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graph_tool.centrality - Centrality measures

This module includes centrality-related algorithms.

Summary

pagerank (g[, damping, prop, epslon, ...]) Calculate the PageRank of each vertex.
betweenness (g[, vprop, eprop, weight, ...]) Calculate the betweenness centrality for each vertex and edge.
central_point_dominance (g, betweenness) Calculate the central point dominance of the graph, given the betweenness centrality of each vertex.
eigentrust (g, trust_map[, vprop, norm, epslon, ...]) Calculate the eigentrust centrality of each vertex in the graph.
absolute_trust (g, trust_map, source[, target, vprop]) Calculate the absolute trust centrality of each vertex in the graph, from a given source.

Contents

graph_tool.centrality.pagerank(g, damping=0.80000000000000004, prop=None, epslon=9.9999999999999995e-07, max_iter=None, ret_iter=False)

Calculate the PageRank of each vertex.

Parameters:

g : Graph

Graph to be used.

damping : float, optional (default: 0.8)

Damping factor.

prop : PropertyMap, optional (default: None)

Vertex property map to store the PageRank values.

epslon : 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 : PropertyMap

A vertex property map containing the PageRank values.

See also

betweenness
betweenness centrality
eigentrust
eigentrust centrality
absolute_trust
absolute trust centrality

Notes

The value of PageRank [pagerank-wikipedia] of vertex v PR(v) is given interactively by the relation:

PR(v) = \frac{1-d}{N} + d \sum_{w \in \Gamma^{-}(v)} \frac{PR (w)}{d^{+}(w)}

where \Gamma^{-}(v) are the in-neighbours of v, 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 epslon. It has a topology-dependent running time.

If enabled during compilation, this algorithm runs in parallel.

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

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
[ 0.63876901  1.13528868  0.31465963  0.55855277  0.2         0.75605741
  0.42628689  0.53066254  0.55004112  0.91717076  0.71164749  0.32015438
  0.67275227  1.08207389  1.14412231  0.9049167   1.32002     1.4692142
  0.76549771  0.71510277  0.23732927  0.40844911  0.2         0.27912876
  0.71309781  0.32015438  1.3376236   0.31352887  0.59346569  0.33381039
  0.67300081  0.73318264  0.65812653  0.73409673  0.93051993  0.83241145
  1.59816568  0.43979363  0.2512247   1.15663357  0.2         0.35977148
  0.72182022  1.01267711  0.76304859  0.49247376  0.49384283  1.8436647
  0.64312224  1.00778243  0.62287633  1.15215387  0.56176895  0.7166227
  0.56506109  0.67104337  0.95570565  0.27996953  0.79975983  0.33631497
  1.09471419  0.33631497  0.2512247   2.09126732  0.68157485  0.2
  0.37140185  0.65619459  1.27370737  0.48383225  1.36125161  0.2
  0.78300573  1.03427279  0.56904755  1.66077917  1.73302035  0.28749261
  0.83143045  1.04969728  0.70090048  0.55991433  0.68440994  0.2
  0.34018009  0.45485484  0.28        1.2015438   2.11850885  1.24990775
  0.59914308  0.59989185  0.73535564  0.78168417  0.55390281  0.38627667
  1.42274704  0.51105348  0.92550979  1.27968065]
graph_tool.centrality.betweenness(g, vprop=None, eprop=None, weight=None, norm=True)

Calculate the betweenness centrality for each vertex and edge.

Parameters:

g : Graph

Graph to be used.

vprop : PropertyMap, optional (default: None)

Vertex property map to store the vertex betweenness values.

eprop : PropertyMap, optional (default: None)

Edge property map to store the edge betweenness values.

weight : 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
absolute_trust
absolute trust centrality

Notes

Betweenness centrality of a vertex C_B(v) is defined as,

C_B(v)= \sum_{s \neq v \neq t \in V \atop s \neq t} \frac{\sigma_{st}(v)}{\sigma_{st}}

where \sigma_{st} is the number of shortest geodesic paths from s to t, and \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 (n-1)(n-2)/2.

The algorithm used here is defined in [brandes-faster-2001], and has a complexity of O(VE) for unweighted graphs and O(VE + V(V+E) \log V) for weighted graphs. The space complexity is O(VE).

If enabled during compilation, this algorithm runs in parallel.

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

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
[ 0.03395047  0.07911989  0.00702948  0.02337119  0.          0.02930099
  0.01684377  0.02558675  0.03440095  0.02886187  0.03124262  0.00975953
  0.01307953  0.03938858  0.07266505  0.01313647  0.          0.06450598
  0.0575418   0.00525468  0.00466089  0.01803829  0.          0.00050161
  0.0085034   0.02362432  0.05620574  0.00097157  0.04006816  0.01301474
  0.02154916  0.          0.06009194  0.02780363  0.08963522  0.04049657
  0.06993559  0.02082698  0.00288318  0.03264322  0.          0.03641759
  0.01083859  0.03750864  0.04079359  0.02092599  0.          0.02153655
  0.          0.05674631  0.03861911  0.05473282  0.00904367  0.03249097
  0.00894043  0.0192741   0.03379204  0.02125998  0.0018321   0.0013495
  0.0336502   0.0210088   0.00125318  0.0489189   0.05254974  0.
  0.00432189  0.04866168  0.06444727  0.02508525  0.02533085  0.
  0.05308703  0.02539854  0.02270809  0.044889    0.04766016  0.0086368
  0.01501699  0.          0.03107868  0.0054221   0.          0.
  0.00596081  0.01183977  0.00159761  0.11435876  0.03988501  0.05128991
  0.04558135  0.02303469  0.05092032  0.04700221  0.00927644  0.00841903
  0.          0.03243633  0.04514374  0.05170213]
graph_tool.centrality.central_point_dominance(g, betweenness)

Calculate the central point dominance of the graph, given the betweenness centrality of each vertex.

Parameters:

g : Graph

Graph to be used.

betweenness : 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 v^* be the vertex with the largest relative betweenness centrality; then, the central point dominance [freeman-set-1977] is defined as:

C'_B = \frac{1}{|V|-1} \sum_{v} C_B(v^*) - C_B(v)

where C_B(v) is the normalized betweenness centrality of vertex v. The value of C_B lies in the range [0,1].

The algorithm has a complexity of O(V).

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)

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)
0.0884414811909
graph_tool.centrality.eigentrust(g, trust_map, vprop=None, norm=False, epslon=9.9999999999999995e-07, max_iter=0, ret_iter=False)

Calculate the eigentrust centrality of each vertex in the graph.

Parameters:

g : Graph

Graph to be used.

trust_map : 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.

epslon : 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
absolute_trust
absolute trust centrality

Notes

The eigentrust [kamvar-eigentrust-2003] values t_i correspond the following limit

\mathbf{t} = \lim_{n\to\infty} \left(C^T\right)^n \mathbf{c}

where c_i = 1/|V| and the elements of the matrix C are the normalized trust values:

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.

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

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()
[  5.51422638e-03   1.12397965e-02   2.34959294e-04   6.32738574e-03
   0.00000000e+00   6.34804836e-03   2.67885424e-03   4.02497751e-03
   1.67943467e-02   6.46196106e-03   1.92402451e-02   9.04032352e-04
   9.70843104e-03   1.40319816e-02   1.04995777e-02   2.86712231e-02
   2.47285894e-02   2.38394469e-02   7.06936059e-03   9.45794717e-03
   2.09970054e-05   1.64768298e-03   0.00000000e+00   1.19346706e-03
   6.88434371e-03   5.36337333e-03   2.08428677e-02   2.85813783e-03
   1.10564670e-02   3.16345060e-04   5.25737238e-03   5.43761445e-03
   7.98048389e-03   7.95939648e-03   2.23891858e-02   5.68630666e-03
   2.09300588e-02   4.28902068e-03   1.70833078e-03   2.37814042e-02
   0.00000000e+00   1.20805010e-03   1.29713483e-02   5.73021992e-03
   8.71093674e-03   7.77661067e-03   8.76489806e-04   2.38519385e-02
   3.53225723e-03   8.46948906e-03   5.09874234e-03   2.44547150e-02
   1.32342629e-02   1.80085559e-03   4.37189381e-03   1.18195253e-02
   1.62748861e-02   1.83200678e-04   1.09745025e-02   1.47544090e-03
   3.34512926e-02   1.58885132e-03   1.13128910e-03   3.04944830e-02
   4.22684975e-03   0.00000000e+00   9.89654274e-04   4.25927156e-03
   2.34516214e-02   4.91370905e-03   2.29366664e-02   0.00000000e+00
   6.83407601e-03   1.60508753e-02   1.62762068e-03   3.94324856e-02
   2.84109571e-02   8.81167727e-04   2.16999908e-02   1.28688125e-02
   1.10825963e-02   2.64915564e-03   2.88711928e-03   0.00000000e+00
   4.24392252e-03   9.38398819e-03   0.00000000e+00   1.74508371e-02
   3.26594153e-02   4.07188867e-02   3.20678152e-03   6.35046287e-03
   8.07061556e-03   5.08505374e-03   3.27300367e-03   3.30989070e-03
   2.30651195e-02   4.20338525e-03   5.04332662e-03   3.58731532e-02]
graph_tool.centrality.absolute_trust(g, trust_map, source, target=None, vprop=None)

Calculate the absolute trust centrality of each vertex in the graph, from a given source.

Parameters:

g : Graph

Graph to be used.

trust_map : PropertyMap

Edge property map with the values of trust associated with each edge. The values must lie in the range [0,1].

source : Vertex

Source vertex. All trust values are computed relative to this vertex.

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 : PropertyMap (optional, default: None)

A vertex property map where the values of trust for each source must be stored.
Returns:

absolute_trust : PropertyMap or float

A vertex property map containing the absolute trust vector from the source vertex to the rest of the network. If target is 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 absolute trust between vertices i and j is defined as

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 A_{ij} is the adjacency matrix, c_{ij} is the direct trust from i to j, and w_G(i\to j) is the weight of the path with maximum weight from i to j, computed as

w_G(i\to j) = \prod_{e\in i\to j} c_e.

The algorithm measures the absolute 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. The resulting complexity is therefore O(N^2\log N) for all targets, and O(N\log N) for a single target.

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.absolute_trust(g, trust, source=g.vertex(0))
>>> print t.a
[ 0.16260667  0.04129912  0.13735376  0.19146125  0.          0.09147461
  0.10371912  0.12465511  0.24631221  0.0603916   0.2375385   0.06637879
  0.08897662  0.0800988   0.05250601  0.66759022  0.09368793  0.08275437
  0.13674709  0.15553915  0.01376162  0.417068    0.          0.06096886
  0.08746817  0.39380693  0.09215297  0.09575144  0.15594162  0.04008874
  0.05483972  0.05691086  0.13571077  0.32376012  0.22477937  0.06347962
  0.10445085  0.19447845  0.38007043  0.13810585  0.          0.08451096
  0.06648153  0.18479174  0.13003649  0.14850631  0.00320603  0.1074644
  0.12088162  0.06792678  0.08472666  0.2002143   0.25963204  0.37838425
  0.03089371  0.18389694  0.39420339  0.03348093  0.11483196  0.0656204
  0.14206403  0.07066434  0.25168986  0.07040126  0.04870569  0.
  0.09861349  0.03882069  0.1105267   0.07951823  0.08748441  0.
  0.08393443  0.11121719  0.21903223  0.25529628  0.0414386   0.03695558
  0.17664854  0.05143033  0.11735779  0.06525968  0.19600919  0.          0.1220922
  0.33330041  0.          0.28595961  0.14526678  0.12514885  0.089524
  0.40738962  0.03719195  0.54409979  0.06247424  0.10660136  0.11674385
  0.13218144  0.02214988  0.23215937]

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© Copyright 2009, Tiago de Paula Peixoto <tiago@forked.de>. Last updated on Feb 21, 2010. Created using Sphinx 0.6.3.