removed old notebooks and unseful scripts
Luca Lombardo
2 years ago
parent
2df71072a6
commit
3e08fee18d
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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": null,
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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 import linalg\n",
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"from scipy.sparse.linalg import norm\n",
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"from scipy.optimize import least_squares"
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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 TESTING\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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"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": null,
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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": null,
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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": null,
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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": null,
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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": null,
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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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"\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 \n",
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"\n",
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"# maybe I could have done it with a lambda more concisely? idk"
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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": null,
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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": null,
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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": null,
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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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"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": null,
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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": null,
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"metadata": {},
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"outputs": [],
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"source": [
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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 matrix vector products\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 \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": "markdown",
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"metadata": {},
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"source": [
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"Testing setup"
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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": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"# list of alpha values, from 0.85 to 0.99 with step 0.01\n",
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"a = []\n",
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"for i in range(85,100):\n",
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" a.append(i/100)\n",
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"\n",
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"max_mv = 1000\n",
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"\n",
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"# run the algorithm for different values of tau from 10^-5 to 10^-9 with step 10^-1\n",
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"for i in range(5,10):\n",
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" tau = 10**(-i)\n",
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" print(\"\\ntau = \", tau)\n",
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" mv, x, r, total_time = Algorithm1(Pt, v, tau, max_mv, a)\n",
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" df = df.append({'alpha': a, 'iterations': mv, 'tau': tau, 'time': total_time}, ignore_index=True) "
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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": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"# save the results in a csv file\n",
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"df.to_csv('../data/results/algo1/different_tau.csv', index=False)"
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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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"Plotting the results of the algorithm for different values of tau, and fixed alpha"
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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": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"x = df['tau'][::-1].tolist()\n",
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"y = df['products m-v'].tolist()\n",
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"\n",
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"fig = go.Figure(data=go.Scatter(x=x, y=y, mode='lines+markers'),\n",
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" layout=go.Layout(title='products needed for the convergence', xaxis_title='tau', yaxis_title='products matrix vector'))\n",
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"\n",
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"# save the figure as a html file\n",
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"fig.write_html(\"../data/results/algo1/taus_over_prods.html\")\n",
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"print(\"The plot has been saved in the folder data/results/algo1\")\n",
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"\n",
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"##### RESULTS OVER TIME #####\n",
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"\n",
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"x1 = df['tau'][::-1].tolist()\n",
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"y1 = df['time'].tolist()\n",
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"\n",
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"fig = go.Figure(data=go.Scatter(x=x1, y=y1, mode='lines+markers'),\n",
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" layout=go.Layout(title='Time needed for the convergence', xaxis_title='tau', yaxis_title='time (seconds)'))\n",
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"\n",
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"# save the plot in a html file\n",
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"fig.write_html(\"../data/results/algo1/taus_over_time.html\")\n",
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"print(\"The plot has been saved in the folder data/results/algo1\")\n"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3.10.6 64-bit",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.10.6"
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},
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"orig_nbformat": 4,
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"vscode": {
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"interpreter": {
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"hash": "916dbcbb3f70747c44a77c7bcd40155683ae19c65e1c03b4aa3499c5328201f1"
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}
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}
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}
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@ -1,184 +0,0 @@
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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": null,
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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 import linalg\n",
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"from scipy.sparse.linalg import norm\n",
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"from scipy.optimize import least_squares"
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]
|
|
||||||
},
|
|
||||||
{
|
|
||||||
"cell_type": "markdown",
|
|
||||||
"metadata": {},
|
|
||||||
"source": [
|
|
||||||
"# Algorithm 2 testing"
|
|
||||||
]
|
|
||||||
},
|
|
||||||
{
|
|
||||||
"cell_type": "code",
|
|
||||||
"execution_count": null,
|
|
||||||
"metadata": {},
|
|
||||||
"outputs": [],
|
|
||||||
"source": [
|
|
||||||
"# def Arnoldi(A, v, m): \n",
|
|
||||||
"# beta = norm(v)\n",
|
|
||||||
"# v = v/beta # dimension of v is n x 1\n",
|
|
||||||
"# H = sp.sparse.lil_matrix((m,m)) # dimension of H is m x m \n",
|
|
||||||
"# V = sp.sparse.lil_matrix((A.shape[0],m))\n",
|
|
||||||
"# V[:,0] = v # each column of V is a vector v\n",
|
|
||||||
"\n",
|
|
||||||
"# for j in range(m):\n",
|
|
||||||
"# print(\"j = \", j)\n",
|
|
||||||
"# w = A @ v \n",
|
|
||||||
"# for i in range(j):\n",
|
|
||||||
"# tmp = v.T @ w \n",
|
|
||||||
"# H[i,j] = tmp[0,0]\n",
|
|
||||||
"# w = w - H[i,j]*v \n",
|
|
||||||
" \n",
|
|
||||||
"# H[j,j-1] = norm(w) \n",
|
|
||||||
"\n",
|
|
||||||
"# if H[j,j-1] == 0: \n",
|
|
||||||
"# print(\"Arnoldi breakdown\")\n",
|
|
||||||
"# m = j\n",
|
|
||||||
"# v = 0\n",
|
|
||||||
"# break\n",
|
|
||||||
"# else:\n",
|
|
||||||
"# if j < m-1:\n",
|
|
||||||
"# v = w/H[j,j-1]\n",
|
|
||||||
"# # for i in range(A.shape[0]):\n",
|
|
||||||
"# # V[i,j+1] = v[i,0]\n",
|
|
||||||
"# V[:,j+1] = v\n",
|
|
||||||
"\n",
|
|
||||||
"# print(j, \" iterations completed\")\n",
|
|
||||||
"# print(\"V = \", V.shape)\n",
|
|
||||||
"# print(\"H = \", H.shape)\n",
|
|
||||||
"# print(\"v = \", v.shape)\n",
|
|
||||||
"# print(\"beta = \", beta)\n",
|
|
||||||
"\n",
|
|
||||||
"# return V, H, v, beta, j"
|
|
||||||
]
|
|
||||||
},
|
|
||||||
{
|
|
||||||
"cell_type": "markdown",
|
|
||||||
"metadata": {},
|
|
||||||
"source": [
|
|
||||||
"Defined as Algorithm 2 in the paper. It's needed since it's called by Algorithm 4"
|
|
||||||
]
|
|
||||||
},
|
|
||||||
{
|
|
||||||
"cell_type": "code",
|
|
||||||
"execution_count": null,
|
|
||||||
"metadata": {},
|
|
||||||
"outputs": [],
|
|
||||||
"source": [
|
|
||||||
"def Arnoldi(A,v0,m):\n",
|
|
||||||
" v = v0\n",
|
|
||||||
" beta = norm(v)\n",
|
|
||||||
" v = v/beta\n",
|
|
||||||
" H = sp.sparse.lil_matrix((m+1,m)) \n",
|
|
||||||
" V = sp.sparse.lil_matrix((A.shape[0],m+1))\n",
|
|
||||||
" V[:,0] = v # each column of V is a vector v\n",
|
|
||||||
"\n",
|
|
||||||
" for j in range(m):\n",
|
|
||||||
" # print(\"j = \", j)\n",
|
|
||||||
" w = A @ v \n",
|
|
||||||
" for i in range(j):\n",
|
|
||||||
" tmp = v.T @ w # tmp is a 1x1 matrix, so it's O(1) in memory\n",
|
|
||||||
" H[i,j] = tmp[0,0] \n",
|
|
||||||
" w = w - H[i,j]*v \n",
|
|
||||||
" \n",
|
|
||||||
" H[j+1,j] = norm(w)\n",
|
|
||||||
"\n",
|
|
||||||
" if H[j+1,j] == 0:\n",
|
|
||||||
" print(\"Arnoldi breakdown\")\n",
|
|
||||||
" m = j\n",
|
|
||||||
" v = 0\n",
|
|
||||||
" break\n",
|
|
||||||
" else:\n",
|
|
||||||
" if j < m-1:\n",
|
|
||||||
" v = w/H[j+1,j]\n",
|
|
||||||
" V[:,j+1] = v\n",
|
|
||||||
"\n",
|
|
||||||
" print(j, \" iterations completed\")\n",
|
|
||||||
" print(\"V = \", V.shape)\n",
|
|
||||||
" print(\"H = \", H.shape)\n",
|
|
||||||
" print(\"v = \", v.shape)\n",
|
|
||||||
" print(\"beta = \", beta)\n",
|
|
||||||
"\n",
|
|
||||||
" return V, H, v, beta, j \n",
|
|
||||||
" "
|
|
||||||
]
|
|
||||||
},
|
|
||||||
{
|
|
||||||
"cell_type": "markdown",
|
|
||||||
"metadata": {},
|
|
||||||
"source": [
|
|
||||||
"Creating a small test case"
|
|
||||||
]
|
|
||||||
},
|
|
||||||
{
|
|
||||||
"cell_type": "code",
|
|
||||||
"execution_count": null,
|
|
||||||
"metadata": {},
|
|
||||||
"outputs": [],
|
|
||||||
"source": [
|
|
||||||
"m = 100\n",
|
|
||||||
"n = 110\n",
|
|
||||||
"A = sp.sparse.rand(n,n, density=0.1, format='lil')\n",
|
|
||||||
"# generate a probability vector, with all the entries as 1/n\n",
|
|
||||||
"v = sp.sparse.lil_matrix((n,1))\n",
|
|
||||||
"for i in range(n):\n",
|
|
||||||
" v[i] = 1/n"
|
|
||||||
]
|
|
||||||
},
|
|
||||||
{
|
|
||||||
"cell_type": "code",
|
|
||||||
"execution_count": null,
|
|
||||||
"metadata": {},
|
|
||||||
"outputs": [],
|
|
||||||
"source": [
|
|
||||||
"Arnoldi(A,v,m)"
|
|
||||||
]
|
|
||||||
}
|
|
||||||
],
|
|
||||||
"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
|
|
||||||
}
|
|
||||||
@ -1,291 +0,0 @@
|
|||||||
#!/usr/bin/env python3
|
|
||||||
|
|
||||||
# Importing the libraries
|
|
||||||
import os
|
|
||||||
import wget
|
|
||||||
import gzip
|
|
||||||
import time
|
|
||||||
import warnings
|
|
||||||
import scipy as sp
|
|
||||||
import numpy as np
|
|
||||||
import pandas as pd
|
|
||||||
import networkx as nx
|
|
||||||
from os.path import exists
|
|
||||||
from scipy.sparse import *
|
|
||||||
from scipy.sparse.linalg import norm
|
|
||||||
import plotly.graph_objs as go
|
|
||||||
|
|
||||||
warnings.simplefilter(action='ignore', category=FutureWarning)
|
|
||||||
# some stupid pandas function that doesn't work
|
|
||||||
|
|
||||||
class Utilities:
|
|
||||||
# Importing the dataset
|
|
||||||
def load_data():
|
|
||||||
# Loading the dataset
|
|
||||||
dataset = int(input("Choose the dataset:\n [1] web-Stanford (use this one for now)\n [2] web-BerkStan: \nEnter an option: "))
|
|
||||||
|
|
||||||
if dataset == 1:
|
|
||||||
if exists('../data/web-Stanford.txt'):
|
|
||||||
dataset = '../data/web-Stanford.txt'
|
|
||||||
else:
|
|
||||||
print("\nThe file doesn't exist, download it from https://snap.stanford.edu/data/web-Stanford.html")
|
|
||||||
|
|
||||||
# if there is no folder data, create it
|
|
||||||
if not exists('../data'):
|
|
||||||
os.makedirs('../data')
|
|
||||||
|
|
||||||
# Downloading the dataset
|
|
||||||
url = 'https://snap.stanford.edu/data/web-Stanford.txt.gz'
|
|
||||||
wget.download(url, '../data/web-Stanford.txt.gz')
|
|
||||||
# Unzipping the dataset
|
|
||||||
with gzip.open('../data/web-Stanford.txt.gz', 'rb') as f_in:
|
|
||||||
with open('../data/web-Stanford.txt', 'wb') as f_out:
|
|
||||||
f_out.write(f_in.read())
|
|
||||||
|
|
||||||
# delete the zipped file
|
|
||||||
os.remove('../data/web-Stanford.txt.gz')
|
|
||||||
|
|
||||||
dataset = '../data/web-Stanford.txt'
|
|
||||||
print("\nDataset downloaded\n")
|
|
||||||
|
|
||||||
elif dataset == 2:
|
|
||||||
if exists('../data/web-BerkStan.txt'):
|
|
||||||
dataset = '../data/web-BerkStan.txt'
|
|
||||||
else:
|
|
||||||
print("\nThe file doesn't exist, download it from https://snap.stanford.edu/data/web-BerkStan.html")
|
|
||||||
|
|
||||||
# if there is no folder data, create it
|
|
||||||
if not exists('../data'):
|
|
||||||
os.makedirs('../data')
|
|
||||||
|
|
||||||
# Downloading the dataset
|
|
||||||
url = 'https://snap.stanford.edu/data/web-BerkStan.txt.gz'
|
|
||||||
wget.download(url, '../data/web-BerkStan.txt.gz')
|
|
||||||
# Unzipping the dataset
|
|
||||||
with gzip.open('../data/web-BerkStan.txt.gz', 'rb') as f_in:
|
|
||||||
with open('../data/web-BerkStan.txt', 'wb') as f_out:
|
|
||||||
f_out.write(f_in.read())
|
|
||||||
|
|
||||||
# delete the zipped file
|
|
||||||
os.remove('../data/web-BerkStan.txt.gz')
|
|
||||||
|
|
||||||
dataset = '../data/web-BerkStan.txt'
|
|
||||||
print("\nDataset downloaded\n")
|
|
||||||
|
|
||||||
return dataset
|
|
||||||
|
|
||||||
# Creating the graph from the dataset
|
|
||||||
def create_graph(dataset):
|
|
||||||
print("\nCreating the graph...")
|
|
||||||
G = nx.read_edgelist(dataset, create_using=nx.DiGraph(), nodetype=int)
|
|
||||||
n = G.number_of_nodes()
|
|
||||||
print("Graph created based on the dataset\n")
|
|
||||||
return G, n
|
|
||||||
|
|
||||||
# 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).
|
|
||||||
def create_matrix(G):
|
|
||||||
print("Creating the transition probability matrix...")
|
|
||||||
P = sp.sparse.lil_matrix((n,n))
|
|
||||||
for i in G.nodes():
|
|
||||||
for j in G[i]: #G[i] is the list of nodes connected to i, it's neighbors
|
|
||||||
P[i-1,j-1] = 1/len(G[i])
|
|
||||||
print("Transition probability matrix created\n")
|
|
||||||
return P
|
|
||||||
|
|
||||||
# The vector is filled with d(i) = 1 if the i row of the matrix P is filled with zeros, other wise is 0
|
|
||||||
def dangling_nodes(P,n):
|
|
||||||
print("Creating the list of dangling nodes...")
|
|
||||||
d = sp.sparse.lil_matrix((n,1))
|
|
||||||
for i in range(n):
|
|
||||||
if P[i].sum() == 0:
|
|
||||||
d[i] = 1
|
|
||||||
print("List of dangling nodes created\n")
|
|
||||||
return d
|
|
||||||
|
|
||||||
# For now it is set to equally distribute the probability to all the nodes
|
|
||||||
def probability_vector(n):
|
|
||||||
print("Creating the probability vector...")
|
|
||||||
v = sp.sparse.lil_matrix((n,1))
|
|
||||||
for i in range(n):
|
|
||||||
v[i] = 1/n
|
|
||||||
print("Probability vector created\n")
|
|
||||||
return v
|
|
||||||
|
|
||||||
def transition_matrix(P, v, d):
|
|
||||||
print("Creating the transition matrix...")
|
|
||||||
Pt = P + v @ (d.T)
|
|
||||||
print("Transition matrix created\n")
|
|
||||||
return Pt
|
|
||||||
|
|
||||||
# it can vary from 0 to 1, the higher the value the more the probability to jump to a random page
|
|
||||||
def alpha():
|
|
||||||
a = []
|
|
||||||
for i in range(85,100):
|
|
||||||
a.append(i/100)
|
|
||||||
return a
|
|
||||||
|
|
||||||
# Class for plotting the results obtained. For now it can only the first algorithm. To be updated once all the algorithms are implemented and the test cases are well defined
|
|
||||||
class Plotting:
|
|
||||||
def tau_over_iterations(df):
|
|
||||||
x = df['tau'][::-1].tolist()
|
|
||||||
y = df['products m-v'].tolist()
|
|
||||||
|
|
||||||
fig = go.Figure(data=go.Scatter(x=x, y=y, mode='lines+markers'),
|
|
||||||
layout=go.Layout(title='products needed for the convergence', xaxis_title='tau', yaxis_title='products matrix vector'))
|
|
||||||
|
|
||||||
# save the figure as a html file
|
|
||||||
fig.write_html("../data/results/algo1/taus_over_prods.html")
|
|
||||||
print("The plot has been saved in the folder data/results/algo1")
|
|
||||||
|
|
||||||
def tau_over_time(df):
|
|
||||||
x1 = df['tau'][::-1].tolist()
|
|
||||||
y1 = df['time'].tolist()
|
|
||||||
|
|
||||||
fig = go.Figure(data=go.Scatter(x=x1, y=y1, mode='lines+markers'),
|
|
||||||
layout=go.Layout(title='Time needed for the convergence', xaxis_title='tau', yaxis_title='time (seconds)'))
|
|
||||||
|
|
||||||
# save the plot in a html file
|
|
||||||
fig.write_html("../data/results/algo1/taus_over_time.html")
|
|
||||||
print("The plot has been saved in the folder data/results/algo1")
|
|
||||||
|
|
||||||
class Algorithms:
|
|
||||||
# Power method adapted to the PageRank problem with different damping factors. Referred as Algorithm 1 in the paper
|
|
||||||
def algo1(Pt, v, tau, max_mv, a: list):
|
|
||||||
start_time = time.time()
|
|
||||||
|
|
||||||
print("STARTING ALGORITHM 1...")
|
|
||||||
u = Pt @ v - v
|
|
||||||
mv = 1 # number of matrix-vector multiplications
|
|
||||||
r = sp.sparse.lil_matrix((n,1))
|
|
||||||
Res = sp.sparse.lil_matrix((len(a),1))
|
|
||||||
x = sp.sparse.lil_matrix((n,1))
|
|
||||||
|
|
||||||
for i in range(len(a)):
|
|
||||||
r = a[i]*(u)
|
|
||||||
normed_r = norm(r)
|
|
||||||
Res[i] = normed_r
|
|
||||||
|
|
||||||
if Res[i] > tau:
|
|
||||||
x = r + v
|
|
||||||
|
|
||||||
while max(Res) > tau and mv < max_mv:
|
|
||||||
u = Pt @ u
|
|
||||||
mv += 1
|
|
||||||
|
|
||||||
for i in range(len(a)):
|
|
||||||
if Res[i] >= tau:
|
|
||||||
r = (a[i]**(mv+1))*(u)
|
|
||||||
Res[i] = norm(r)
|
|
||||||
|
|
||||||
if Res[i] > tau:
|
|
||||||
x = r + x
|
|
||||||
|
|
||||||
if mv == max_mv:
|
|
||||||
print("The algorithm didn't converge in ", max_mv, " iterations")
|
|
||||||
else:
|
|
||||||
print("The algorithm converged with ", mv, " matrix-vector multiplications executed")
|
|
||||||
|
|
||||||
total_time = time.time() - start_time
|
|
||||||
total_time = round(total_time, 2)
|
|
||||||
|
|
||||||
print("The algorithm took ", total_time, " seconds to run\n")
|
|
||||||
|
|
||||||
return mv, x, r, total_time
|
|
||||||
|
|
||||||
# Refers to Algorithm 2 in the paper
|
|
||||||
def Arnoldi(A,v0,m):
|
|
||||||
v = v0
|
|
||||||
beta = norm(v)
|
|
||||||
v = v/beta
|
|
||||||
H = sp.sparse.lil_matrix((m+1,m))
|
|
||||||
V = sp.sparse.lil_matrix((A.shape[0],m+1))
|
|
||||||
V[:,0] = v # each column of V is a vector v
|
|
||||||
|
|
||||||
for j in range(m):
|
|
||||||
# print("j = ", j)
|
|
||||||
w = A @ v
|
|
||||||
for i in range(j):
|
|
||||||
tmp = v.T @ w # tmp is a 1x1 matrix, so it's O(1) in memory
|
|
||||||
H[i,j] = tmp[0,0]
|
|
||||||
w = w - H[i,j]*v
|
|
||||||
|
|
||||||
H[j+1,j] = norm(w)
|
|
||||||
|
|
||||||
if H[j+1,j] == 0:
|
|
||||||
print("Arnoldi breakdown")
|
|
||||||
m = j
|
|
||||||
v = 0
|
|
||||||
break
|
|
||||||
else:
|
|
||||||
if j < m-1:
|
|
||||||
v = w/H[j+1,j]
|
|
||||||
V[:,j+1] = v
|
|
||||||
|
|
||||||
print(j, " iterations completed")
|
|
||||||
print("V = ", V.shape)
|
|
||||||
print("H = ", H.shape)
|
|
||||||
print("v = ", v.shape)
|
|
||||||
print("beta = ", beta)
|
|
||||||
|
|
||||||
return V, H, v, beta, j
|
|
||||||
|
|
||||||
def algo4():
|
|
||||||
# TO DO
|
|
||||||
pass
|
|
||||||
|
|
||||||
class runners:
|
|
||||||
def ShiftedPowerMethod(tau):
|
|
||||||
dataset = Utilities.load_data()
|
|
||||||
max_mv = 100
|
|
||||||
G, n = Utilities.create_graph(dataset)
|
|
||||||
P = Utilities.create_matrix(G)
|
|
||||||
d = Utilities.dangling_nodes(P,n)
|
|
||||||
v = Utilities.probability_vector(n)
|
|
||||||
Pt = Utilities.transition_matrix(P, v, d)
|
|
||||||
a = Utilities.alpha()
|
|
||||||
|
|
||||||
mv, x, r, total_time = Algorithms.algo1(Pt, v, tau, max_mv, a)
|
|
||||||
|
|
||||||
print("total time = ", total_time)
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
# pandas dataframe to store the results
|
|
||||||
df = pd.DataFrame(columns=['alpha', 'products m-v', 'tau', 'time'])
|
|
||||||
|
|
||||||
# Main
|
|
||||||
if __name__ == "__main__":
|
|
||||||
dataset = Utilities.load_data()
|
|
||||||
# maximum number of iterations, asked to the user
|
|
||||||
max_mv = int(input("\nInsert the maximum number of matrix-vector operations: "))
|
|
||||||
|
|
||||||
G, n = Utilities.create_graph(dataset)
|
|
||||||
P = Utilities.create_matrix(G)
|
|
||||||
d = Utilities.dangling_nodes(P,n)
|
|
||||||
v = Utilities.probability_vector(n)
|
|
||||||
Pt = Utilities.transition_matrix(P, v, d)
|
|
||||||
a = Utilities.alpha()
|
|
||||||
|
|
||||||
# run the algorithm for different values of tau from 10^-5 to 10^-9 with step 10^-1
|
|
||||||
for i in range(5,10):
|
|
||||||
tau = 10**(-i)
|
|
||||||
print("\ntau = ", tau)
|
|
||||||
mv, x, r, total_time = Algorithms.algo1(Pt, v, tau, max_mv, a)
|
|
||||||
|
|
||||||
# store the results in the dataframe
|
|
||||||
df = df.append({'alpha': a, 'products m-v': mv, 'tau': tau, 'time': total_time}, ignore_index=True)
|
|
||||||
|
|
||||||
# save the results in a csv file
|
|
||||||
df.to_csv('../data/results/algo1/different_tau.csv', index=False)
|
|
||||||
|
|
||||||
# plot the results
|
|
||||||
Plotting.tau_over_iterations(df)
|
|
||||||
Plotting.tau_over_time(df)
|
|
||||||
|
|
||||||
# print in the terminal the columns of the dataframe iterations, tau and time
|
|
||||||
print("Computations done. Here are the results:")
|
|
||||||
print("\n", df[['products m-v', 'tau', 'time']])
|
|
||||||
Loading…
Reference in New Issue