Source code for networkx.algorithms.link_analysis.hits_alg

"""Hubs and authorities analysis of graph structure.
"""
#    Copyright (C) 2008-2012 by
#    Aric Hagberg <hagberg@lanl.gov>
#    Dan Schult <dschult@colgate.edu>
#    Pieter Swart <swart@lanl.gov>
#    All rights reserved.
#    BSD license.
#    NetworkX:http://networkx.github.io/
import networkx as nx
from networkx.exception import NetworkXError
__author__ = """Aric Hagberg (hagberg@lanl.gov)"""
__all__ = ['hits','hits_numpy','hits_scipy','authority_matrix','hub_matrix']

[docs]def hits(G,max_iter=100,tol=1.0e-8,nstart=None,normalized=True): """Return HITS hubs and authorities values for nodes. The HITS algorithm computes two numbers for a node. Authorities estimates the node value based on the incoming links. Hubs estimates the node value based on outgoing links. Parameters ---------- G : graph A NetworkX graph 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 each node for power method iteration. normalized : bool (default=True) Normalize results by the sum of all of the values. Returns ------- (hubs,authorities) : two-tuple of dictionaries Two dictionaries keyed by node containing the hub and authority values. Examples -------- >>> G=nx.path_graph(4) >>> h,a=nx.hits(G) 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 HITS algorithm was designed for directed graphs but this algorithm does not check if the input graph is directed and will execute on undirected graphs. References ---------- .. [1] A. Langville and C. Meyer, "A survey of eigenvector methods of web information retrieval." http://citeseer.ist.psu.edu/713792.html .. [2] Jon Kleinberg, Authoritative sources in a hyperlinked environment Journal of the ACM 46 (5): 604-32, 1999. doi:10.1145/324133.324140. http://www.cs.cornell.edu/home/kleinber/auth.pdf. """ if type(G) == nx.MultiGraph or type(G) == nx.MultiDiGraph: raise Exception("hits() not defined for graphs with multiedges.") if len(G) == 0: return {},{} # choose fixed starting vector if not given if nstart is None: h=dict.fromkeys(G,1.0/G.number_of_nodes()) else: h=nstart # normalize starting vector s=1.0/sum(h.values()) for k in h: h[k]*=s i=0 while True: # power iteration: make up to max_iter iterations hlast=h h=dict.fromkeys(hlast.keys(),0) a=dict.fromkeys(hlast.keys(),0) # this "matrix multiply" looks odd because it is # doing a left multiply a^T=hlast^T*G for n in h: for nbr in G[n]: a[nbr]+=hlast[n]*G[n][nbr].get('weight',1) # now multiply h=Ga for n in h: for nbr in G[n]: h[n]+=a[nbr]*G[n][nbr].get('weight',1) # normalize vector s=1.0/max(h.values()) for n in h: h[n]*=s # normalize vector s=1.0/max(a.values()) for n in a: a[n]*=s # check convergence, l1 norm err=sum([abs(h[n]-hlast[n]) for n in h]) if err < tol: break if i>max_iter: raise NetworkXError(\ "HITS: power iteration failed to converge in %d iterations."%(i+1)) i+=1 if normalized: s = 1.0/sum(a.values()) for n in a: a[n] *= s s = 1.0/sum(h.values()) for n in h: h[n] *= s return h,a
[docs]def authority_matrix(G,nodelist=None): """Return the HITS authority matrix.""" M=nx.to_numpy_matrix(G,nodelist=nodelist) return M.T*M
[docs]def hub_matrix(G,nodelist=None): """Return the HITS hub matrix.""" M=nx.to_numpy_matrix(G,nodelist=nodelist) return M*M.T
[docs]def hits_numpy(G,normalized=True): """Return HITS hubs and authorities values for nodes. The HITS algorithm computes two numbers for a node. Authorities estimates the node value based on the incoming links. Hubs estimates the node value based on outgoing links. Parameters ---------- G : graph A NetworkX graph normalized : bool (default=True) Normalize results by the sum of all of the values. Returns ------- (hubs,authorities) : two-tuple of dictionaries Two dictionaries keyed by node containing the hub and authority values. Examples -------- >>> G=nx.path_graph(4) >>> h,a=nx.hits(G) Notes ----- The eigenvector calculation uses NumPy's interface to LAPACK. The HITS algorithm was designed for directed graphs but this algorithm does not check if the input graph is directed and will execute on undirected graphs. References ---------- .. [1] A. Langville and C. Meyer, "A survey of eigenvector methods of web information retrieval." http://citeseer.ist.psu.edu/713792.html .. [2] Jon Kleinberg, Authoritative sources in a hyperlinked environment Journal of the ACM 46 (5): 604-32, 1999. doi:10.1145/324133.324140. http://www.cs.cornell.edu/home/kleinber/auth.pdf. """ try: import numpy as np except ImportError: raise ImportError(\ "hits_numpy() requires NumPy: http://scipy.org/") if len(G) == 0: return {},{} H=nx.hub_matrix(G,G.nodes()) e,ev=np.linalg.eig(H) m=e.argsort()[-1] # index of maximum eigenvalue h=np.array(ev[:,m]).flatten() A=nx.authority_matrix(G,G.nodes()) e,ev=np.linalg.eig(A) m=e.argsort()[-1] # index of maximum eigenvalue a=np.array(ev[:,m]).flatten() if normalized: h = h/h.sum() a = a/a.sum() else: h = h/h.max() a = a/a.max() hubs=dict(zip(G.nodes(),map(float,h))) authorities=dict(zip(G.nodes(),map(float,a))) return hubs,authorities
[docs]def hits_scipy(G,max_iter=100,tol=1.0e-6,normalized=True): """Return HITS hubs and authorities values for nodes. The HITS algorithm computes two numbers for a node. Authorities estimates the node value based on the incoming links. Hubs estimates the node value based on outgoing links. Parameters ---------- G : graph A NetworkX graph 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 each node for power method iteration. normalized : bool (default=True) Normalize results by the sum of all of the values. Returns ------- (hubs,authorities) : two-tuple of dictionaries Two dictionaries keyed by node containing the hub and authority values. Examples -------- >>> G=nx.path_graph(4) >>> h,a=nx.hits(G) Notes ----- This implementation uses SciPy sparse matrices. 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 HITS algorithm was designed for directed graphs but this algorithm does not check if the input graph is directed and will execute on undirected graphs. References ---------- .. [1] A. Langville and C. Meyer, "A survey of eigenvector methods of web information retrieval." http://citeseer.ist.psu.edu/713792.html .. [2] Jon Kleinberg, Authoritative sources in a hyperlinked environment Journal of the ACM 46 (5): 604-632, 1999. doi:10.1145/324133.324140. http://www.cs.cornell.edu/home/kleinber/auth.pdf. """ try: import scipy.sparse import numpy as np except ImportError: raise ImportError(\ "hits_scipy() requires SciPy: http://scipy.org/") if len(G) == 0: return {},{} M=nx.to_scipy_sparse_matrix(G,nodelist=G.nodes()) (n,m)=M.shape # should be square A=M.T*M # authority matrix x=scipy.ones((n,1))/n # initial guess # power iteration on authority matrix i=0 while True: xlast=x x=A*x x=x/x.max() # check convergence, l1 norm err=scipy.absolute(x-xlast).sum() if err < tol: break if i>max_iter: raise NetworkXError(\ "HITS: power iteration failed to converge in %d iterations."%(i+1)) i+=1 a=np.asarray(x).flatten() # h=M*a h=np.asarray(M*a).flatten() if normalized: h = h/h.sum() a = a/a.sum() hubs=dict(zip(G.nodes(),map(float,h))) authorities=dict(zip(G.nodes(),map(float,a))) return hubs,authorities
# 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")