mirror of
https://github.com/lukefleed/ShfitedPowGMRES.git
synced 2026-10-06 05:54:53 +00:00
minor fixes
This commit is contained in:
Luca Lombardo
+3
-1
@@ -275,4 +275,6 @@ TSWLatexianTemp*
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# Makeindex log files
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*.lpz
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data/*.txt
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data/
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__pycache__/
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Binary file not shown.
Vendored
+1
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@@ -35,7 +35,7 @@ for solving PageRank with multiple damping factors}
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\maketitle
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\begin{abstract}
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\noindent In the years following its publication in 1968, the PageRank model has been studied deeply to be extended in fields such as chemistry, biology and social network analysis. The aim of this project is the implementation of a modified version of the Power method to solve the PageRank problem with multiple damping factors. The proposed method is based on the combination of the Power method with the shifted \texttt{GMRES} method.
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\noindent In the years following its publication in 1998, the PageRank model has been studied deeply to be extended in fields such as chemistry, biology and social network analysis. The aim of this project is the implementation of a modified version of the Power method to solve the PageRank problem with multiple damping factors. The proposed method is based on the combination of the Power method with the shifted \texttt{GMRES} method.
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% During the years since it's first version, the PageRank model has been extended to many fields far beyond search engine rankings, such as chemistry, biology, bioinformatics, social network analysis, to name a few. Due to the large dimension of PageRank problems, in the past decade or so, considerable research efforts have been devoted to their efficient solution especially for the difficult cases where the damping factors are close to 1. However, there exists few research work concerning about the solution of the case where several PageRank problems with the same network structure and various damping factors need to be solved. In this paper, we generalize the Power method to solving the PageRank problem with multiple damping factors. We demonstrate that the solution has almost the equative cost of solving the most difficult PageRank system of the sequence, and the residual vectors of the PageRank systems after running this method are collinear. Based upon these results, we develop a more efficient method that combines this Power method with the shifted GMRES method. For further accelerating the solving phase, we present a seed system choosing strategy combined with an extrapolation technique, and analyze their effect. Numerical experiments demonstrate the potential of the proposed iterative solver for accelerating realistic PageRank computations with multiple damping factors.
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\end{abstract}
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Executable
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#!/usr/bin/env python3
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# Importing the libraries
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import os
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import wget
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import gzip
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import time
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import warnings
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import scipy as sp
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import numpy as np
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import pandas as pd
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import networkx as nx
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from os.path import exists
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from scipy.sparse import *
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from scipy.sparse.linalg import norm
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import plotly.graph_objs as go
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warnings.simplefilter(action='ignore', category=FutureWarning)
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# some stupid pandas function that doesn't work
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class Utilities:
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# Importing the dataset
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def load_data():
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# Loading the dataset
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dataset = int(input("Choose the dataset:\n [1] web-Stanford (use this one for now)\n [2] web-BerkStan: \nEnter an option: "))
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if dataset == 1:
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if exists('../data/web-Stanford.txt'):
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dataset = '../data/web-Stanford.txt'
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else:
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print("\nThe file doesn't exist, download it from https://snap.stanford.edu/data/web-Stanford.html")
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# if there is no folder data, create it
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if not exists('../data'):
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os.makedirs('../data')
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# Downloading the dataset
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url = 'https://snap.stanford.edu/data/web-Stanford.txt.gz'
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wget.download(url, '../data/web-Stanford.txt.gz')
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# Unzipping the dataset
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with gzip.open('../data/web-Stanford.txt.gz', 'rb') as f_in:
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with open('../data/web-Stanford.txt', 'wb') as f_out:
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f_out.write(f_in.read())
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# delete the zipped file
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os.remove('../data/web-Stanford.txt.gz')
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dataset = '../data/web-Stanford.txt'
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print("\nDataset downloaded\n")
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elif dataset == 2:
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if exists('../data/web-BerkStan.txt'):
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dataset = '../data/web-BerkStan.txt'
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else:
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print("\nThe file doesn't exist, download it from https://snap.stanford.edu/data/web-BerkStan.html")
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# if there is no folder data, create it
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if not exists('../data'):
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os.makedirs('../data')
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# Downloading the dataset
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url = 'https://snap.stanford.edu/data/web-BerkStan.txt.gz'
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wget.download(url, '../data/web-BerkStan.txt.gz')
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# Unzipping the dataset
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with gzip.open('../data/web-BerkStan.txt.gz', 'rb') as f_in:
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with open('../data/web-BerkStan.txt', 'wb') as f_out:
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f_out.write(f_in.read())
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# delete the zipped file
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os.remove('../data/web-BerkStan.txt.gz')
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dataset = '../data/web-BerkStan.txt'
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print("\nDataset downloaded\n")
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return dataset
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# Creating the graph from the dataset
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def create_graph(dataset):
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print("\nCreating the graph...")
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G = nx.read_edgelist(dataset, create_using=nx.DiGraph(), nodetype=int)
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n = G.number_of_nodes()
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print("Graph created based on the dataset\n")
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return G, n
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# 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).
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def create_matrix(G):
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print("Creating the transition probability matrix...")
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P = sp.sparse.lil_matrix((n,n))
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for i in G.nodes():
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for j in G[i]: #G[i] is the list of nodes connected to i, it's neighbors
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P[i-1,j-1] = 1/len(G[i])
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print("Transition probability matrix created\n")
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return P
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# The vector is filled with d(i) = 1 if the i row of the matrix P is filled with zeros, other wise is 0
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def dangling_nodes(P,n):
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print("Creating the list of dangling nodes...")
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d = sp.sparse.lil_matrix((n,1))
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for i in range(n):
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if P[i].sum() == 0:
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d[i] = 1
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print("List of dangling nodes created\n")
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return d
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# For now it is set to equally distribute the probability to all the nodes
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def probability_vector(n):
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print("Creating the probability vector...")
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v = sp.sparse.lil_matrix((n,1))
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for i in range(n):
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v[i] = 1/n
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print("Probability vector created\n")
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return v
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def transition_matrix(P, v, d):
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print("Creating the transition matrix...")
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Pt = P + v @ (d.T)
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print("Transition matrix created\n")
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return Pt
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# it can vary from 0 to 1, the higher the value the more the probability to jump to a random page
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def alpha():
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a = []
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for i in range(85,100):
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a.append(i/100)
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return a
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# 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
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class Plotting:
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def tau_over_iterations(df):
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x = df['tau'][::-1].tolist()
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y = df['products m-v'].tolist()
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fig = go.Figure(data=go.Scatter(x=x, y=y, mode='lines+markers'),
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layout=go.Layout(title='products needed for the convergence', xaxis_title='tau', yaxis_title='products matrix vector'))
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# save the figure as a html file
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fig.write_html("../data/results/algo1/taus_over_prods.html")
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print("The plot has been saved in the folder data/results/algo1")
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def tau_over_time(df):
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x1 = df['tau'][::-1].tolist()
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y1 = df['time'].tolist()
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fig = go.Figure(data=go.Scatter(x=x1, y=y1, mode='lines+markers'),
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layout=go.Layout(title='Time needed for the convergence', xaxis_title='tau', yaxis_title='time (seconds)'))
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# save the plot in a html file
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fig.write_html("../data/results/algo1/taus_over_time.html")
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print("The plot has been saved in the folder data/results/algo1")
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class Algorithms:
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# Power method adapted to the PageRank problem with different damping factors. Referred as Algorithm 1 in the paper
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def algo1(Pt, v, tau, max_mv, a: list):
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start_time = time.time()
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print("STARTING ALGORITHM 1...")
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u = Pt @ v - v
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mv = 1 # number of matrix-vector multiplications
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r = sp.sparse.lil_matrix((n,1))
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Res = sp.sparse.lil_matrix((len(a),1))
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x = sp.sparse.lil_matrix((n,1))
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for i in range(len(a)):
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r = a[i]*(u)
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normed_r = norm(r)
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Res[i] = normed_r
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if Res[i] > tau:
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x = r + v
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while max(Res) > tau and mv < max_mv:
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u = Pt @ u
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mv += 1
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for i in range(len(a)):
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if Res[i] >= tau:
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r = (a[i]**(mv+1))*(u)
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Res[i] = norm(r)
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if Res[i] > tau:
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x = r + x
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if mv == max_mv:
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print("The algorithm didn't converge in ", max_mv, " iterations")
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else:
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print("The algorithm converged with ", mv, " matrix-vector multiplications executed")
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total_time = time.time() - start_time
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total_time = round(total_time, 2)
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print("The algorithm took ", total_time, " seconds to run\n")
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return mv, x, r, total_time
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# Refers to Algorithm 2 in the paper
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def Arnoldi(A,v0,m):
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v = v0
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beta = norm(v)
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v = v/beta
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H = sp.sparse.lil_matrix((m+1,m))
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V = sp.sparse.lil_matrix((A.shape[0],m+1))
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V[:,0] = v # each column of V is a vector v
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for j in range(m):
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# print("j = ", j)
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w = A @ v
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for i in range(j):
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tmp = v.T @ w # tmp is a 1x1 matrix, so it's O(1) in memory
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H[i,j] = tmp[0,0]
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w = w - H[i,j]*v
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H[j+1,j] = norm(w)
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if H[j+1,j] == 0:
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print("Arnoldi breakdown")
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m = j
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v = 0
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break
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else:
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if j < m-1:
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v = w/H[j+1,j]
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V[:,j+1] = v
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print(j, " iterations completed")
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print("V = ", V.shape)
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print("H = ", H.shape)
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print("v = ", v.shape)
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print("beta = ", beta)
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return V, H, v, beta, j
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def algo4():
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# TO DO
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pass
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class runners:
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def ShiftedPowerMethod(tau):
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dataset = Utilities.load_data()
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max_mv = 100
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G, n = Utilities.create_graph(dataset)
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P = Utilities.create_matrix(G)
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d = Utilities.dangling_nodes(P,n)
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v = Utilities.probability_vector(n)
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Pt = Utilities.transition_matrix(P, v, d)
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a = Utilities.alpha()
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mv, x, r, total_time = Algorithms.algo1(Pt, v, tau, max_mv, a)
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print("total time = ", total_time)
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# pandas dataframe to store the results
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df = pd.DataFrame(columns=['alpha', 'products m-v', 'tau', 'time'])
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# Main
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if __name__ == "__main__":
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dataset = Utilities.load_data()
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# maximum number of iterations, asked to the user
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max_mv = int(input("\nInsert the maximum number of matrix-vector operations: "))
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G, n = Utilities.create_graph(dataset)
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P = Utilities.create_matrix(G)
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d = Utilities.dangling_nodes(P,n)
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v = Utilities.probability_vector(n)
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Pt = Utilities.transition_matrix(P, v, d)
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a = Utilities.alpha()
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# run the algorithm for different values of tau from 10^-5 to 10^-9 with step 10^-1
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for i in range(5,10):
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tau = 10**(-i)
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print("\ntau = ", tau)
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mv, x, r, total_time = Algorithms.algo1(Pt, v, tau, max_mv, a)
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# store the results in the dataframe
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df = df.append({'alpha': a, 'products m-v': mv, 'tau': tau, 'time': total_time}, ignore_index=True)
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# save the results in a csv file
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df.to_csv('../data/results/algo1/different_tau.csv', index=False)
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# plot the results
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Plotting.tau_over_iterations(df)
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Plotting.tau_over_time(df)
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# print in the terminal the columns of the dataframe iterations, tau and time
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print("Computations done. Here are the results:")
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print("\n", df[['products m-v', 'tau', 'time']])
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Vendored
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