n",
       " 0.014446227929373995,\n",
       " 0.003210272873194221,\n",
       " 0.001070090957731407,\n",
       " 0.002140181915462814,\n",
       " 0.0037453183520599247,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.002140181915462814,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.0037453183520599247,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.003210272873194221,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.006955591225254146,\n",
       " 0.001070090957731407,\n",
       " 0.003210272873194221,\n",
       " 0.0016051364365971105,\n",
       " 0.004280363830925628,\n",
       " 0.003210272873194221,\n",
       " 0.002675227394328518,\n",
       " 0.004815409309791332,\n",
       " 0.004815409309791332,\n",
       " 0.003210272873194221,\n",
       " 0.0037453183520599247,\n",
       " 0.0016051364365971105,\n",
       " 0.008560727661851257,\n",
       " 0.0005350454788657035,\n",
       " 0.029962546816479398,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0037453183520599247,\n",
       " 0.001070090957731407,\n",
       " 0.002140181915462814,\n",
       " 0.0037453183520599247,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.005350454788657036,\n",
       " 0.003210272873194221,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.0037453183520599247,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.002675227394328518,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.008025682182985553,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.003210272873194221,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.007490636704119849,\n",
       " 0.002675227394328518,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.003210272873194221,\n",
       " 0.0005350454788657035,\n",
       " 0.010165864098448368,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.005885500267522739,\n",
       " 0.0005350454788657035,\n",
       " 0.002675227394328518,\n",
       " 0.003210272873194221,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.003210272873194221,\n",
       " 0.002140181915462814,\n",
       " 0.0016051364365971105,\n",
       " 0.012306046013911182,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.002675227394328518,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.003210272873194221,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.0037453183520599247,\n",
       " 0.002140181915462814,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.006420545746388442,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.002675227394328518,\n",
       " 0.002140181915462814,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.002140181915462814,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.001070090957731407,\n",
       " 0.002140181915462814,\n",
       " 0.001070090957731407,\n",
       " 0.0037453183520599247,\n",
       " 0.006955591225254146,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.002675227394328518,\n",
       " 0.001070090957731407,\n",
       " 0.002675227394328518,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.004280363830925628,\n",
       " 0.002675227394328518,\n",
       " 0.003210272873194221,\n",
       " 0.006955591225254146,\n",
       " 0.005350454788657036,\n",
       " 0.005885500267522739,\n",
       " 0.002140181915462814,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.002675227394328518,\n",
       " 0.002140181915462814,\n",
       " 0.002675227394328518,\n",
       " 0.002675227394328518,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.003210272873194221,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.011235955056179775,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.003210272873194221,\n",
       " 0.0037453183520599247,\n",
       " 0.0037453183520599247,\n",
       " 0.003210272873194221,\n",
       " 0.003210272873194221,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.002140181915462814,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.002675227394328518,\n",
       " 0.002140181915462814,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.0005350454788657035,\n",
       " 0.002675227394328518,\n",
       " 0.001070090957731407,\n",
       " 0.002675227394328518,\n",
       " 0.002140181915462814,\n",
       " 0.006955591225254146,\n",
       " 0.0016051364365971105,\n",
       " 0.002675227394328518,\n",
       " 0.005885500267522739,\n",
       " 0.0016051364365971105,\n",
       " 0.002140181915462814,\n",
       " 0.0016051364365971105,\n",
       " 0.002675227394328518,\n",
       " 0.002140181915462814,\n",
       " 0.003210272873194221,\n",
       " 0.003210272873194221,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.005350454788657036,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0037453183520599247,\n",
       " 0.0005350454788657035,\n",
       " 0.003210272873194221,\n",
       " 0.0005350454788657035,\n",
       " 0.008560727661851257,\n",
       " 0.002140181915462814,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.005885500267522739,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.006955591225254146,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.002675227394328518,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.002675227394328518,\n",
       " 0.002675227394328518,\n",
       " 0.0037453183520599247,\n",
       " 0.006420545746388442,\n",
       " 0.002675227394328518,\n",
       " 0.002140181915462814,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0037453183520599247,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.005885500267522739,\n",
       " 0.002140181915462814,\n",
       " 0.003210272873194221,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.002675227394328518,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.003210272873194221,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.002140181915462814,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.005885500267522739,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.004815409309791332,\n",
       " 0.0005350454788657035,\n",
       " 0.009630818619582664,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.002675227394328518,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0037453183520599247,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.002140181915462814,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0037453183520599247,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.005885500267522739,\n",
       " 0.001070090957731407,\n",
       " 0.007490636704119849,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.004815409309791332,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.002675227394328518,\n",
       " 0.003210272873194221,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.003210272873194221,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.002140181915462814,\n",
       " 0.002675227394328518,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0037453183520599247,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.004280363830925628,\n",
       " 0.002675227394328518,\n",
       " 0.003210272873194221,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0037453183520599247,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.002675227394328518,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.002140181915462814,\n",
       " 0.002140181915462814,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.001070090957731407,\n",
       " 0.003210272873194221,\n",
       " 0.003210272873194221,\n",
       " 0.005885500267522739,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.002140181915462814,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.003210272873194221,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.005350454788657036,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.002140181915462814,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.001070090957731407,\n",
       " 0.003210272873194221,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.005885500267522739,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.002675227394328518,\n",
       " 0.0005350454788657035,\n",
       " 0.003210272873194221,\n",
       " 0.001070090957731407,\n",
       " 0.003210272873194221,\n",
       " 0.006420545746388442,\n",
       " 0.0005350454788657035,\n",
       " 0.002675227394328518,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.002140181915462814,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.003210272873194221,\n",
       " 0.002140181915462814,\n",
       " 0.003210272873194221,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.002675227394328518,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.004280363830925628,\n",
       " 0.0005350454788657035,\n",
       " 0.002675227394328518,\n",
       " 0.002140181915462814,\n",
       " 0.003210272873194221,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.003210272873194221,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.003210272873194221,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.002675227394328518,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.002140181915462814,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.004815409309791332,\n",
       " 0.002140181915462814,\n",
       " 0.002140181915462814,\n",
       " 0.001070090957731407,\n",
       " 0.002675227394328518,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.002675227394328518,\n",
       " 0.0005350454788657035,\n",
       " 0.002675227394328518,\n",
       " 0.001070090957731407,\n",
       " 0.004280363830925628,\n",
       " 0.004280363830925628,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.004280363830925628,\n",
       " 0.002140181915462814,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.002140181915462814,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.0016051364365971105,\n",
       " 0.002675227394328518,\n",
       " 0.002675227394328518,\n",
       " 0.002140181915462814,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.004280363830925628,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.002140181915462814,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.002675227394328518,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.004815409309791332,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.002675227394328518,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.0037453183520599247,\n",
       " 0.002675227394328518,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.0016051364365971105,\n",
       " 0.003210272873194221,\n",
       " 0.0005350454788657035,\n",
       " 0.001070090957731407,\n",
       " 0.001070090957731407,\n",
       " 0.002675227394328518,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.002140181915462814,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.004815409309791332,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0016051364365971105,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.0005350454788657035,\n",
       " 0.003210272873194221,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " 0.0016051364365971105,\n",
       " 0.003210272873194221,\n",
       " 0.002140181915462814,\n",
       " 0.0016051364365971105,\n",
       " 0.0016051364365971105,\n",
       " 0.001070090957731407,\n",
       " ...]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Degree centrality distribution:\n",
    "list(nx.degree_centrality(G).values())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "How are protein-protein networks formed? Are they formed by an [Erdos-Renyi](https://en.wikipedia.org/wiki/Erdős–Rényi_model) process, or something else?\n",
    "\n",
    ">In the G(n, p) model, a graph is constructed by connecting nodes randomly. Each edge is included in the graph with probability p independent from every other edge.\n",
    "\n",
    "If protein-protein networks are formed by an E-R process, then we would expect that properties of the protein-protein graph would look statistically similar to those of an actual E-R graph."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Exercise\n",
    "\n",
    "Make a histogram of the degree centralities for the protein-protein interaction graph, and the E-R graph."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYkAAAEACAYAAABGYoqtAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\ngg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1040f6390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ppG_deg_centralities = list(nx.degree_centrality(G).values())\n",
    "n, bins, patches = plt.hist(ppG_deg_centralities)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The construction of an E-R graph requires a value for `n` and `p`. \n",
    "\n",
    "A reasonable number for `n` is the number of nodes in our protein-protein graph.\n",
    "\n",
    "A reasonable value for `p` might be the [density](https://www.quora.com/What-is-graph-density) of the protein-protein graph."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYIAAAEACAYAAAC+gnFaAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n13is4QAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x112454978>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "erG = nx.erdos_renyi_graph(n=len(G.nodes()), p=nx.density(G))\n",
    "erG_deg_centralities = list(nx.degree_centrality(erG).values())\n",
    "n, bins, patches = plt.hist(erG_deg_centralities)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "From visualizing these two distributions, it is clear that they look very different. How do we quantify this difference, and statistically test whether the protein-protein graph could have arisen under an Erdos-Renyi model?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "One thing we might observe is that the variance, that is the \"spread\" around the mean, differs between the E-R model compared to our data. Therefore, we can compare variance of the data to the distribtion of variances under an E-R model.\n",
    "\n",
    "This is essentially following the logic of statistical inference by 'hacking' (not to be confused with the statistical bad practice of p-hacking)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Exercise\n",
    "\n",
    "Fill in the skeleton code below to simulate 100 E-R graphs."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "# 1. Generate 100 E-R graph degree centrality variance measurements and store them.\n",
    "# Takes ~50 seconds or so.\n",
    "n_sims = 100\n",
    "er_vars = np.zeros(n_sims)  # variances for n simulaed E-R graphs.\n",
    "for i in range(n_sims):\n",
    "    erG = nx.erdos_renyi_graph(n=len(G.nodes()), p=nx.density(G))\n",
    "    erG_deg_centralities = list(nx.degree_centrality(erG).values())\n",
    "    er_vars[i] = np.var(erG_deg_centralities)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "# 2. Compute the test statistic that is going to be used for the hypothesis test.\n",
    "# Hint: numpy has a \"var\" function implemented that computes the variance of a distribution of data.\n",
    "ppG_var = np.var(ppG_deg_centralities)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.LineCollection at 0x1007c5e48>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZMAAAEACAYAAAB27puMAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nRK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1007c5e80>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Do a quick visual check\n",
    "n, bins, patches = plt.hist(er_vars)\n",
    "plt.vlines(ppG_var, ymin=0, ymax=max(n), color='red', lw=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Visually, it should be quite evident that the protein-protein graph did not come from an E-R distribution. Statistically, we can also use the hypothesis test procedure to quantitatively test this, using our simulated E-R data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Conduct the hypothesis test.\n",
    "ppG_var > np.percentile(er_vars, 99)  # we can only use the 99th percentile, because there are only 100 data points."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Another way to do this is to use the 2-sample Kolmogorov-Smirnov test implemented in the `scipy.stats` module. From the docs:\n",
    "\n",
    "> This tests whether 2 samples are drawn from the same distribution. Note\n",
    "that, like in the case of the one-sample K-S test, the distribution is\n",
    "assumed to be continuous.\n",
    ">\n",
    "> This is the two-sided test, one-sided tests are not implemented.\n",
    "The test uses the two-sided asymptotic Kolmogorov-Smirnov distribution.\n",
    ">\n",
    "> If the K-S statistic is small or the p-value is high, then we cannot\n",
    "reject the hypothesis that the distributions of the two samples\n",
    "are the same.\n",
    "\n",
    "As an example to convince yourself that this test works, run the synthetic examples below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ks_2sampResult(statistic=0.090000000000000024, pvalue=0.79418377128477269)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Scenario 1: Data come from the same distributions.\n",
    "# Notice the size of the p-value.\n",
    "dist1 = npr.random(size=(100))\n",
    "dist2 = npr.random(size=(100))\n",
    "\n",
    "ks_2samp(dist1, dist2)\n",
    "# Note how the p-value, which ranges between 0 and 1, is likely to be greater than a commonly-accepted\n",
    "# threshold of 0.05"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ks_2sampResult(statistic=0.78000000000000003, pvalue=7.1742366934117429e-28)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Scenario 2: Data come from different distributions. \n",
    "# Note the size of the KS statistic, and the p-value.\n",
    "\n",
    "dist1 = norm(3, 1).rvs(100)\n",
    "dist2 = norm(5, 1).rvs(100)\n",
    "\n",
    "ks_2samp(dist1, dist2)\n",
    "# Note how the p-value is likely to be less than 0.05, and even more stringent cut-offs of 0.01 or 0.001."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Exercise\n",
    "\n",
    "Now, conduct the K-S test for one synthetic graph and the data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ks_2sampResult(statistic=0.22834224598930486, pvalue=4.1056079163841703e-43)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Now try it on the data distribution\n",
    "ks_2samp(erG_deg_centralities, ppG_deg_centralities)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Networks may be high-dimensional objects, but the logic for inference on network data essentially follows the same logic as for 'regular' data:\n",
    "\n",
    "- Identify a model of 'randomness' that may model how your data may have been generated.\n",
    "- Compute a \"test statistic\" for your data and the model.\n",
    "- Compute the probability of observing the data's test statistic under the model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Further Reading\n",
    "\n",
    "Jake Vanderplas' \"Statistics for Hackers\" slides: https://speakerdeck.com/jakevdp/statistics-for-hackers\n",
    "\n",
    "Allen Downey's \"There is Only One Test\": http://allendowney.blogspot.com/2011/05/there-is-only-one-test.html"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "celltoolbar": "Slideshow",
  "kernelspec": {
   "display_name": "Python [default]",
   "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": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}