ó ,ľ[c@ sdZddlmZddlmZmZddlmZddlZ ddl m Z ddd gZ e d ƒdededd „ƒZe d ƒdeddd „ƒZdeddd„Zd„Zd„Zd„Zd„Zd„Zeded„Zedd„ZdS(s Betweenness centrality measures.i˙˙˙˙(tdivision(theappushtheappop(tcountN(tpy_random_statetbetweenness_centralitytedge_betweenness_centralitytedge_betweennessic C stj|dƒ}|dkr'|}n|j|jƒ|ƒ}x‰|D]}|dkrst||ƒ\} } } nt|||ƒ\} } } |rŻt|| | | |ƒ}qFt|| | | |ƒ}qFWt |t |ƒd|d|j ƒd|d|ƒ}|S(sn 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 (default=None) 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. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness`. Note that this is only used if k is not None. 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 [4]_ for the original first published version and [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] 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 .. [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 .. [4] Linton C. Freeman: A set of measures of centrality based on betweenness. Sociometry 40: 35–41, 1977 http://moreno.ss.uci.edu/23.pdf gt normalizedtdirectedtkt endpointsN( tdicttfromkeystNonetsampletnodest"_single_source_shortest_path_basict"_single_source_dijkstra_path_basict_accumulate_endpointst_accumulate_basict_rescaletlent is_directed( tGR RtweightR tseedt betweennessRtstStPtsigma((sIlib/python2.7/site-packages/networkx/algorithms/centrality/betweenness.pyRsY    ic C s tj|dƒ}|jtj|jƒdƒƒ|dkrF|}n|j|jƒ|ƒ}xh|D]`}|dkr’t||ƒ\}} } nt|||ƒ\}} } t ||| | |ƒ}qeWx|D] } || =qĐWt |t |ƒd|d|j ƒƒ}|S(s™Compute 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(e) =\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. 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 $2/(n(n-1))$ for graphs, and $1/(n(n-1))$ for directed graphs where $n$ is the number of nodes in G. weight : None or string, optional (default=None) If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness`. Note that this is only used if k is not None. 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. 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