testing over the algorithm 1 with different taus
Luca Lombardo
2 years ago
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6651a16f28
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990d59e3ab
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import networkx as nx\n",
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"import time\n",
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"import math\n",
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"import pandas as pd\n",
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"import scipy as sp\n",
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"import plotly.express as px\n",
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"import plotly.graph_objs as go\n",
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"from scipy.sparse import *\n",
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"from scipy.sparse.linalg import norm"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Let's create two graphs from the list of edges downloaded from the Snap database. "
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [],
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"source": [
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"G1 = nx.read_edgelist('../data/web-Stanford.txt', create_using=nx.DiGraph(), nodetype=int)\n",
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"\n",
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"# G2 = nx.read_edgelist('../data/web-BerkStan.txt', create_using=nx.DiGraph(), nodetype=int)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Creating the transition probability matrix"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {},
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"outputs": [],
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"source": [
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"# square matrix of size n x n, where n is the number of nodes in the graph. The matrix is filled with zeros and the (i,j) element is x if the node i is connected to the node j. Where x is 1/(number of nodes connected to i).\n",
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"\n",
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"def create_matrix(G):\n",
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" n = G.number_of_nodes()\n",
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" P = sp.sparse.lil_matrix((n,n))\n",
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" for i in G.nodes():\n",
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" for j in G[i]: #G[i] is the list of nodes connected to i, it's neighbors\n",
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" P[i-1,j-1] = 1/len(G[i])\n",
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" return P"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"To ensure that the random process has a unique stationary distribution and it will not stagnate, the transition matrix P is usually modified to be an irreducible stochastic matrix A (called the Google matrix) as follows\n",
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"\n",
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"$$ A = \\alpha \\tilde{P} + (1-\\alpha)v e^T$$\n",
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"\n",
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"Where $\\tilde{P}$ is defined as \n",
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"\n",
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"$$ \\tilde{P} = P + v d^T$$\n",
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"\n",
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"Where $d \\in \\mathbb{N}^{n \\times 1}$ s a binary vector tracing the indices of dangling web-pages with no hyperlinks, i.e., $d(i ) = 1$ if the `ith` page has no hyperlink, $v \\in \\mathbb{R}^{n \\times 1}$ is a probability vector, $e = [1, 1, . . . , 1]^T$ , and $0 < \\alpha < 1$ is the so-called damping factor that represents the probability in the model that the surfer transfer by clicking a hyperlink rather than other ways"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {},
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"outputs": [],
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"source": [
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"n = G1.number_of_nodes()\n",
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"P = create_matrix(G1) "
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"the vector `d` solves the dangling nodes problem"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"metadata": {},
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"outputs": [],
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"source": [
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"# define d as a nx1 sparse matrix, where n is the number of nodes in the graph. The vector is filled with d(i) = 1 if the i row of the matrix P is filled with zeros, other wise is 0\n",
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"\n",
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"# d is the vector of dangling nodes\n",
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"d = sp.sparse.lil_matrix((n,1))\n",
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"for i in range(n):\n",
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" if P[i].sum() == 0:\n",
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" d[i] = 1"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The vector v is a probability vector, the sum of its elements bust be one"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"metadata": {},
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"outputs": [],
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"source": [
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"# define v as the probability vector of size n x 1, where n is the number of nodes in the graph. The vector is filled with 1/n\n",
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"# https://en.wikipedia.org/wiki/Probability_vector\n",
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"\n",
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"v = sp.sparse.lil_matrix((n,1))\n",
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"for i in range(n):\n",
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" v[i] = 1/n "
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Now we can compute the transition matrix\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"metadata": {},
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"outputs": [],
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"source": [
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"Pt = P + v.dot(d.T)\n",
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"\n",
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"# Pt is a sparse matrix too"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 13,
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"metadata": {},
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"outputs": [],
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"source": [
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"# e is a nx1 sparse matrix filled with ones\n",
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"e = sp.sparse.lil_matrix((1,n))\n",
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"for i in range(n):\n",
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" e[0,i] = 1"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 14,
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"metadata": {},
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"outputs": [],
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"source": [
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"# # v*eT is a nxn sparse matrix filled all with 1/n, let's call it B\n",
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"\n",
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"# B = sp.sparse.lil_matrix((n,n))\n",
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"# for i in range(n):\n",
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"# for j in range(n):\n",
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"# B[i,j] = 1/n\n",
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"\n",
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"# A = alpha*Pt + (1-alpha)*B"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Algorithm 1 Shifted-Power method for PageRank with multiple damping factors:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 15,
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"metadata": {},
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"outputs": [],
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"source": [
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"# pandas dataframe to store the results\n",
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"df = pd.DataFrame(columns=['alpha', 'iterations', 'tau', 'time'])"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 16,
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"metadata": {},
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"outputs": [],
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"source": [
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"# this should return mv (the number of iteration needed for the convergence), and two vector called x and r. Where x is the vector of the pagerank and r is the residual vector\n",
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"\n",
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"def Algorithm1(Pt, v, tau, max_mv, a: list):\n",
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" \n",
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" start_time = time.time()\n",
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"\n",
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" u = Pt.dot(v) - v \n",
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" mv = 1 # number of iteration\n",
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" r = sp.sparse.lil_matrix((n,1)) \n",
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" Res = sp.sparse.lil_matrix((len(a),1))\n",
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" x = sp.sparse.lil_matrix((n,1)) \n",
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"\n",
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" for i in range(len(a)):\n",
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" r = a[i]*(u) \n",
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" normed_r = norm(r)\n",
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" Res[i] = normed_r \n",
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"\n",
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" if Res[i] > tau:\n",
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" x = r + v \n",
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"\n",
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" while max(Res) > tau and mv < max_mv:\n",
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" u = Pt*u # should it be the same u of the beginning?\n",
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" mv += 1 \n",
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"\n",
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" for i in range(len(a)):\n",
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" if Res[i] >= tau: \n",
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" r = (a[i]**(mv+1))*(u)\n",
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" Res[i] = norm(r)\n",
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"\n",
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" if Res[i] > tau:\n",
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" x = r + x\n",
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"\n",
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" if mv == max_mv:\n",
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" print(\"The algorithm didn't converge in \", max_mv, \" iterations\")\n",
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" else:\n",
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" print(\"The algorithm converged in \", mv, \" iterations\")\n",
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"\n",
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" total_time = time.time() - start_time\n",
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" print(\"The algorithm took \", total_time, \" seconds\")\n",
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" \n",
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" return mv, x, r, total_time "
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]
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},
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{
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"cell_type": "code",
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"execution_count": 17,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"\n",
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"tau = 1e-05\n",
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"The algorithm converged in 19 iterations\n",
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"The algorithm took 12.630054473876953 seconds\n",
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"\n",
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"tau = 1e-06\n"
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]
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},
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{
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"name": "stderr",
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"output_type": "stream",
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"text": [
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"/tmp/ipykernel_1733107/908708813.py:13: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.\n",
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" df = df.append({'alpha': a, 'iterations': mv, 'tau': tau, 'time': total_time}, ignore_index=True)\n"
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]
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},
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"The algorithm converged in 56 iterations\n",
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"The algorithm took 32.28381681442261 seconds\n",
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"\n",
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"tau = 1e-07\n"
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]
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},
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{
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"name": "stderr",
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"output_type": "stream",
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"text": [
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"/tmp/ipykernel_1733107/908708813.py:13: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.\n",
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" df = df.append({'alpha': a, 'iterations': mv, 'tau': tau, 'time': total_time}, ignore_index=True)\n"
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]
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},
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"The algorithm converged in 113 iterations\n",
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"The algorithm took 75.9386818408966 seconds\n",
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"\n",
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"tau = 1e-08\n"
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]
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},
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{
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"name": "stderr",
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"output_type": "stream",
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"text": [
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"/tmp/ipykernel_1733107/908708813.py:13: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.\n",
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" df = df.append({'alpha': a, 'iterations': mv, 'tau': tau, 'time': total_time}, ignore_index=True)\n"
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]
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},
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"The algorithm converged in 275 iterations\n",
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"The algorithm took 107.68565726280212 seconds\n",
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"\n",
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"tau = 1e-09\n"
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]
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},
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{
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"name": "stderr",
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"output_type": "stream",
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"text": [
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"/tmp/ipykernel_1733107/908708813.py:13: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.\n",
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" df = df.append({'alpha': a, 'iterations': mv, 'tau': tau, 'time': total_time}, ignore_index=True)\n"
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]
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},
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"The algorithm converged in 454 iterations\n",
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"The algorithm took 152.8943109512329 seconds\n"
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]
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},
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{
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"name": "stderr",
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"output_type": "stream",
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"text": [
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"/tmp/ipykernel_1733107/908708813.py:13: FutureWarning: The frame.append method is deprecated and will be removed from pandas in a future version. Use pandas.concat instead.\n",
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" df = df.append({'alpha': a, 'iterations': mv, 'tau': tau, 'time': total_time}, ignore_index=True)\n"
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]
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},
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{
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"ename": "OSError",
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"evalue": "Cannot save file into a non-existent directory: '../data/algo1'",
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"output_type": "error",
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"traceback": [
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"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
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"\u001b[0;31mOSError\u001b[0m Traceback (most recent call last)",
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"\u001b[0;32m/tmp/ipykernel_1733107/908708813.py\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m 14\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 15\u001b[0m \u001b[0;31m# save the results in a csv file\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 16\u001b[0;31m \u001b[0mdf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mto_csv\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'../data/algo1/different_tau.csv'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mindex\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mFalse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 17\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
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"\u001b[0;32m~/.local/lib/python3.10/site-packages/pandas/core/generic.py\u001b[0m in \u001b[0;36mto_csv\u001b[0;34m(self, path_or_buf, sep, na_rep, float_format, columns, header, index, index_label, mode, encoding, compression, quoting, quotechar, line_terminator, chunksize, date_format, doublequote, escapechar, decimal, errors, storage_options)\u001b[0m\n\u001b[1;32m 3549\u001b[0m )\n\u001b[1;32m 3550\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3551\u001b[0;31m return DataFrameRenderer(formatter).to_csv(\n\u001b[0m\u001b[1;32m 3552\u001b[0m \u001b[0mpath_or_buf\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 3553\u001b[0m \u001b[0mline_terminator\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mline_terminator\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
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"\u001b[0;32m~/.local/lib/python3.10/site-packages/pandas/io/formats/format.py\u001b[0m in \u001b[0;36mto_csv\u001b[0;34m(self, path_or_buf, encoding, sep, columns, index_label, mode, compression, quoting, quotechar, line_terminator, chunksize, date_format, doublequote, escapechar, errors, storage_options)\u001b[0m\n\u001b[1;32m 1178\u001b[0m \u001b[0mformatter\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfmt\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 1179\u001b[0m )\n\u001b[0;32m-> 1180\u001b[0;31m \u001b[0mcsv_formatter\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msave\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 1181\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 1182\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mcreated_buffer\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
|
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"\u001b[0;32m~/.local/lib/python3.10/site-packages/pandas/io/formats/csvs.py\u001b[0m in \u001b[0;36msave\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m 239\u001b[0m \"\"\"\n\u001b[1;32m 240\u001b[0m \u001b[0;31m# apply compression and byte/text conversion\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 241\u001b[0;31m with get_handle(\n\u001b[0m\u001b[1;32m 242\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfilepath_or_buffer\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 243\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmode\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
|
||||
"\u001b[0;32m~/.local/lib/python3.10/site-packages/pandas/io/common.py\u001b[0m in \u001b[0;36mget_handle\u001b[0;34m(path_or_buf, mode, encoding, compression, memory_map, is_text, errors, storage_options)\u001b[0m\n\u001b[1;32m 692\u001b[0m \u001b[0;31m# Only for write methods\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 693\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;34m\"r\"\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mmode\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mis_path\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 694\u001b[0;31m \u001b[0mcheck_parent_directory\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mhandle\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 695\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 696\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mcompression\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
|
||||
"\u001b[0;32m~/.local/lib/python3.10/site-packages/pandas/io/common.py\u001b[0m in \u001b[0;36mcheck_parent_directory\u001b[0;34m(path)\u001b[0m\n\u001b[1;32m 566\u001b[0m \u001b[0mparent\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mPath\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mpath\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mparent\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 567\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mparent\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_dir\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 568\u001b[0;31m \u001b[0;32mraise\u001b[0m \u001b[0mOSError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34mrf\"Cannot save file into a non-existent directory: '{parent}'\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 569\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 570\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
|
||||
"\u001b[0;31mOSError\u001b[0m: Cannot save file into a non-existent directory: '../data/algo1'"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"# list of alpha values, from 0.85 to 0.99 with step 0.01\n",
|
||||
"a = []\n",
|
||||
"for i in range(85,100):\n",
|
||||
" a.append(i/100)\n",
|
||||
"\n",
|
||||
"max_mv = 1000\n",
|
||||
"\n",
|
||||
"# run the algorithm for different values of tau from 10^-5 to 10^-9 with step 10^-1\n",
|
||||
"for i in range(5,10):\n",
|
||||
" tau = 10**(-i)\n",
|
||||
" print(\"\\ntau = \", tau)\n",
|
||||
" mv, x, r, total_time = Algorithm1(Pt, v, tau, max_mv, a)\n",
|
||||
" df = df.append({'alpha': a, 'iterations': mv, 'tau': tau, 'time': total_time}, ignore_index=True) \n",
|
||||
"\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 18,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"# save the results in a csv file\n",
|
||||
"df.to_csv('../data/results/algo1/different_tau.csv', index=False)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### Plotting the results of the algorithm for different values of tau, and fixed alpha"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 21,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"x = df['tau'][::-1].tolist()\n",
|
||||
"y = df['iterations'].tolist()\n",
|
||||
"\n",
|
||||
"fig1 = go.Figure(data=go.Scatter(x=x, y=y, mode='lines+markers'), \n",
|
||||
" layout=go.Layout(title='Iterations needed for the convergence', xaxis_title='tau', yaxis_title='iterations'))\n",
|
||||
" \n",
|
||||
"# save the plot in a html file\n",
|
||||
"fig1.write_html(\"../data/results/algo1/taus_over_iterations.html\")\n",
|
||||
"\n",
|
||||
"##### RESULTS OVER TIME #####\n",
|
||||
"\n",
|
||||
"x1 = df['tau'][::-1].tolist()\n",
|
||||
"y1 = df['time'].tolist()\n",
|
||||
"\n",
|
||||
"fig2 = go.Figure(data=go.Scatter(x=x1, y=y1, mode='lines+markers'),\n",
|
||||
" layout=go.Layout(title='Time needed for the convergence', xaxis_title='tau', yaxis_title='time (seconds)'))\n",
|
||||
"\n",
|
||||
"# save the plot in a html file\n",
|
||||
"fig2.write_html(\"../data/results/algo1/taus_over_time.html\")\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"To view the graph just use the command\n",
|
||||
"\n",
|
||||
"```bash\n",
|
||||
"firefox taus_over_iterations.html \n",
|
||||
"```\n",
|
||||
"or \n",
|
||||
"\n",
|
||||
"```bash\n",
|
||||
"firefox taus_over_time.html\n",
|
||||
"```\n",
|
||||
"\n",
|
||||
"_In the right folder_"
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3.10.6 64-bit",
|
||||
"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": "3.10.6"
|
||||
},
|
||||
"orig_nbformat": 4,
|
||||
"vscode": {
|
||||
"interpreter": {
|
||||
"hash": "916dbcbb3f70747c44a77c7bcd40155683ae19c65e1c03b4aa3499c5328201f1"
|
||||
}
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 2
|
||||
}
|
||||
@ -1,368 +0,0 @@
|
||||
{
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"import numpy as np\n",
|
||||
"import networkx as nx\n",
|
||||
"import time\n",
|
||||
"import math\n",
|
||||
"import scipy as sp\n",
|
||||
"from scipy.sparse import *\n",
|
||||
"from scipy.sparse.linalg import norm"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Let's create two graphs from the list of edges downloaded from the Snap database. "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"G1 = nx.read_edgelist('../data/web-Stanford.txt', create_using=nx.DiGraph(), nodetype=int)\n",
|
||||
"\n",
|
||||
"G2 = nx.read_edgelist('../data/web-BerkStan.txt', create_using=nx.DiGraph(), nodetype=int)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### Creating the transition probability matrix"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"# square matrix of size n x n, where n is the number of nodes in the graph. The matrix is filled with zeros and the (i,j) element is x if the node i is connected to the node j. Where x is 1/(number of nodes connected to i).\n",
|
||||
"\n",
|
||||
"def create_matrix(G):\n",
|
||||
" n = G.number_of_nodes()\n",
|
||||
" P = sp.sparse.lil_matrix((n,n))\n",
|
||||
" for i in G.nodes():\n",
|
||||
" for j in G[i]: #G[i] is the list of nodes connected to i, it's neighbors\n",
|
||||
" P[i-1,j-1] = 1/len(G[i])\n",
|
||||
" return P"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"To ensure that the random process has a unique stationary distribution and it will not stagnate, the transition matrix P is usually modified to be an irreducible stochastic matrix A (called the Google matrix) as follows\n",
|
||||
"\n",
|
||||
"$$ A = \\alpha \\tilde{P} + (1-\\alpha)v e^T$$\n",
|
||||
"\n",
|
||||
"Where $\\tilde{P}$ is defined as \n",
|
||||
"\n",
|
||||
"$$ \\tilde{P} = P + v d^T$$\n",
|
||||
"\n",
|
||||
"Where $d \\in \\mathbb{N}^{n \\times 1}$ s a binary vector tracing the indices of dangling web-pages with no hyperlinks, i.e., $d(i ) = 1$ if the `ith` page has no hyperlink, $v \\in \\mathbb{R}^{n \\times 1}$ is a probability vector, $e = [1, 1, . . . , 1]^T$ , and $0 < \\alpha < 1$ is the so-called damping factor that represents the probability in the model that the surfer transfer by clicking a hyperlink rather than other ways"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"n = G2.number_of_nodes()\n",
|
||||
"P = create_matrix(G2) "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"the vector `d` solves the dangling nodes problem"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"# define d as a nx1 sparse matrix, where n is the number of nodes in the graph. The vector is filled with d(i) = 1 if the i row of the matrix P is filled with zeros, other wise is 0\n",
|
||||
"\n",
|
||||
"# d is the vector of dangling nodes\n",
|
||||
"d = sp.sparse.lil_matrix((n,1))\n",
|
||||
"for i in range(n):\n",
|
||||
" if P[i].sum() == 0:\n",
|
||||
" d[i] = 1"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"The vector v is a probability vector, the sum of its elements bust be one"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"# define v as the probability vector of size n x 1, where n is the number of nodes in the graph. The vector is filled with 1/n\n",
|
||||
"# https://en.wikipedia.org/wiki/Probability_vector\n",
|
||||
"\n",
|
||||
"v = sp.sparse.lil_matrix((n,1))\n",
|
||||
"for i in range(n):\n",
|
||||
" v[i] = 1/n "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Now we can compute the transition matrix\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"Pt = P + v.dot(d.T)\n",
|
||||
"\n",
|
||||
"# Pt is a sparse matrix too"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"# e is a nx1 sparse matrix filled with ones\n",
|
||||
"e = sp.sparse.lil_matrix((1,n))\n",
|
||||
"for i in range(n):\n",
|
||||
" e[0,i] = 1"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"# # v*eT is a nxn sparse matrix filled all with 1/n, let's call it B\n",
|
||||
"\n",
|
||||
"# B = sp.sparse.lil_matrix((n,n))\n",
|
||||
"# for i in range(n):\n",
|
||||
"# for j in range(n):\n",
|
||||
"# B[i,j] = 1/n\n",
|
||||
"\n",
|
||||
"# A = alpha*Pt + (1-alpha)*B"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Algorithm 1 Shifted-Power method for PageRank with multiple damping factors:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"# list of alpha values, from 0.85 to 0.99 with step 0.01\n",
|
||||
"a = []\n",
|
||||
"for i in range(85,100):\n",
|
||||
" a.append(i/100)\n",
|
||||
"\n",
|
||||
"tau = 10**-6\n",
|
||||
"max_mv = 100\n",
|
||||
"\n",
|
||||
"# this should return mv (the number of iteration needed for the convergence), and two vector called x and r. Where x is the vector of the pagerank and r is the residual vector\n",
|
||||
"\n",
|
||||
"def Algorithm1(Pt, v, tau, max_mv, a: list):\n",
|
||||
" # take time of the performance\n",
|
||||
" start_time = time.time()\n",
|
||||
"\n",
|
||||
"\n",
|
||||
" u = Pt.dot(v) - v \n",
|
||||
" mv = 1 # number of iteration\n",
|
||||
" r = sp.sparse.lil_matrix((n,1)) \n",
|
||||
" Res = sp.sparse.lil_matrix((len(a),1))\n",
|
||||
" x = sp.sparse.lil_matrix((n,1)) \n",
|
||||
"\n",
|
||||
" for i in range(len(a)):\n",
|
||||
" r = a[i]*(u) \n",
|
||||
" normed_r = norm(r)\n",
|
||||
" Res[i] = normed_r \n",
|
||||
"\n",
|
||||
" if Res[i] > tau:\n",
|
||||
" x = r + v \n",
|
||||
"\n",
|
||||
" print(\"STARTING THE WHILE LOOP\\n\")\n",
|
||||
"\n",
|
||||
" # take the maximum value of the sparse matrix Res\n",
|
||||
"\n",
|
||||
"\n",
|
||||
" while max(Res) > tau and mv < max_mv:\n",
|
||||
" u = Pt*u # should it be the same u of the beginning?\n",
|
||||
" mv += 1 \n",
|
||||
" print(\"mv = \", mv)\n",
|
||||
" print(\"max(Res) = \", max(Res))\n",
|
||||
"\n",
|
||||
" for i in range(len(a)):\n",
|
||||
" if Res[i] >= tau: \n",
|
||||
" r = (a[i]**(mv+1))*(u)\n",
|
||||
" Res[i] = norm(r)\n",
|
||||
"\n",
|
||||
" if Res[i] > tau:\n",
|
||||
" x = r + x\n",
|
||||
"\n",
|
||||
" print(\"\\nEND OF THE WHILE LOOP\\n\")\n",
|
||||
"\n",
|
||||
" if mv == max_mv:\n",
|
||||
" print(\"The algorithm didn't converge in \", max_mv, \" iterations\")\n",
|
||||
" else:\n",
|
||||
" print(\"The algorithm converged in \", mv, \" iterations\")\n",
|
||||
"\n",
|
||||
" print(\"\\nThe execution time is %s seconds\" % (time.time() - start_time))\n",
|
||||
" \n",
|
||||
" return mv, x, r "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"mv, x, r = Algorithm1(Pt, v, tau, max_mv, a)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### Algorithm 2 Arnoldi process"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def Algorithm2():\n",
|
||||
" pass"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Algorithm 4 shifted-GMRES method for PageRank with multiple damping factors: "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def Algorithm4(Pt, v, m , a: list, tau , maxit, x: list):\n",
|
||||
" iter = 1\n",
|
||||
" \n",
|
||||
" e1 = sp.sparse.lil_matrix((len(a),1))\n",
|
||||
" e1[0] = 1\n",
|
||||
"\n",
|
||||
" # identity matrix sparse\n",
|
||||
" I = sp.sparse.lil_matrix((len(a),len(a)))\n",
|
||||
" for i in range(n):\n",
|
||||
" I[i,i] = 1\n",
|
||||
"\n",
|
||||
" # create the page rank vector x\n",
|
||||
" x = sp.sparse.lil_matrix((n,1))\n",
|
||||
" for i in range(n):\n",
|
||||
" x[i] = 1/n\n",
|
||||
"\n",
|
||||
" # create the vector r \n",
|
||||
" r = sp.sparse.lil_matrix((len(a),1))\n",
|
||||
" for i in range(len(a)):\n",
|
||||
" r[i] = ((1-a[i])/a[i]).dot(v) - ((1/a[i]).dot(I) - Pt).dot(x[i]).dot(e1)\n",
|
||||
" \n",
|
||||
" # create the vector Res\n",
|
||||
" Res = sp.sparse.lil_matrix((len(a),1))\n",
|
||||
" for i in range(len(a)):\n",
|
||||
" Res[i] = a[i] * np.linalg.norm(r[i])\n",
|
||||
"\n",
|
||||
" mv = 0\n",
|
||||
"\n",
|
||||
" while max(Res) > tau and mv < maxit:\n",
|
||||
" # find the k that satisfy the condition res[k] = max(res[i])\n",
|
||||
" k = 0\n",
|
||||
" for i in range(len(a)):\n",
|
||||
" if Res[i] == max(Res):\n",
|
||||
" k = i\n",
|
||||
" break\n",
|
||||
" \n",
|
||||
" # compute a new vector called delta where delta[i] = (res[i]*a[k])/(res[k]*a[i])\n",
|
||||
" delta = sp.sparse.lil_matrix((len(a),1))\n",
|
||||
" for i in range(len(a)) and i != k:\n",
|
||||
" delta[i] = (Res[i]*a[k])/(Res[k]*a[i])\n",
|
||||
"\n",
|
||||
" # run algorithm 2\n",
|
||||
" # TO DO\n",
|
||||
"\n",
|
||||
" # j depends on the algorithm 2\n",
|
||||
" mv = mv + j\n",
|
||||
"\n",
|
||||
" \n",
|
||||
" # .............\n",
|
||||
"\n",
|
||||
" \n",
|
||||
" "
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3.10.6 64-bit",
|
||||
"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": "3.10.6"
|
||||
},
|
||||
"orig_nbformat": 4,
|
||||
"vscode": {
|
||||
"interpreter": {
|
||||
"hash": "916dbcbb3f70747c44a77c7bcd40155683ae19c65e1c03b4aa3499c5328201f1"
|
||||
}
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 2
|
||||
}
|
||||
Loading…
Reference in New Issue