#! /usr/bin/env python # -*- coding: utf-8 -*- # # graph_tool -- a general graph manipulation python module # # Copyright (C) 2007-2010 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 eigentrust trust_transitivity Contents ++++++++ """ from .. dl_import import dl_import dl_import("import libgraph_tool_centrality") from .. core import _prop, ungroup_vector_property import sys import numpy __all__ = ["pagerank", "betweenness", "central_point_dominance", "eigentrust", "trust_transitivity"] def pagerank(g, damping=0.8, 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: 0.8) Damping factor. prop : :class:~graph_tool.PropertyMap, optional (default: None) Vertex property map to store the PageRank values. 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.PropertyMap A vertex property map containing the PageRank values. See Also -------- betweenness: betweenness centrality eigentrust: eigentrust centrality trust_transitivity: pervasive trust transitivity Notes ----- The value of PageRank [pagerank-wikipedia]_ of vertex v :math:PR(v) is given interactively by the relation: .. math:: PR(v) = \frac{1-d}{N} + d \sum_{w \in \Gamma^{-}(v)} \frac{PR (w)}{d^{+}(w)} where :math:\Gamma^{-}(v) are the in-neighbours of v, :math:d^{+}(w) is the out-degree of w, and d is a damping factor. The implemented algorithm progressively iterates the above condition, 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 -------- >>> from numpy.random import poisson, seed >>> seed(42) >>> g = gt.random_graph(100, lambda: (poisson(3), poisson(3))) >>> pr = gt.pagerank(g) >>> print pr.a [ 0.87011681 1.73449398 0.47587866 0.4534494 0.2 1.26596887 0.60964865 0.68064477 0.8137542 0.86269096 0.51833002 0.49194604 0.74875795 0.52831993 0.601438 0.63921165 1.32489495 0.68360746 1.02608206 0.90903761 1.1026286 0.56290713 0.2 0.30840086 0.90726785 0.35583967 0.95582862 0.232 0.41090313 0.88734742 0.47424296 0.66138242 1.26313184 0.7459428 0.84110051 0.9497316 1.0589998 0.94412292 0.26433617 0.86197354 0.2 0.25333333 0.65974242 0.69889305 1.02798531 0.77618244 0.57905885 1.12828577 0.232 1.18366748 0.38929224 1.72424164 0.47966878 1.0931673 0.45937603 1.09479766 0.80274459 0.44782081 1.04618114 0.25333333 0.82295953 0.40210109 0.72779393 0.75075946 0.41742276 0.2 0.8984279 0.92941713 0.69682427 0.69340983 1.02679348 0.2 0.67750539 0.85622403 0.77232588 1.09093307 1.14410169 0.59413937 0.54456339 0.64371752 0.40275133 0.72976606 1.40446885 0.2 0.31831299 0.3734494 0.2562224 1.05807688 1.02419007 0.82747632 0.49646186 0.72960178 0.48621114 1.42147072 0.65622314 0.31664379 1.55387576 0.58439879 2.03922765 1.47802266] 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 """ if max_iter == None: max_iter = 0 if prop == None: prop = g.new_vertex_property("double") ic = libgraph_tool_centrality.\ get_pagerank(g._Graph__graph, _prop("v", g, prop), damping, epsilon, max_iter) if ret_iter: return prop, ic else: return prop def betweenness(g, 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. vprop : :class:~graph_tool.PropertyMap, optional (default: None) Vertex property map to store the vertex betweenness values. eprop : :class:~graph_tool.PropertyMap, optional (default: None) Edge property map to store the edge betweenness values. weight : :class:~graph_tool.PropertyMap, 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 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 geodesic paths from s to t, and :math:\sigma_{st}(v) is the number of shortest geodesic paths from s to t that pass through a vertex 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. 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 enabled during compilation, this algorithm runs in parallel. Examples -------- >>> from numpy.random import poisson, seed >>> seed(42) >>> g = gt.random_graph(100, lambda: (poisson(3), poisson(3))) >>> vb, eb = gt.betweenness(g) >>> print vb.a [ 2.65012897e-02 1.04414799e-01 2.73374899e-02 1.52782183e-02 0.00000000e+00 2.74548352e-02 3.54680121e-02 3.72671558e-02 2.39732112e-02 2.34942149e-02 2.97950758e-02 4.08351383e-02 4.31702840e-02 1.90317902e-02 3.66879750e-02 8.65571818e-03 0.00000000e+00 3.74046494e-02 4.22428130e-02 2.10503176e-02 1.39558854e-02 8.40349783e-03 0.00000000e+00 4.45784374e-03 3.38671970e-02 1.72390157e-02 4.82232543e-02 1.03071532e-04 1.42200266e-02 4.82793598e-02 1.82020235e-02 0.00000000e+00 7.04969679e-02 2.31267158e-02 6.42817952e-02 3.71139131e-02 3.81618985e-02 4.06231715e-02 2.16376594e-03 2.44758076e-02 0.00000000e+00 6.86198722e-03 1.36132952e-02 1.73886977e-02 2.30213129e-02 4.44999980e-02 0.00000000e+00 1.40589569e-02 0.00000000e+00 4.74213177e-02 2.65427674e-02 1.05684330e-01 6.30552365e-03 2.86320444e-02 4.50079022e-03 7.76843152e-02 2.88642900e-02 3.52207159e-02 2.01852506e-02 9.26784855e-04 4.35733012e-02 1.84745904e-02 1.35102237e-02 2.69638287e-02 1.88247064e-02 0.00000000e+00 2.03784688e-02 4.14981678e-02 1.79538495e-02 1.12983577e-02 3.23765203e-02 0.00000000e+00 3.99771399e-02 2.85164571e-03 2.18967289e-02 3.96111705e-02 3.40096863e-02 1.72800650e-02 1.36861815e-02 0.00000000e+00 1.19328203e-02 1.71726485e-02 0.00000000e+00 0.00000000e+00 6.33251858e-03 4.64324980e-03 1.33084980e-03 9.89021626e-02 3.52934995e-02 2.96267777e-02 1.73480268e-02 3.07545000e-02 2.47891161e-02 3.32486832e-02 7.45403501e-03 1.46792267e-02 0.00000000e+00 3.35642472e-02 8.78597450e-02 3.94517740e-02] 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 """ if vprop == None: vprop = g.new_vertex_property("double") if eprop == None: eprop = g.new_edge_property("double") if weight != None and weight.value_type() != eprop.value_type(): nw = g.new_edge_property(eprop.value_type()) g.copy_property(weight, nw) weight = nw libgraph_tool_centrality.\ get_betweenness(g._Graph__graph, _prop("e", g, weight), _prop("e", g, eprop), _prop("v", g, vprop), norm) return vprop, eprop 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.PropertyMap 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 -------- >>> from numpy.random import poisson, seed >>> seed(42) >>> g = gt.random_graph(100, lambda: (poisson(3), poisson(3))) >>> vb, eb = gt.betweenness(g) >>> print gt.central_point_dominance(g, vb) 0.0813233725942 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 """ return libgraph_tool_centrality.\ get_central_point_dominance(g._Graph__graph, _prop("v", g, betweenness)) 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.PropertyMap Edge property map with the values of trust associated with each edge. The values must lie in the range [0,1]. vprop : PropertyMap, 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 : 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 -------- >>> from numpy.random import poisson, random, seed >>> seed(42) >>> g = gt.random_graph(100, lambda: (poisson(3), poisson(3))) >>> trust = g.new_edge_property("double") >>> trust.get_array()[:] = random(g.num_edges())*42 >>> t = gt.eigentrust(g, trust, norm=True) >>> print t.get_array() [ 0.02100449 0.01735932 0.00227182 0.00342703 0. 0.01739914 0.00658874 0.00592764 0.00879695 0.01483758 0.00390145 0.00939709 0.01038803 0.00896039 0.0080222 0.00583084 0.01510505 0.01106463 0.02048866 0.0179936 0.02196625 0.00604554 0. 0.00038504 0.01704679 0.00431482 0.00538866 0. 0.00163772 0.02009726 0.00254747 0.00440903 0.02305541 0.01061566 0.00583414 0.01521545 0.01894677 0.00941793 0.00259066 0.00454916 0. 0. 0.00411855 0.01005776 0.029152 0.01500648 0.00797009 0.02057446 0. 0.02100182 0.00519358 0.02503401 0.00368714 0.02176737 0.00111934 0.02763714 0.00615445 0.00163793 0.01998869 0. 0.00831816 0.00692008 0.00439715 0.01287125 0.00534507 0. 0.00805071 0.02094972 0.00622514 0.00285397 0.01009464 0. 0.00360911 0.00653993 0.00800227 0.01521205 0.02901848 0.01693622 0.00323205 0.00748302 0.00443795 0.0076314 0.01147831 0. 0.00129362 0.00173367 0.00188625 0.02110825 0.01349257 0.00956502 0.00694694 0.01780551 0.00344632 0.02869166 0.00388418 0.0016279 0.01691452 0.00783781 0.02795918 0.03327071] 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 """ if vprop == 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.PropertyMap Edge property map with the values of trust associated with each edge. The values must lie in the range [0,1]. source : 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 : 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.PropertyMap (optional, default: None) A vertex property map where the values of transitive trust must be stored. Returns ------- trust_transitivity : :class:~graph_tool.PropertyMap 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-neighbours 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(N^2\log N) for all targets, and :math:O(N\log N) for a single target. For a given target, the complexity for obtaining the trust from all given sources is :math:O(kN\log N), where :math:k is the in-degree of the target. Thus, the complexity for obtaining the complete trust matrix is :math:O(EN\log N), where :math:E is the number of edges in the network. If enabled during compilation, this algorithm runs in parallel. Examples -------- >>> from numpy.random import poisson, random, seed >>> seed(42) >>> g = gt.random_graph(100, lambda: (poisson(3), poisson(3))) >>> trust = g.new_edge_property("double") >>> trust.a = random(g.num_edges()) >>> t = gt.trust_transitivity(g, trust, source=g.vertex(0)) >>> print t.a [ 1. 0.09649648 0.01375374 0.09864347 0. 0.52668732 0.02655169 0.05771735 0.25651251 0.13071344 0.1258206 0.13065921 0.12051013 0.13754053 0.26727787 0.06951245 0.38774441 0.25343023 0.21297027 0.59232433 0.10843174 0.02810649 0. 0.04000351 0.13784095 0.06125175 0.04156937 0. 0.05771925 0.04967184 0.11251086 0.25172931 0.1982562 0.28225643 0.05339001 0.10629504 0.04440744 0.05815895 0.097983 0.03333347 0. 0. 0.10845473 0.13751647 0.27567139 0.03946153 0.25063883 0.0755547 0. 0.25167962 0.33205973 0.08237051 0.12983804 0.02587608 0.09694727 0.16435599 0.09445501 0.07402817 0.06425702 0. 0.22420236 0.11284837 0.05567628 0.0561254 0.36563496 0. 0.09358333 0.06315609 0.3853858 0.01338133 0.08506159 0. 0.23226712 0.0841518 0.07274848 0.17553984 0.14032908 0.15737553 0.13703351 0.25035262 0.03570828 0.04341688 0.11955905 0. 0.01757771 0.04990193 0.10457395 0.41668972 0.04546921 0.04404905 0.24922167 0.09752267 0.03872946 0.26113888 0.04677363 0.03220735 0.03928181 0.08696124 0.21697483 0.1388346 ] References ---------- .. [richters-trust-2010] Oliver Richters, Tiago P. Peixoto, "Trust transitivity in social networks", :arXiv:1012.1358, 2010 """ if vprop == None: vprop = g.new_vertex_property("vector") if target == None: target = -1 else: target = g.vertex_index[target] if source == 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