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#!/usr/bin/env python
# Copyright (C) 2004-2008 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
# NetworkX:http://networkx.lanl.gov/.
__author__ = """Aric Hagberg (hagberg@lanl.gov)"""
import networkx
from networkx.exception import NetworkXError
__all__ = ['pagerank','pagerank_numpy','pagerank_scipy','google_matrix']
def pagerank(G,alpha=0.85,max_iter=100,tol=1.0e-8,nstart=None):
"""Return the PageRank of the nodes in the graph.
PageRank computes the largest eigenvector of the stochastic
adjacency matrix of G.
Parameters
-----------
G : graph
A networkx graph
alpha : float, optional
Parameter for PageRank, default=0.85
max_iter : interger, optional
Maximum number of iterations in power method.
tol : float, optional
Error tolerance used to check convergence in power method iteration.
nstart : dictionary, optional
Starting value of PageRank iteration for each node.
Returns
-------
nodes : dictionary
Dictionary of nodes with value as PageRank
Examples
--------
>>> G=nx.DiGraph(nx.path_graph(4))
>>> pr=nx.pagerank(G,alpha=0.9)
Notes
-----
The eigenvector calculation is done by the power iteration method
and has no guarantee of convergence. The iteration will stop
after max_iter iterations or an error tolerance of
number_of_nodes(G)*tol has been reached.
The PageRank algorithm was designed for directed graphs but this
algorithm does not check if the input graph is directed and will
execute on undirected graphs.
For an overview see:
A. Langville and C. Meyer, "A survey of eigenvector methods of web
information retrieval." http://citeseer.ist.psu.edu/713792.html
"""
import networkx
if type(G) == networkx.MultiGraph or type(G) == networkx.MultiDiGraph:
raise Exception("pagerank() not defined for graphs with multiedges.")
# create a copy in (right) stochastic form
W=networkx.stochastic_graph(G)
# choose fixed starting vector if not given
if nstart is None:
x=dict.fromkeys(W,1.0/W.number_of_nodes())
else:
x=nstart
# normalize starting vector to 1
s=1.0/sum(x.values())
for k in x: x[k]*=s
nnodes=W.number_of_nodes()
# "dangling" nodes, no links out from them
out_degree=W.out_degree(with_labels=True)
# dangle=[n for n in W if sum(W[n].values())==0.0]
dangle=[n for n in W if out_degree[n]==0.0]
# pagerank power iteration: make up to max_iter iterations
for i in range(max_iter):
xlast=x
x=dict.fromkeys(xlast.keys(),0)
danglesum=alpha/nnodes*sum(xlast[n] for n in dangle)
teleportsum=(1.0-alpha)/nnodes*sum(xlast.values())
for n in x:
# this matrix multiply looks odd because it is
# doing a left multiply x^T=xlast^T*W
for nbr in W[n]:
x[nbr]+=alpha*xlast[n]*W[n][nbr]['weight']
x[n]+=danglesum+teleportsum
# normalize vector to 1
s=1.0/sum(x.values())
for n in x: x[n]*=s
# check convergence, l1 norm
err=sum([abs(x[n]-xlast[n]) for n in x])
if err < tol:
return x
raise NetworkXError("pagerank: power iteration failed to converge in %d iterations."%(i+1))
def google_matrix(G,alpha=0.85,nodelist=None):
import numpy
import networkx
M=networkx.to_numpy_matrix(G,nodelist=nodelist)
(n,m)=M.shape # should be square
# add constant to dangling nodes' row
dangling=numpy.where(M.sum(axis=1)==0)
for d in dangling[0]:
M[d]=1.0/n
# normalize
M=M/M.sum(axis=1)
# add "teleportation"
P=alpha*M+(1-alpha)*numpy.ones((n,n))/n
return P
def pagerank_numpy(G,alpha=0.85,max_iter=100,tol=1.0e-6,nodelist=None):
"""Return a NumPy array of the PageRank of G.
"""
import numpy
import networkx
M=networkx.google_matrix(G,alpha,nodelist)
(n,m)=M.shape # should be square
x=numpy.ones((n))/n
for i in range(max_iter):
xlast=x
x=numpy.dot(x,M)
# check convergence, l1 norm
err=numpy.abs(x-xlast).sum()
if err < n*tol:
return numpy.asarray(x).flatten()
raise NetworkXError("pagerank: power iteration failed to converge in %d iterations."%(i+1))
def pagerank_scipy(G,alpha=0.85,max_iter=100,tol=1.0e-6,nodelist=None):
"""Return a SciPy sparse array of the PageRank of G.
"""
import scipy.sparse
import networkx
M=networkx.to_scipy_sparse_matrix(G,nodelist=nodelist)
(n,m)=M.shape # should be square
S=scipy.array(M.sum(axis=1)).flatten()
index=scipy.where(S<>0)[0]
for i in index:
M[i,:]*=1.0/S[i]
x=scipy.ones((n))/n # initial guess
dangle=scipy.array(scipy.where(M.sum(axis=1)==0,1.0/n,0)).flatten()
for i in range(max_iter):
xlast=x
x=alpha*(x*M+scipy.dot(dangle,xlast))+(1-alpha)*xlast.sum()/n
# check convergence, l1 norm
err=scipy.absolute(x-xlast).sum()
if err < n*tol:
return x
raise NetworkXError("pagerank_scipy: power iteration failed to converge in %d iterations."%(i+1))
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