NetworkX

Source code for networkx.algorithms.link_analysis.pagerank_alg

"""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")