ó ś£Õ\c@s„dZdZdddgZddlZddlmZddlmZ e dd „Z e dd „Z eddd „Zd „Zd „Ze d„Ze d„ZdS(s7 Betweenness centrality measures for subsets of nodes. sAric Hagberg (hagberg@lanl.gov)tbetweenness_centrality_subsett"edge_betweenness_centrality_subsettbetweenness_centrality_sourcei’’’’N(t"_single_source_dijkstra_path_basic(t"_single_source_shortest_path_basicc Cs«tj|dƒ}xk|D]c}|dkrFt||ƒ\}}} nt|||ƒ\}}} t|||| ||ƒ}qWt|t|ƒd|d|jƒƒ}|S(s¼Compute betweenness centrality for a subset of nodes. .. math:: c_B(v) =\sum_{s\in S, t \in T} rac{\sigma(s, t|v)}{\sigma(s, t)} where `S` is the set of sources, `T` is the set of targets, `\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 sources: list of nodes Nodes to use as sources for shortest paths in betweenness targets: list of nodes Nodes to use as targets for shortest paths in betweenness 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. Returns ------- nodes : dictionary Dictionary of nodes with betweenness centrality as the value. See Also -------- edge_betweenness_centrality load_centrality Notes ----- The basic algorithm is from [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. The normalization might seem a little strange but it is the same as in betweenness_centrality() and is designed to make betweenness_centrality(G) be the same as betweenness_centrality_subset(G,sources=G.nodes(),targets=G.nodes()). References ---------- .. [1] Ulrik Brandes, A Faster Algorithm for Betweenness Centrality. 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 gt normalizedtdirectedN( tdicttfromkeystNonet shortest_pathtdijkstrat_accumulate_subsett_rescaletlent is_directed( tGtsourcesttargetsRtweighttbtstStPtsigma((si/Users/dxp/prism/prism-games/prism-examples/smgs/car/networkx/algorithms/centrality/betweenness_subset.pyRsF  'c Csātj|dƒ}|jtj|jƒdƒƒxk|D]c}|dkret||ƒ\}}} nt|||ƒ\}}} t|||| ||ƒ}q8Wx|D] } || =q¦Wt|t |ƒd|d|j ƒƒ}|S(smCompute betweenness centrality for edges for a subset of nodes. .. math:: c_B(v) =\sum_{s\in S,t \in T} rac{\sigma(s, t|e)}{\sigma(s, t)} where `S` is the set of sources, `T` is the set of targets, `\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 sources: list of nodes Nodes to use as sources for shortest paths in betweenness targets: list of nodes Nodes to use as targets for shortest paths in betweenness 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 basic algorithm is from [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. The normalization might seem a little strange but it is the same as in edge_betweenness_centrality() and is designed to make edge_betweenness_centrality(G) be the same as edge_betweenness_centrality_subset(G,sources=G.nodes(),targets=G.nodes()). References ---------- .. [1] Ulrik Brandes, A Faster Algorithm for Betweenness Centrality. 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. 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