#! /usr/bin/env python # -*- coding: utf-8 -*- # # graph_tool -- a general graph manipulation python module # # Copyright (C) 2006-2018 Tiago de Paula Peixoto # # This program is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 3 of the License, or # (at your option) any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # You should have received a copy of the GNU General Public License # along with this program. If not, see . """ graph_tool.centrality - Centrality measures ----------------------------------------------- This module includes centrality-related algorithms. Summary +++++++ .. autosummary:: :nosignatures: pagerank betweenness central_point_dominance closeness eigenvector katz hits eigentrust trust_transitivity Contents ++++++++ """ from __future__ import division, absolute_import, print_function from .. dl_import import dl_import dl_import("from . import libgraph_tool_centrality") from .. import _prop, ungroup_vector_property, Vector_size_t from .. topology import shortest_distance import sys import numpy import numpy.linalg __all__ = ["pagerank", "betweenness", "central_point_dominance", "closeness", "eigentrust", "eigenvector", "katz", "hits", "trust_transitivity"] def pagerank(g, damping=0.85, pers=None, weight=None, prop=None, epsilon=1e-6, max_iter=None, ret_iter=False): r"""Calculate the PageRank of each vertex. Parameters ---------- g : :class:~graph_tool.Graph Graph to be used. damping : float, optional (default: 0.85) Damping factor. pers : :class:~graph_tool.VertexPropertyMap, optional (default: None) Personalization vector. If omitted, a constant value of :math:1/N will be used. weight : :class:~graph_tool.EdgePropertyMap, optional (default: None) Edge weights. If omitted, a constant value of 1 will be used. prop : :class:~graph_tool.VertexPropertyMap, optional (default: None) Vertex property map to store the PageRank values. If supplied, it will be used uninitialized. epsilon : float, optional (default: 1e-6) Convergence condition. The iteration will stop if the total delta of all vertices are below this value. max_iter : int, optional (default: None) If supplied, this will limit the total number of iterations. ret_iter : bool, optional (default: False) If true, the total number of iterations is also returned. Returns ------- pagerank : :class:~graph_tool.VertexPropertyMap A vertex property map containing the PageRank values. See Also -------- betweenness: betweenness centrality eigentrust: eigentrust centrality eigenvector: eigenvector centrality hits: authority and hub centralities trust_transitivity: pervasive trust transitivity Notes ----- The value of PageRank [pagerank-wikipedia]_ of vertex v, :math:PR(v), is given iteratively by the relation: .. math:: PR(v) = \frac{1-d}{N} + d \sum_{u \in \Gamma^{-}(v)} \frac{PR (u)}{d^{+}(u)} where :math:\Gamma^{-}(v) are the in-neighbors of v, :math:d^{+}(u) is the out-degree of u, and d is a damping factor. If a personalization property :math:p(v) is given, the definition becomes: .. math:: PR(v) = (1-d)p(v) + d \sum_{u \in \Gamma^{-}(v)} \frac{PR (u)}{d^{+}(u)} If edge weights are also given, the equation is then generalized to: .. math:: PR(v) = (1-d)p(v) + d \sum_{u \in \Gamma^{-}(v)} \frac{PR (u) w_{u\to v}}{d^{+}(u)} where :math:d^{+}(u)=\sum_{y}A_{u,y}w_{u\to y} is redefined to be the sum of the weights of the out-going edges from u. If a node has out-degree zero, it is assumed to connect to every other node with a weight proportional to :math:p(v) or a constant if no personalization is given. The implemented algorithm progressively iterates the above equations, until it no longer changes, according to the parameter epsilon. It has a topology-dependent running time. If enabled during compilation, this algorithm runs in parallel. Examples -------- .. testsetup:: pagerank import matplotlib .. doctest:: pagerank >>> g = gt.collection.data["polblogs"] >>> g = gt.GraphView(g, vfilt=gt.label_largest_component(g)) >>> pr = gt.pagerank(g) >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=pr, ... vertex_size=gt.prop_to_size(pr, mi=5, ma=15), ... vorder=pr, vcmap=matplotlib.cm.gist_heat, ... output="polblogs_pr.pdf") <...> .. testcode:: pagerank :hide: gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=pr, vertex_size=gt.prop_to_size(pr, mi=5, ma=15), vorder=pr, vcmap=matplotlib.cm.gist_heat, output="polblogs_pr.png") .. figure:: polblogs_pr.* :align: center PageRank values of the a political blogs network of [adamic-polblogs]_. Now with a personalization vector, and edge weights: .. doctest:: pagerank >>> d = g.degree_property_map("total") >>> periphery = d.a <= 2 >>> p = g.new_vertex_property("double") >>> p.a[periphery] = 100 >>> pr = gt.pagerank(g, pers=p) >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=pr, ... vertex_size=gt.prop_to_size(pr, mi=5, ma=15), ... vorder=pr, vcmap=matplotlib.cm.gist_heat, ... output="polblogs_pr_pers.pdf") <...> .. testcode:: pagerank :hide: gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=pr, vertex_size=gt.prop_to_size(pr, mi=5, ma=15), vcmap=matplotlib.cm.gist_heat, vorder=pr, output="polblogs_pr_pers.png") .. figure:: polblogs_pr_pers.* :align: center Personalized PageRank values of the a political blogs network of [adamic-polblogs]_, where vertices with very low degree are given artificially high scores. References ---------- .. [pagerank-wikipedia] http://en.wikipedia.org/wiki/Pagerank .. [lawrence-pagerank-1998] P. Lawrence, B. Sergey, M. Rajeev, W. Terry, "The pagerank citation ranking: Bringing order to the web", Technical report, Stanford University, 1998 .. [Langville-survey-2005] A. N. Langville, C. D. Meyer, "A Survey of Eigenvector Methods for Web Information Retrieval", SIAM Review, vol. 47, no. 1, pp. 135-161, 2005, :DOI:10.1137/S0036144503424786 .. [adamic-polblogs] L. A. Adamic and N. Glance, "The political blogosphere and the 2004 US Election", in Proceedings of the WWW-2005 Workshop on the Weblogging Ecosystem (2005). :DOI:10.1145/1134271.1134277 """ if max_iter is None: max_iter = 0 if prop is None: prop = g.new_vertex_property("double") N = len(prop.fa) prop.fa = pers.fa[:N] if pers is not None else 1. / g.num_vertices() ic = libgraph_tool_centrality.\ get_pagerank(g._Graph__graph, _prop("v", g, prop), _prop("v", g, pers), _prop("e", g, weight), damping, epsilon, max_iter) if ret_iter: return prop, ic else: return prop def betweenness(g, pivots=None, vprop=None, eprop=None, weight=None, norm=True): r"""Calculate the betweenness centrality for each vertex and edge. Parameters ---------- g : :class:~graph_tool.Graph Graph to be used. pivots : list or :class:~numpy.ndarray, optional (default: None) If provided, the betweenness will be estimated using the vertices in this list as pivots. If the list contains all nodes (the default) the algorithm will be exact, and if the vertices are randomly chosen the result will be an unbiased estimator. vprop : :class:~graph_tool.VertexPropertyMap, optional (default: None) Vertex property map to store the vertex betweenness values. eprop : :class:~graph_tool.EdgePropertyMap, optional (default: None) Edge property map to store the edge betweenness values. weight : :class:~graph_tool.EdgePropertyMap, optional (default: None) Edge property map corresponding to the weight value of each edge. norm : bool, optional (default: True) Whether or not the betweenness values should be normalized. Returns ------- vertex_betweenness : A vertex property map with the vertex betweenness values. edge_betweenness : An edge property map with the edge betweenness values. See Also -------- central_point_dominance: central point dominance of the graph pagerank: PageRank centrality eigentrust: eigentrust centrality eigenvector: eigenvector centrality hits: authority and hub centralities trust_transitivity: pervasive trust transitivity Notes ----- Betweenness centrality of a vertex :math:C_B(v) is defined as, .. math:: C_B(v)= \sum_{s \neq v \neq t \in V \atop s \neq t} \frac{\sigma_{st}(v)}{\sigma_{st}} where :math:\sigma_{st} is the number of shortest paths from s to t, and :math:\sigma_{st}(v) is the number of shortest paths from s to t that pass through a vertex :math:v. This may be normalised by dividing through the number of pairs of vertices not including v, which is :math:(n-1)(n-2)/2, for undirected graphs, or :math:(n-1)(n-2) for directed ones. The algorithm used here is defined in [brandes-faster-2001]_, and has a complexity of :math:O(VE) for unweighted graphs and :math:O(VE + V(V+E)\log V) for weighted graphs. The space complexity is :math:O(VE). If the pivots parameter is given, the complexity will be instead :math:O(PE) for unweighted graphs and :math:O(PE + P(V+E)\log V) for weighted graphs, where :math:P is the number of pivot vertices. If enabled during compilation, this algorithm runs in parallel. Examples -------- .. testsetup:: betweenness import matplotlib .. doctest:: betweenness >>> g = gt.collection.data["polblogs"] >>> g = gt.GraphView(g, vfilt=gt.label_largest_component(g)) >>> vp, ep = gt.betweenness(g) >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=vp, ... vertex_size=gt.prop_to_size(vp, mi=5, ma=15), ... edge_pen_width=gt.prop_to_size(ep, mi=0.5, ma=5), ... vcmap=matplotlib.cm.gist_heat, ... vorder=vp, output="polblogs_betweenness.pdf") <...> .. testcode:: betweenness :hide: gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=vp, vertex_size=gt.prop_to_size(vp, mi=5, ma=15), edge_pen_width=gt.prop_to_size(ep, mi=0.5, ma=5), vcmap=matplotlib.cm.gist_heat, vorder=vp, output="polblogs_betweenness.png") .. figure:: polblogs_betweenness.* :align: center Betweenness values of the a political blogs network of [adamic-polblogs]_. References ---------- .. [betweenness-wikipedia] http://en.wikipedia.org/wiki/Centrality#Betweenness_centrality .. [brandes-faster-2001] U. Brandes, "A faster algorithm for betweenness centrality", Journal of Mathematical Sociology, 2001, :doi:10.1080/0022250X.2001.9990249 .. [brandes-centrality-2007] U. Brandes, C. Pich, "Centrality estimation in large networks", Int. J. Bifurcation Chaos 17, 2303 (2007). :DOI:10.1142/S0218127407018403 .. [adamic-polblogs] L. A. Adamic and N. Glance, "The political blogosphere and the 2004 US Election", in Proceedings of the WWW-2005 Workshop on the Weblogging Ecosystem (2005). :DOI:10.1145/1134271.1134277 """ if vprop is None: vprop = g.new_vertex_property("double") if eprop is None: eprop = g.new_edge_property("double") if weight is not None and weight.value_type() != eprop.value_type(): nw = g.new_edge_property(eprop.value_type()) g.copy_property(weight, nw) weight = nw if pivots is not None: pivots = numpy.asarray(pivots, dtype="uint64") else: pivots = g.get_vertices() vpivots = Vector_size_t(len(pivots)) vpivots.a = pivots libgraph_tool_centrality.\ get_betweenness(g._Graph__graph, vpivots, _prop("e", g, weight), _prop("e", g, eprop), _prop("v", g, vprop), norm) return vprop, eprop def closeness(g, weight=None, source=None, vprop=None, norm=True, harmonic=False): r""" Calculate the closeness centrality for each vertex. Parameters ---------- g : :class:~graph_tool.Graph Graph to be used. weight : :class:~graph_tool.EdgePropertyMap, optional (default: None) Edge property map corresponding to the weight value of each edge. source : :class:~graph_tool.Vertex, optional (default: None) If specified, the centrality is computed for this vertex alone. vprop : :class:~graph_tool.VertexPropertyMap, optional (default: None) Vertex property map to store the vertex centrality values. norm : bool, optional (default: True) Whether or not the centrality values should be normalized. harmonic : bool, optional (default: False) If true, the sum of the inverse of the distances will be computed, instead of the inverse of the sum. Returns ------- vertex_closeness : :class:~graph_tool.VertexPropertyMap A vertex property map with the vertex closeness values. See Also -------- central_point_dominance: central point dominance of the graph pagerank: PageRank centrality eigentrust: eigentrust centrality eigenvector: eigenvector centrality hits: authority and hub centralities trust_transitivity: pervasive trust transitivity Notes ----- The closeness centrality of a vertex :math:i is defined as, .. math:: c_i = \frac{1}{\sum_j d_{ij}} where :math:d_{ij} is the (possibly directed and/or weighted) distance from :math:i to :math:j. In case there is no path between the two vertices, here the distance is taken to be zero. If harmonic == True, the definition becomes .. math:: c_i = \sum_j\frac{1}{d_{ij}}, but now, in case there is no path between the two vertices, we take :math:d_{ij} \to\infty such that :math:1/d_{ij}=0. If norm == True, the values of :math:c_i are normalized by :math:n_i-1 where :math:n_i is the size of the (out-) component of :math:i. If harmonic == True, they are instead simply normalized by :math:V-1. The algorithm complexity of :math:O(V(V + E)) for unweighted graphs and :math:O(V(V+E) \log V) for weighted graphs. If the option source is specified, this drops to :math:O(V + E) and :math:O((V+E)\log V) respectively. If enabled during compilation, this algorithm runs in parallel. Examples -------- .. testsetup:: closeness import matplotlib .. doctest:: closeness >>> g = gt.collection.data["polblogs"] >>> g = gt.GraphView(g, vfilt=gt.label_largest_component(g)) >>> c = gt.closeness(g) >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=c, ... vertex_size=gt.prop_to_size(c, mi=5, ma=15), ... vcmap=matplotlib.cm.gist_heat, ... vorder=c, output="polblogs_closeness.pdf") <...> .. testcode:: closeness :hide: gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=c, vertex_size=gt.prop_to_size(c, mi=5, ma=15), vcmap=matplotlib.cm.gist_heat, vorder=c, output="polblogs_closeness.png") .. figure:: polblogs_closeness.* :align: center Closeness values of the a political blogs network of [adamic-polblogs]_. References ---------- .. [closeness-wikipedia] https://en.wikipedia.org/wiki/Closeness_centrality .. [opsahl-node-2010] Opsahl, T., Agneessens, F., Skvoretz, J., "Node centrality in weighted networks: Generalizing degree and shortest paths". Social Networks 32, 245-251, 2010 :DOI:10.1016/j.socnet.2010.03.006 .. [adamic-polblogs] L. A. Adamic and N. Glance, "The political blogosphere and the 2004 US Election", in Proceedings of the WWW-2005 Workshop on the Weblogging Ecosystem (2005). :DOI:10.1145/1134271.1134277 """ if source is None: if vprop is None: vprop = g.new_vertex_property("double") libgraph_tool_centrality.\ closeness(g._Graph__graph, _prop("e", g, weight), _prop("v", g, vprop), harmonic, norm) return vprop else: max_dist = g.num_vertices() + 1 dist = shortest_distance(g, source=source, weights=weight, max_dist=max_dist) dists = dist.fa[(dist.fa < max_dist) * (dist.fa > 0)] if harmonic: c = (1. / dists).sum() if norm: c /= g.num_vertices() - 1 else: c = 1. / dists.sum() if norm: c *= len(dists) return c def central_point_dominance(g, betweenness): r"""Calculate the central point dominance of the graph, given the betweenness centrality of each vertex. Parameters ---------- g : :class:~graph_tool.Graph Graph to be used. betweenness : :class:~graph_tool.VertexPropertyMap Vertex property map with the betweenness centrality values. The values must be normalized. Returns ------- cp : float The central point dominance. See Also -------- betweenness: betweenness centrality Notes ----- Let :math:v^* be the vertex with the largest relative betweenness centrality; then, the central point dominance [freeman-set-1977]_ is defined as: .. math:: C'_B = \frac{1}{|V|-1} \sum_{v} C_B(v^*) - C_B(v) where :math:C_B(v) is the normalized betweenness centrality of vertex v. The value of :math:C_B lies in the range [0,1]. The algorithm has a complexity of :math:O(V). Examples -------- >>> g = gt.collection.data["polblogs"] >>> g = gt.GraphView(g, vfilt=gt.label_largest_component(g)) >>> vp, ep = gt.betweenness(g) >>> print(gt.central_point_dominance(g, vp)) 0.11610685614... References ---------- .. [freeman-set-1977] Linton C. Freeman, "A Set of Measures of Centrality Based on Betweenness", Sociometry, Vol. 40, No. 1, pp. 35-41, 1977, :doi:10.2307/3033543 """ return libgraph_tool_centrality.\ get_central_point_dominance(g._Graph__graph, _prop("v", g, betweenness)) def eigenvector(g, weight=None, vprop=None, epsilon=1e-6, max_iter=None): r""" Calculate the eigenvector centrality of each vertex in the graph, as well as the largest eigenvalue. Parameters ---------- g : :class:~graph_tool.Graph Graph to be used. weight : :class:~graph_tool.EdgePropertyMap (optional, default: None) Edge property map with the edge weights. vprop : :class:~graph_tool.VertexPropertyMap, optional (default: None) Vertex property map where the values of eigenvector must be stored. If provided, it will be used uninitialized. epsilon : float, optional (default: 1e-6) Convergence condition. The iteration will stop if the total delta of all vertices are below this value. max_iter : int, optional (default: None) If supplied, this will limit the total number of iterations. Returns ------- eigenvalue : float The largest eigenvalue of the (weighted) adjacency matrix. eigenvector : :class:~graph_tool.VertexPropertyMap A vertex property map containing the eigenvector values. See Also -------- betweenness: betweenness centrality pagerank: PageRank centrality hits: authority and hub centralities trust_transitivity: pervasive trust transitivity Notes ----- The eigenvector centrality :math:\mathbf{x} is the eigenvector of the (weighted) adjacency matrix with the largest eigenvalue :math:\lambda, i.e. it is the solution of .. math:: \mathbf{A}\mathbf{x} = \lambda\mathbf{x}, where :math:\mathbf{A} is the (weighted) adjacency matrix and :math:\lambda is the largest eigenvalue. The algorithm uses the power method which has a topology-dependent complexity of :math:O\left(N\times\frac{-\log\epsilon}{\log|\lambda_1/\lambda_2|}\right), where :math:N is the number of vertices, :math:\epsilon is the epsilon parameter, and :math:\lambda_1 and :math:\lambda_2 are the largest and second largest eigenvalues of the (weighted) adjacency matrix, respectively. If enabled during compilation, this algorithm runs in parallel. Examples -------- .. testsetup:: eigenvector np.random.seed(42) import matplotlib .. doctest:: eigenvector >>> g = gt.collection.data["polblogs"] >>> g = gt.GraphView(g, vfilt=gt.label_largest_component(g)) >>> w = g.new_edge_property("double") >>> w.a = np.random.random(len(w.a)) * 42 >>> ee, x = gt.eigenvector(g, w) >>> print(ee) 724.302745... >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=x, ... vertex_size=gt.prop_to_size(x, mi=5, ma=15), ... vcmap=matplotlib.cm.gist_heat, ... vorder=x, output="polblogs_eigenvector.pdf") <...> .. testcode:: eigenvector :hide: gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=x, vertex_size=gt.prop_to_size(x, mi=5, ma=15), vcmap=matplotlib.cm.gist_heat, vorder=x, output="polblogs_eigenvector.png") .. figure:: polblogs_eigenvector.* :align: center Eigenvector values of the a political blogs network of [adamic-polblogs]_, with random weights attributed to the edges. References ---------- .. [eigenvector-centrality] http://en.wikipedia.org/wiki/Centrality#Eigenvector_centrality .. [power-method] http://en.wikipedia.org/wiki/Power_iteration .. [langville-survey-2005] A. N. Langville, C. D. Meyer, "A Survey of Eigenvector Methods for Web Information Retrieval", SIAM Review, vol. 47, no. 1, pp. 135-161, 2005, :DOI:10.1137/S0036144503424786 .. [adamic-polblogs] L. A. Adamic and N. Glance, "The political blogosphere and the 2004 US Election", in Proceedings of the WWW-2005 Workshop on the Weblogging Ecosystem (2005). :DOI:10.1145/1134271.1134277 """ if vprop is None: vprop = g.new_vertex_property("double") vprop.fa = 1. / g.num_vertices() if max_iter is None: max_iter = 0 ee = libgraph_tool_centrality.\ get_eigenvector(g._Graph__graph, _prop("e", g, weight), _prop("v", g, vprop), epsilon, max_iter) return ee, vprop def katz(g, alpha=0.01, beta=None, weight=None, vprop=None, epsilon=1e-6, max_iter=None, norm=True): r""" Calculate the Katz centrality of each vertex in the graph. Parameters ---------- g : :class:~graph_tool.Graph Graph to be used. weight : :class:~graph_tool.EdgePropertyMap (optional, default: None) Edge property map with the edge weights. alpha : float, optional (default: 0.01) Free parameter :math:\alpha. This must be smaller than the inverse of the largest eigenvalue of the adjacency matrix. beta : :class:~graph_tool.VertexPropertyMap, optional (default: None) Vertex property map where the local personalization values. If not provided, the global value of 1 will be used. vprop : :class:~graph_tool.VertexPropertyMap, optional (default: None) Vertex property map where the values of eigenvector must be stored. If provided, it will be used uninitialized. epsilon : float, optional (default: 1e-6) Convergence condition. The iteration will stop if the total delta of all vertices are below this value. max_iter : int, optional (default: None) If supplied, this will limit the total number of iterations. norm : bool, optional (default: True) Whether or not the centrality values should be normalized. Returns ------- centrality : :class:~graph_tool.VertexPropertyMap A vertex property map containing the Katz centrality values. See Also -------- betweenness: betweenness centrality pagerank: PageRank centrality eigenvector: eigenvector centrality hits: authority and hub centralities trust_transitivity: pervasive trust transitivity Notes ----- The Katz centrality :math:\mathbf{x} is the solution of the nonhomogeneous linear system .. math:: \mathbf{x} = \alpha\mathbf{A}\mathbf{x} + \mathbf{\beta}, where :math:\mathbf{A} is the (weighted) adjacency matrix and :math:\mathbf{\beta} is the personalization vector (if not supplied, :math:\mathbf{\beta} = \mathbf{1} is assumed). The algorithm uses successive iterations of the equation above, which has a topology-dependent convergence complexity. If enabled during compilation, this algorithm runs in parallel. Examples -------- .. testsetup:: katz np.random.seed(42) import matplotlib .. doctest:: katz >>> g = gt.collection.data["polblogs"] >>> g = gt.GraphView(g, vfilt=gt.label_largest_component(g)) >>> w = g.new_edge_property("double") >>> w.a = np.random.random(len(w.a)) >>> x = gt.katz(g, weight=w) >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=x, ... vertex_size=gt.prop_to_size(x, mi=5, ma=15), ... vcmap=matplotlib.cm.gist_heat, ... vorder=x, output="polblogs_katz.pdf") <...> .. testcode:: katz :hide: gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=x, vertex_size=gt.prop_to_size(x, mi=5, ma=15), vcmap=matplotlib.cm.gist_heat, vorder=x, output="polblogs_katz.png") .. figure:: polblogs_katz.* :align: center Katz centrality values of the a political blogs network of [adamic-polblogs]_, with random weights attributed to the edges. References ---------- .. [katz-centrality] http://en.wikipedia.org/wiki/Katz_centrality .. [katz-new] L. Katz, "A new status index derived from sociometric analysis", Psychometrika 18, Number 1, 39-43, 1953, :DOI:10.1007/BF02289026 .. [adamic-polblogs] L. A. Adamic and N. Glance, "The political blogosphere and the 2004 US Election", in Proceedings of the WWW-2005 Workshop on the Weblogging Ecosystem (2005). :DOI:10.1145/1134271.1134277 """ if vprop is None: vprop = g.new_vertex_property("double") if max_iter is None: max_iter = 0 libgraph_tool_centrality.\ get_katz(g._Graph__graph, _prop("e", g, weight), _prop("v", g, vprop), _prop("v", g, beta), float(alpha), epsilon, max_iter) if norm: vprop.fa = vprop.fa / numpy.linalg.norm(vprop.fa) return vprop def hits(g, weight=None, xprop=None, yprop=None, epsilon=1e-6, max_iter=None): r""" Calculate the authority and hub centralities of each vertex in the graph. Parameters ---------- g : :class:~graph_tool.Graph Graph to be used. weight : :class:~graph_tool.EdgePropertyMap (optional, default: None) Edge property map with the edge weights. xprop : :class:~graph_tool.VertexPropertyMap, optional (default: None) Vertex property map where the authority centrality must be stored. yprop : :class:~graph_tool.VertexPropertyMap, optional (default: None) Vertex property map where the hub centrality must be stored. epsilon : float, optional (default: 1e-6) Convergence condition. The iteration will stop if the total delta of all vertices are below this value. max_iter : int, optional (default: None) If supplied, this will limit the total number of iterations. Returns ------- eig : float The largest eigenvalue of the cocitation matrix. x : :class:~graph_tool.VertexPropertyMap A vertex property map containing the authority centrality values. y : :class:~graph_tool.VertexPropertyMap A vertex property map containing the hub centrality values. See Also -------- betweenness: betweenness centrality eigenvector: eigenvector centrality pagerank: PageRank centrality trust_transitivity: pervasive trust transitivity Notes ----- The Hyperlink-Induced Topic Search (HITS) centrality assigns hub (:math:\mathbf{y}) and authority (:math:\mathbf{x}) centralities to the vertices, following: .. math:: \begin{align} \mathbf{x} &= \alpha\mathbf{A}\mathbf{y} \\ \mathbf{y} &= \beta\mathbf{A}^T\mathbf{x} \end{align} where :math:\mathbf{A} is the (weighted) adjacency matrix and :math:\lambda = 1/(\alpha\beta) is the largest eigenvalue of the cocitation matrix, :math:\mathbf{A}\mathbf{A}^T. (Without loss of generality, we set :math:\beta=1 in the algorithm.) The algorithm uses the power method which has a topology-dependent complexity of :math:O\left(N\times\frac{-\log\epsilon}{\log|\lambda_1/\lambda_2|}\right), where :math:N is the number of vertices, :math:\epsilon is the epsilon parameter, and :math:\lambda_1 and :math:\lambda_2 are the largest and second largest eigenvalues of the (weighted) cocitation matrix, respectively. If enabled during compilation, this algorithm runs in parallel. Examples -------- .. testsetup:: hits import matplotlib .. doctest:: hits >>> g = gt.collection.data["polblogs"] >>> g = gt.GraphView(g, vfilt=gt.label_largest_component(g)) >>> ee, x, y = gt.hits(g) >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=x, ... vertex_size=gt.prop_to_size(x, mi=5, ma=15), ... vcmap=matplotlib.cm.gist_heat, ... vorder=x, output="polblogs_hits_auths.pdf") <...> >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=y, ... vertex_size=gt.prop_to_size(y, mi=5, ma=15), ... vcmap=matplotlib.cm.gist_heat, ... vorder=y, output="polblogs_hits_hubs.pdf") <...> .. testcode:: hits :hide: gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=x, vertex_size=gt.prop_to_size(x, mi=5, ma=15), vcmap=matplotlib.cm.gist_heat, vorder=x, output="polblogs_hits_auths.png") gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=y, vertex_size=gt.prop_to_size(y, mi=5, ma=15), vcmap=matplotlib.cm.gist_heat, vorder=y, output="polblogs_hits_hubs.png") .. figure:: polblogs_hits_auths.* :align: center HITS authority values of the a political blogs network of [adamic-polblogs]_. .. figure:: polblogs_hits_hubs.* :align: center HITS hub values of the a political blogs network of [adamic-polblogs]_. References ---------- .. [hits-algorithm] http://en.wikipedia.org/wiki/HITS_algorithm .. [kleinberg-authoritative] J. Kleinberg, "Authoritative sources in a hyperlinked environment", Journal of the ACM 46 (5): 604-632, 1999, :DOI:10.1145/324133.324140. .. [power-method] http://en.wikipedia.org/wiki/Power_iteration .. [adamic-polblogs] L. A. Adamic and N. Glance, "The political blogosphere and the 2004 US Election", in Proceedings of the WWW-2005 Workshop on the Weblogging Ecosystem (2005). :DOI:10.1145/1134271.1134277 """ if xprop is None: xprop = g.new_vertex_property("double") if yprop is None: yprop = g.new_vertex_property("double") if max_iter is None: max_iter = 0 l = libgraph_tool_centrality.\ get_hits(g._Graph__graph, _prop("e", g, weight), _prop("v", g, xprop), _prop("v", g, yprop), epsilon, max_iter) return 1. / l, xprop, yprop def eigentrust(g, trust_map, vprop=None, norm=False, epsilon=1e-6, max_iter=0, ret_iter=False): r""" Calculate the eigentrust centrality of each vertex in the graph. Parameters ---------- g : :class:~graph_tool.Graph Graph to be used. trust_map : :class:~graph_tool.EdgePropertyMap Edge property map with the values of trust associated with each edge. The values must lie in the range [0,1]. vprop : :class:~graph_tool.VertexPropertyMap, optional (default: None) Vertex property map where the values of eigentrust must be stored. norm : bool, optional (default: False) Norm eigentrust values so that the total sum equals 1. epsilon : float, optional (default: 1e-6) Convergence condition. The iteration will stop if the total delta of all vertices are below this value. max_iter : int, optional (default: None) If supplied, this will limit the total number of iterations. ret_iter : bool, optional (default: False) If true, the total number of iterations is also returned. Returns ------- eigentrust : :class:~graph_tool.VertexPropertyMap A vertex property map containing the eigentrust values. See Also -------- betweenness: betweenness centrality pagerank: PageRank centrality trust_transitivity: pervasive trust transitivity Notes ----- The eigentrust [kamvar-eigentrust-2003]_ values :math:t_i correspond the following limit .. math:: \mathbf{t} = \lim_{n\to\infty} \left(C^T\right)^n \mathbf{c} where :math:c_i = 1/|V| and the elements of the matrix :math:C are the normalized trust values: .. math:: c_{ij} = \frac{\max(s_{ij},0)}{\sum_{j} \max(s_{ij}, 0)} The algorithm has a topology-dependent complexity. If enabled during compilation, this algorithm runs in parallel. Examples -------- .. testsetup:: eigentrust np.random.seed(42) import matplotlib .. doctest:: eigentrust >>> g = gt.collection.data["polblogs"] >>> g = gt.GraphView(g, vfilt=gt.label_largest_component(g)) >>> w = g.new_edge_property("double") >>> w.a = np.random.random(len(w.a)) * 42 >>> t = gt.eigentrust(g, w) >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=t, ... vertex_size=gt.prop_to_size(t, mi=5, ma=15), ... vcmap=matplotlib.cm.gist_heat, ... vorder=t, output="polblogs_eigentrust.pdf") <...> .. testcode:: eigentrust :hide: gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=t, vertex_size=gt.prop_to_size(t, mi=5, ma=15), vcmap=matplotlib.cm.gist_heat, vorder=t, output="polblogs_eigentrust.png") .. figure:: polblogs_eigentrust.* :align: center Eigentrust values of the a political blogs network of [adamic-polblogs]_, with random weights attributed to the edges. References ---------- .. [kamvar-eigentrust-2003] S. D. Kamvar, M. T. Schlosser, H. Garcia-Molina "The eigentrust algorithm for reputation management in p2p networks", Proceedings of the 12th international conference on World Wide Web, Pages: 640 - 651, 2003, :doi:10.1145/775152.775242 .. [adamic-polblogs] L. A. Adamic and N. Glance, "The political blogosphere and the 2004 US Election", in Proceedings of the WWW-2005 Workshop on the Weblogging Ecosystem (2005). :DOI:10.1145/1134271.1134277 """ if vprop is None: vprop = g.new_vertex_property("double") i = libgraph_tool_centrality.\ get_eigentrust(g._Graph__graph, _prop("e", g, trust_map), _prop("v", g, vprop), epsilon, max_iter) if norm: vprop.get_array()[:] /= sum(vprop.get_array()) if ret_iter: return vprop, i else: return vprop def trust_transitivity(g, trust_map, source=None, target=None, vprop=None): r""" Calculate the pervasive trust transitivity between chosen (or all) vertices in the graph. Parameters ---------- g : :class:~graph_tool.Graph Graph to be used. trust_map : :class:~graph_tool.EdgePropertyMap Edge property map with the values of trust associated with each edge. The values must lie in the range [0,1]. source : :class:~graph_tool.Vertex (optional, default: None) Source vertex. All trust values are computed relative to this vertex. If left unspecified, the trust values for all sources are computed. target : :class:~graph_tool.Vertex (optional, default: None) The only target for which the trust value will be calculated. If left unspecified, the trust values for all targets are computed. vprop : :class:~graph_tool.VertexPropertyMap (optional, default: None) A vertex property map where the values of transitive trust must be stored. Returns ------- trust_transitivity : :class:~graph_tool.VertexPropertyMap or float A vertex vector property map containing, for each source vertex, a vector with the trust values for the other vertices. If only one of source or target is specified, this will be a single-valued vertex property map containing the trust vector from/to the source/target vertex to/from the rest of the network. If both source and target are specified, the result is a single float, with the corresponding trust value for the target. See Also -------- eigentrust: eigentrust centrality betweenness: betweenness centrality pagerank: PageRank centrality Notes ----- The pervasive trust transitivity between vertices i and j is defined as .. math:: t_{ij} = \frac{\sum_m A_{m,j} w^2_{G\setminus\{j\}}(i\to m)c_{m,j}} {\sum_m A_{m,j} w_{G\setminus\{j\}}(i\to m)} where :math:A_{ij} is the adjacency matrix, :math:c_{ij} is the direct trust from i to j, and :math:w_G(i\to j) is the weight of the path with maximum weight from i to j, computed as .. math:: w_G(i\to j) = \prod_{e\in i\to j} c_e. The algorithm measures the transitive trust by finding the paths with maximum weight, using Dijkstra's algorithm, to all in-neighbors of a given target. This search needs to be performed repeatedly for every target, since it needs to be removed from the graph first. For each given source, the resulting complexity is therefore :math:O(V^2\log V) for all targets, and :math:O(V\log V) for a single target. For a given target, the complexity for obtaining the trust from all given sources is :math:O(kV\log V), where :math:k is the in-degree of the target. Thus, the complexity for obtaining the complete trust matrix is :math:O(EV\log V), where :math:E is the number of edges in the network. If enabled during compilation, this algorithm runs in parallel. Examples -------- .. testsetup:: trust_transitivity np.random.seed(42) import matplotlib .. doctest:: trust_transitivity >>> g = gt.collection.data["polblogs"] >>> g = gt.GraphView(g, vfilt=gt.label_largest_component(g)) >>> g = gt.Graph(g, prune=True) >>> w = g.new_edge_property("double") >>> w.a = np.random.random(len(w.a)) >>> g.vp["label"][g.vertex(42)] 'blogforamerica.com' >>> t = gt.trust_transitivity(g, w, source=g.vertex(42)) >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=t, ... vertex_size=gt.prop_to_size(t, mi=5, ma=15), ... vcmap=matplotlib.cm.gist_heat, ... vorder=t, output="polblogs_trust_transitivity.pdf") <...> .. testcode:: trust_transitivity :hide: gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=t, vertex_size=gt.prop_to_size(t, mi=5, ma=15), vcmap=matplotlib.cm.gist_heat, vorder=t, output="polblogs_trust_transitivity.png") .. figure:: polblogs_trust_transitivity.* :align: center Trust transitivity values from source vertex 42 of the a political blogs network of [adamic-polblogs]_, with random weights attributed to the edges. References ---------- .. [richters-trust-2010] Oliver Richters and Tiago P. Peixoto, "Trust Transitivity in Social Networks," PLoS ONE 6, no. 4: e1838 (2011), :doi:10.1371/journal.pone.0018384 .. [adamic-polblogs] L. A. Adamic and N. Glance, "The political blogosphere and the 2004 US Election", in Proceedings of the WWW-2005 Workshop on the Weblogging Ecosystem (2005). :DOI:10.1145/1134271.1134277 """ if vprop is None: vprop = g.new_vertex_property("vector") if target is None: target = -1 else: target = g.vertex_index[target] if source is None: source = -1 else: source = g.vertex_index[source] libgraph_tool_centrality.\ get_trust_transitivity(g._Graph__graph, source, target, _prop("e", g, trust_map), _prop("v", g, vprop)) if target != -1 or source != -1: vprop = ungroup_vector_property(vprop, [0])[0] if target != -1 and source != -1: return vprop.a[target] return vprop