\c@sdZddlZddlZddlZdZdddgZdede ddZ eddZ edd Z d Z d d Zd ZdZdZe ddZe dZdS(s" Betweenness centrality measures. iNsAric Hagberg (hagberg@lanl.gov)tbetweenness_centralitytedge_betweenness_centralitytedge_betweennessc Cs tj|d}|dkr'|}n%tj|tj|j|}x|D]}|dkrt||\} } } nt|||\} } } |rt || | | |}qSt || | | |}qSWt |t |d|d|j d|}|S(s Compute the shortest-path betweenness centrality for nodes. Betweenness centrality of a node `v` is the sum of the fraction of all-pairs shortest paths that pass through `v`: .. math:: c_B(v) =\sum_{s,t \in V} \frac{\sigma(s, t|v)}{\sigma(s, t)} where `V` is the set of nodes, `\sigma(s, t)` is the number of shortest `(s, t)`-paths, and `\sigma(s, t|v)` is the number of those paths passing through some node `v` other than `s, t`. If `s = t`, `\sigma(s, t) = 1`, and if `v \in {s, t}`, `\sigma(s, t|v) = 0` [2]_. Parameters ---------- G : graph A NetworkX graph k : int, optional (default=None) If k is not None use k node samples to estimate betweenness. The value of k <= n where n is the number of nodes in the graph. Higher values give better approximation. normalized : bool, optional If True the betweenness values are normalized by `2/((n-1)(n-2))` for graphs, and `1/((n-1)(n-2))` for directed graphs where `n` is the number of nodes in G. weight : None or string, optional If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight. endpoints : bool, optional If True include the endpoints in the shortest path counts. Returns ------- nodes : dictionary Dictionary of nodes with betweenness centrality as the value. See Also -------- edge_betweenness_centrality load_centrality Notes ----- The algorithm is from Ulrik Brandes [1]_. See [2]_ for details on algorithms for variations and related metrics. For approximate betweenness calculations set k=#samples to use k nodes ("pivots") to estimate the betweenness values. For an estimate of the number of pivots needed see [3]_. For weighted graphs the edge weights must be greater than zero. Zero edge weights can produce an infinite number of equal length paths between pairs of nodes. References ---------- .. [1] A Faster Algorithm for Betweenness Centrality. Ulrik Brandes, Journal of Mathematical Sociology 25(2):163-177, 2001. http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf .. [2] Ulrik Brandes: On Variants of Shortest-Path Betweenness Centrality and their Generic Computation. Social Networks 30(2):136-145, 2008. http://www.inf.uni-konstanz.de/algo/publications/b-vspbc-08.pdf .. [3] Ulrik Brandes and Christian Pich: Centrality Estimation in Large Networks. International Journal of Bifurcation and Chaos 17(7):2303-2318, 2007. http://www.inf.uni-konstanz.de/algo/publications/bp-celn-06.pdf gt normalizedtdirectedtkN(tdicttfromkeystNonetrandomtseedtsampletnodest"_single_source_shortest_path_basict"_single_source_dijkstra_path_basict_accumulate_endpointst_accumulate_basict_rescaletlent is_directed( tGRRtweightt endpointsR t betweennessR tstStPtsigma((sb/Users/dxp/prism/prism-games/prism-examples/smgs/car/networkx/algorithms/centrality/betweenness.pyRs"N       c Cstj|d}|jtj|jdxh|D]`}|dkret||\}}}nt|||\}}}t|||||}q8Wx|D] }||=qWt|t |d|d|j }|S(sCompute betweenness centrality for edges. Betweenness centrality of an edge `e` is the sum of the fraction of all-pairs shortest paths that pass through `e`: .. math:: c_B(v) =\sum_{s,t \in V} \frac{\sigma(s, t|e)}{\sigma(s, t)} where `V` is the set of nodes,`\sigma(s, t)` is the number of shortest `(s, t)`-paths, and `\sigma(s, t|e)` is the number of those paths passing through edge `e` [2]_. Parameters ---------- G : graph A NetworkX graph normalized : bool, optional If True the betweenness values are normalized by `2/(n(n-1))` for graphs, and `1/(n(n-1))` for directed graphs where `n` is the number of nodes in G. weight : None or string, optional If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight. Returns ------- edges : dictionary Dictionary of edges with betweenness centrality as the value. See Also -------- betweenness_centrality edge_load Notes ----- The algorithm is from Ulrik Brandes [1]_. For weighted graphs the edge weights must be greater than zero. Zero edge weights can produce an infinite number of equal length paths between pairs of nodes. References ---------- .. [1] A Faster Algorithm for Betweenness Centrality. Ulrik Brandes, Journal of Mathematical Sociology 25(2):163-177, 2001. http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf .. [2] Ulrik Brandes: On Variants of Shortest-Path Betweenness Centrality and their Generic Computation. Social Networks 30(2):136-145, 2008. http://www.inf.uni-konstanz.de/algo/publications/b-vspbc-08.pdf gRRN( RRtupdatetedgesRR Rt_accumulate_edgest _rescale_eRR( RRRRRRRRtn((sb/Users/dxp/prism/prism-games/prism-examples/smgs/car/networkx/algorithms/centrality/betweenness.pyRzs8    cCst|||S(N(R(RRR((sb/Users/dxp/prism/prism-games/prism-examples/smgs/car/networkx/algorithms/centrality/betweenness.pyRsc Csg}i}x|D]}g||s&      eL  !