"""PageRank analysis of graph structure. """
# Copyright (C) 2004-2011 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/
import networkx as nx
from networkx.exception import NetworkXError
__author__ = """Aric Hagberg (hagberg@lanl.gov)"""
__all__ = ['pagerank','pagerank_numpy','pagerank_scipy','google_matrix']
[docs]def pagerank(G,alpha=0.85,personalization=None,
max_iter=100,tol=1.0e-8,nstart=None,weight='weight'):
"""Return the PageRank of the nodes in the graph.
PageRank computes a ranking of the nodes in the graph G based on
the structure of the incoming links. It was originally designed as
an algorithm to rank web pages.
Parameters
-----------
G : graph
A NetworkX graph
alpha : float, optional
Damping parameter for PageRank, default=0.85
personalization: dict, optional
The "personalization vector" consisting of a dictionary with a
key for every graph node and nonzero personalization value for each node.
max_iter : integer, optional
Maximum number of iterations in power method eigenvalue solver.
tol : float, optional
Error tolerance used to check convergence in power method solver.
nstart : dictionary, optional
Starting value of PageRank iteration for each node.
weight : key, optional
Edge data key to use as weight. If None weights are set to 1.
Returns
-------
pagerank : dictionary
Dictionary of nodes with PageRank as value
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 by converting each oriented edge in the
directed graph to two edges.
See Also
--------
pagerank_numpy, pagerank_scipy, google_matrix
References
----------
.. [1] A. Langville and C. Meyer,
"A survey of eigenvector methods of web information retrieval."
http://citeseer.ist.psu.edu/713792.html
.. [2] Page, Lawrence; Brin, Sergey; Motwani, Rajeev and Winograd, Terry,
The PageRank citation ranking: Bringing order to the Web. 1999
http://dbpubs.stanford.edu:8090/pub/showDoc.Fulltext?lang=en&doc=1999-66&format=pdf
"""
if type(G) == nx.MultiGraph or type(G) == nx.MultiDiGraph:
raise Exception("pagerank() not defined for graphs with multiedges.")
if len(G) == 0:
return {}
if not G.is_directed():
D=G.to_directed()
else:
D=G
# create a copy in (right) stochastic form
W=nx.stochastic_graph(D, weight=weight)
scale=1.0/W.number_of_nodes()
# choose fixed starting vector if not given
if nstart is None:
x=dict.fromkeys(W,scale)
else:
x=nstart
# normalize starting vector to 1
s=1.0/sum(x.values())
for k in x: x[k]*=s
# assign uniform personalization/teleportation vector if not given
if personalization is None:
p=dict.fromkeys(W,scale)
else:
p=personalization
# normalize starting vector to 1
s=1.0/sum(p.values())
for k in p:
p[k]*=s
if set(p)!=set(G):
raise NetworkXError('Personalization vector '
'must have a value for every node')
# "dangling" nodes, no links out from them
out_degree=W.out_degree()
dangle=[n for n in W if out_degree[n]==0.0]
i=0
while True: # power iteration: make up to max_iter iterations
xlast=x
x=dict.fromkeys(xlast.keys(),0)
danglesum=alpha*scale*sum(xlast[n] for n in dangle)
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+(1.0-alpha)*p[n]
# normalize vector
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:
break
if i>max_iter:
raise NetworkXError('pagerank: power iteration failed to converge'
'in %d iterations.'%(i+1))
i+=1
return x
[docs]def google_matrix(G, alpha=0.85, personalization=None,
nodelist=None, weight='weight'):
"""Return the Google matrix of the graph.
Parameters
-----------
G : graph
A NetworkX graph
alpha : float
The damping factor
personalization: dict, optional
The "personalization vector" consisting of a dictionary with a
key for every graph node and nonzero personalization value for each node.
nodelist : list, optional
The rows and columns are ordered according to the nodes in nodelist.
If nodelist is None, then the ordering is produced by G.nodes().
weight : key, optional
Edge data key to use as weight. If None weights are set to 1.
Returns
-------
A : NumPy matrix
Google matrix of the graph
See Also
--------
pagerank, pagerank_numpy, pagerank_scipy
"""
try:
import numpy as np
except ImportError:
raise ImportError(\
"google_matrix() requires NumPy: http://scipy.org/")
# choose ordering in matrix
if personalization is None: # use G.nodes() ordering
nodelist=G.nodes()
else: # use personalization "vector" ordering
nodelist=personalization.keys()
if set(nodelist)!=set(G):
raise NetworkXError('Personalization vector dictionary'
'must have a value for every node')
M=nx.to_numpy_matrix(G,nodelist=nodelist,weight=weight)
(n,m)=M.shape # should be square
if n == 0:
return M
# add constant to dangling nodes' row
dangling=np.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"/personalization
e=np.ones((n))
if personalization is not None:
v=np.array(list(personalization.values()),dtype=float)
else:
v=e
v=v/v.sum()
P=alpha*M+(1-alpha)*np.outer(e,v)
return P
[docs]def pagerank_numpy(G, alpha=0.85, personalization=None, weight='weight'):
"""Return the PageRank of the nodes in the graph.
PageRank computes a ranking of the nodes in the graph G based on
the structure of the incoming links. It was originally designed as
an algorithm to rank web pages.
Parameters
-----------
G : graph
A NetworkX graph
alpha : float, optional
Damping parameter for PageRank, default=0.85
personalization: dict, optional
The "personalization vector" consisting of a dictionary with a
key for every graph node and nonzero personalization value for each node.
weight : key, optional
Edge data key to use as weight. If None weights are set to 1.
Returns
-------
pagerank : dictionary
Dictionary of nodes with PageRank as value
Examples
--------
>>> G=nx.DiGraph(nx.path_graph(4))
>>> pr=nx.pagerank_numpy(G,alpha=0.9)
Notes
-----
The eigenvector calculation uses NumPy's interface to the LAPACK
eigenvalue solvers. This will be the fastest and most accurate
for small graphs.
This implementation works with Multi(Di)Graphs.
See Also
--------
pagerank, pagerank_scipy, google_matrix
References
----------
.. [1] A. Langville and C. Meyer,
"A survey of eigenvector methods of web information retrieval."
http://citeseer.ist.psu.edu/713792.html
.. [2] Page, Lawrence; Brin, Sergey; Motwani, Rajeev and Winograd, Terry,
The PageRank citation ranking: Bringing order to the Web. 1999
http://dbpubs.stanford.edu:8090/pub/showDoc.Fulltext?lang=en&doc=1999-66&format=pdf
"""
try:
import numpy as np
except ImportError:
raise ImportError("pagerank_numpy() requires NumPy: http://scipy.org/")
if len(G) == 0:
return {}
# choose ordering in matrix
if personalization is None: # use G.nodes() ordering
nodelist=G.nodes()
else: # use personalization "vector" ordering
nodelist=personalization.keys()
M=google_matrix(G, alpha, personalization=personalization,
nodelist=nodelist, weight=weight)
# use numpy LAPACK solver
eigenvalues,eigenvectors=np.linalg.eig(M.T)
ind=eigenvalues.argsort()
# eigenvector of largest eigenvalue at ind[-1], normalized
largest=np.array(eigenvectors[:,ind[-1]]).flatten().real
norm=float(largest.sum())
centrality=dict(zip(nodelist,map(float,largest/norm)))
return centrality
[docs]def pagerank_scipy(G, alpha=0.85, personalization=None,
max_iter=100, tol=1.0e-6, weight='weight'):
"""Return the PageRank of the nodes in the graph.
PageRank computes a ranking of the nodes in the graph G based on
the structure of the incoming links. It was originally designed as
an algorithm to rank web pages.
Parameters
-----------
G : graph
A NetworkX graph
alpha : float, optional
Damping parameter for PageRank, default=0.85
personalization: dict, optional
The "personalization vector" consisting of a dictionary with a
key for every graph node and nonzero personalization value for each node.
max_iter : integer, optional
Maximum number of iterations in power method eigenvalue solver.
tol : float, optional
Error tolerance used to check convergence in power method solver.
weight : key, optional
Edge data key to use as weight. If None weights are set to 1.
Returns
-------
pagerank : dictionary
Dictionary of nodes with PageRank as value
Examples
--------
>>> G=nx.DiGraph(nx.path_graph(4))
>>> pr=nx.pagerank_scipy(G,alpha=0.9)
Notes
-----
The eigenvector calculation uses power iteration with a SciPy
sparse matrix representation.
See Also
--------
pagerank, pagerank_numpy, google_matrix
References
----------
.. [1] A. Langville and C. Meyer,
"A survey of eigenvector methods of web information retrieval."
http://citeseer.ist.psu.edu/713792.html
.. [2] Page, Lawrence; Brin, Sergey; Motwani, Rajeev and Winograd, Terry,
The PageRank citation ranking: Bringing order to the Web. 1999
http://dbpubs.stanford.edu:8090/pub/showDoc.Fulltext?lang=en&doc=1999-66&format=pdf
"""
try:
import scipy.sparse
except ImportError:
raise ImportError("pagerank_scipy() requires SciPy: http://scipy.org/")
if len(G) == 0:
return {}
# choose ordering in matrix
if personalization is None: # use G.nodes() ordering
nodelist=G.nodes()
else: # use personalization "vector" ordering
nodelist=personalization.keys()
M=nx.to_scipy_sparse_matrix(G,nodelist=nodelist,weight=weight,dtype='f')
(n,m)=M.shape # should be square
S=scipy.array(M.sum(axis=1)).flatten()
# for i, j, v in zip( *scipy.sparse.find(M) ):
# M[i,j] = v / S[i]
S[S>0] = 1.0 / S[S>0]
Q = scipy.sparse.spdiags(S.T, 0, *M.shape, format='csr')
M = Q * M
x=scipy.ones((n))/n # initial guess
dangle=scipy.array(scipy.where(M.sum(axis=1)==0,1.0/n,0)).flatten()
# add "teleportation"/personalization
if personalization is not None:
v=scipy.array(list(personalization.values()),dtype=float)
v=v/v.sum()
else:
v=x
i=0
while i <= max_iter:
# power iteration: make up to max_iter iterations
xlast=x
x=alpha*(x*M+scipy.dot(dangle,xlast))+(1-alpha)*v
x=x/x.sum()
# check convergence, l1 norm
err=scipy.absolute(x-xlast).sum()
if err < n*tol:
return dict(zip(nodelist,map(float,x)))
i+=1
raise NetworkXError('pagerank_scipy: power iteration failed to converge'
'in %d iterations.'%(i+1))
# fixture for nose tests
def setup_module(module):
from nose import SkipTest
try:
import numpy
except:
raise SkipTest("NumPy not available")
try:
import scipy
except:
raise SkipTest("SciPy not available")