class07b_pagerank_python3.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# CSX46 - Class Session 10 - Pagerank centrality\n",
    "\n",
    "In this class session we are going to compute the outgoing-edge PageRank centrality of each gene (vertex) in a human gene regulatory network (a directed graph) from a landmark paper on human gene regulation (Neph et al., Cell, volume 150, pages 1274-1286, 2012; see PDF on Canvas in `Files/Optional Reading`). We will also write our own PageRank function and compare it to the `igraph.Graph.pagerank` method."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using Pandas `read_csv`, read in the ifle `shared/neph_gene_network.txt`, which has two columns of text (first column is the regulator gene, second column is the target gene), into a data frame. The file has no header and is tab-delimited.  Assign the column names of the dataframe to be `regulator` and `target`, respectively."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's load the Python packages that we will need for this exercise"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas\n",
    "import igraph\n",
    "import numpy\n",
    "import matplotlib.pyplot\n",
    "import random"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using `pandas.read_csv`, read the file `shared/neph_gene_network.txt`; name the two columns of the resulting data frame, `regulator` and `target`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "edge_list_neph = pandas.read_csv(\"shared/neph_gene_network.txt\",\n",
    "                                 sep=\"\\t\", \n",
    "                                 names=[\"regulator\",\"target\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Load the edge-list data into a directed `igraph.Graph` object `neph_graph`, using `igraph.Graph.TupleList`. NOTE: igraph's `igraph.Graph.pagerank` computes only *incoming* pagerank centrality, and we want *outgoing* pagerank centrality. So: *make sure to reverse the columns of the data frame when you input the data frame into `Graph.TupleList` since we want the outgoing pagerank centrality not incoming pagerank centrality.* Print out a summary of the graph, using the `igraph.Graph.summary` method:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'IGRAPH DN-- 538 47945 -- \\n+ attr: name (v)'"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "neph_graph = igraph.Graph.TupleList(edge_list_neph[[\"target\",\"regulator\"]].values.tolist(), directed=True)\n",
    "neph_graph.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the pagerank centrality measures of all vertices, using `igraph.Graph.pagerank`.  Use the resulting object to initialize a `numpy.array`, `pageranks`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "pageranks = numpy.array(neph_graph.pagerank())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Which vertex has highest pagerank centrality in the gene regulatory network, and what is its pagerank centrality value? (think `numpy.max` and `numpy.argmax`). Get a `VertexSet` sequence using the `igraph.Graph.vs` property, and then index into that sequence using Python indexing:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.00749112688039444\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "igraph.Vertex(<igraph.Graph object at 0x7f4349d3ec78>, 122, {'name': 'FOS'})"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(numpy.max(pageranks))\n",
    "neph_graph.vs[numpy.argmax(pageranks)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculate the in-degree of all vertices in the graph, and scatter plot `log(degree)` vs. `log(pagerank)`.  (Do you see why we used `in` here?  Note the column swapping we did earlier). Note-- you will have to eliminate one vertex that has zero in-degree."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot\n",
    "ax = matplotlib.pyplot.gca()\n",
    "ax.set_xscale(\"log\")\n",
    "ax.set_yscale(\"log\")\n",
    "degrees = numpy.array(neph_graph.indegree())\n",
    "inds_keep = numpy.where(degrees > 0)\n",
    "ax.scatter(degrees[inds_keep], \n",
    "           pageranks[inds_keep])\n",
    "#ax.scatter(neph_graph.indegree(), pageranks)\n",
    "matplotlib.pyplot.xlabel(\"degree\")\n",
    "matplotlib.pyplot.ylabel(\"pagerank\")\n",
    "matplotlib.pyplot.ylim(3*1e-4, 1e-2)\n",
    "matplotlib.pyplot.xlim(0.8, 600)\n",
    "matplotlib.pyplot.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "See if you can calculate the pagerank centrality yourself, using the matrix inversion (Eq. 7.19 from Newman). Test your function on a small directed graph."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def pagerank(g):\n",
    "    # N is the number of vertices\n",
    "    N = len(g.vs)\n",
    "    \n",
    "    # alpha is the damping parameter\n",
    "    alpha = 0.85\n",
    "    \n",
    "    # beta = (1-alpha)/N\n",
    "    beta = (1-alpha)/N\n",
    "    \n",
    "    # compute the out-degree of each vertex\n",
    "    degree_values = g.degree(mode=\"out\")\n",
    "    \n",
    "    # get a floating-point adjacency matrix M in the Newman format (take transpose from igraph format)\n",
    "    M = numpy.matrix(g.get_adjacency().data).transpose().astype(float)\n",
    "    \n",
    "    # or each column in 0,N-1:\n",
    "    for j in range(0,N):\n",
    "        \n",
    "        # get the out degree of the vertex as a float\n",
    "        degree_value = float(degree_values[j])\n",
    "        \n",
    "        # if degree is nonzero, normalize the column of M\n",
    "        if degree_value > 0:\n",
    "            M[:,j] /= degree_value\n",
    "        else:\n",
    "            # set the column to zero\n",
    "            M[:,j] = 0\n",
    "            \n",
    "    # compute pagerank following Newman Eq. 7.19, where M = AD^(-1)\n",
    "    pr = numpy.linalg.inv(numpy.diag([1.]*N) - alpha * M) * numpy.matrix([beta]*N).transpose()\n",
    "    \n",
    "    # normalize pagerank centrality by its sum\n",
    "    pr /= numpy.sum(pr)\n",
    "    \n",
    "    retlist = pr.transpose().tolist()[0]\n",
    "    return(retlist)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Test out your function on a small 5-vertex Barabasi-Albert graph:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.2770649250864387, 0.21240875912408758, 0.21240875912408758, 0.14905877833269304, 0.14905877833269304]\n",
      "[0.2770649250864387, 0.21240875912408758, 0.21240875912408758, 0.14905877833269301, 0.14905877833269304]\n"
     ]
    }
   ],
   "source": [
    "g = igraph.Graph.Barabasi(n=5, m=2) \n",
    "import cairo\n",
    "igraph.drawing.plot(g)\n",
    "print(pagerank(g))\n",
    "print(g.pagerank())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
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