Source code for networkx.algorithms.link_analysis.hits_alg

"""Hubs and authorities analysis of graph structure.
"""
import networkx as nx

__all__ = ["hits"]


[docs]def hits(G, max_iter=100, tol=1.0e-8, nstart=None, normalized=True): """Returns 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 : integer, 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. Raises ------ PowerIterationFailedConvergence If the algorithm fails to converge to the specified tolerance within the specified number of iterations of the power iteration method. 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. """ import numpy as np import scipy as sp import scipy.sparse.linalg # call as sp.sparse.linalg if len(G) == 0: return {}, {} A = nx.adjacency_matrix(G, nodelist=list(G), dtype=float) if nstart is None: _, _, vt = sp.sparse.linalg.svds(A, k=1, maxiter=max_iter, tol=tol) else: nstart = np.array(list(nstart.values())) _, _, vt = sp.sparse.linalg.svds(A, k=1, v0=nstart, maxiter=max_iter, tol=tol) a = vt.flatten().real h = A @ a if normalized: h /= h.sum() a /= a.sum() hubs = dict(zip(G, map(float, h))) authorities = dict(zip(G, map(float, a))) return hubs, authorities
def _hits_python(G, max_iter=100, tol=1.0e-8, nstart=None, normalized=True): if isinstance(G, (nx.MultiGraph, 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 for _ in range(max_iter): # 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 else: raise nx.PowerIterationFailedConvergence(max_iter) 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 def _hits_numpy(G, normalized=True): """Returns 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) The `hubs` and `authorities` are given by the eigenvectors corresponding to the maximum eigenvalues of the hubs_matrix and the authority_matrix, respectively. The ``hubs`` and ``authority`` matrices are computed from the adjancency matrix: >>> adj_ary = nx.to_numpy_array(G) >>> hubs_matrix = adj_ary @ adj_ary.T >>> authority_matrix = adj_ary.T @ adj_ary `_hits_numpy` maps the eigenvector corresponding to the maximum eigenvalue of the respective matrices to the nodes in `G`: >>> from networkx.algorithms.link_analysis.hits_alg import _hits_numpy >>> hubs, authority = _hits_numpy(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. """ import numpy as np if len(G) == 0: return {}, {} adj_ary = nx.to_numpy_array(G) # Hub matrix H = adj_ary @ adj_ary.T e, ev = np.linalg.eig(H) h = ev[:, np.argmax(e)] # eigenvector corresponding to the maximum eigenvalue # Authority matrix A = adj_ary.T @ adj_ary e, ev = np.linalg.eig(A) a = ev[:, np.argmax(e)] # eigenvector corresponding to the maximum eigenvalue if normalized: h /= h.sum() a /= a.sum() else: h /= h.max() a /= a.max() hubs = dict(zip(G, map(float, h))) authorities = dict(zip(G, map(float, a))) return hubs, authorities def _hits_scipy(G, max_iter=100, tol=1.0e-6, nstart=None, normalized=True): """Returns 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 : integer, 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 -------- >>> from networkx.algorithms.link_analysis.hits_alg import _hits_scipy >>> G = nx.path_graph(4) >>> h, a = _hits_scipy(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. Raises ------ PowerIterationFailedConvergence If the algorithm fails to converge to the specified tolerance within the specified number of iterations of the power iteration method. 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. """ import numpy as np if len(G) == 0: return {}, {} A = nx.to_scipy_sparse_array(G, nodelist=list(G)) (n, _) = A.shape # should be square ATA = A.T @ A # authority matrix # choose fixed starting vector if not given if nstart is None: x = np.ones((n, 1)) / n else: x = np.array([nstart.get(n, 0) for n in list(G)], dtype=float) x /= x.sum() # power iteration on authority matrix i = 0 while True: xlast = x x = ATA @ x x /= x.max() # check convergence, l1 norm err = np.absolute(x - xlast).sum() if err < tol: break if i > max_iter: raise nx.PowerIterationFailedConvergence(max_iter) i += 1 a = x.flatten() h = A @ a if normalized: h /= h.sum() a /= a.sum() hubs = dict(zip(G, map(float, h))) authorities = dict(zip(G, map(float, a))) return hubs, authorities