o 3ήc(@shdZddlZddlmZddlmZmZddgZeddde d fd dZ eddde d fd dZ dS) zBCurrent-flow betweenness centrality measures for subsets of nodes.N)flow_matrix_row)not_implemented_forreverse_cuthill_mckee_ordering*current_flow_betweenness_centrality_subset/edge_current_flow_betweenness_centrality_subsetdirectedTluc sJddl}ddlm}t|std|} t||tt t | } t || } t | d} t | |||dD]>\} \}}|D]5}| |}|D],}| |}| |d|| || |7<| |d|| || |7<qKqCq;|r| d| d }nd }| D]}| ||dd | | |<qfd d | DS) a- Compute current-flow betweenness centrality for subsets of nodes. Current-flow betweenness centrality uses an electrical current model for information spreading in contrast to betweenness centrality which uses shortest paths. Current-flow betweenness centrality is also known as random-walk betweenness centrality [2]_. Parameters ---------- G : graph A NetworkX graph sources: list of nodes Nodes to use as sources for current targets: list of nodes Nodes to use as sinks for current normalized : bool, optional (default=True) If True the betweenness values are normalized by b=b/(n-1)(n-2) where n is the number of nodes in G. weight : string or None, optional (default=None) Key for edge data used as the edge weight. If None, then use 1 as each edge weight. The weight reflects the capacity or the strength of the edge. dtype: data type (float) Default data type for internal matrices. Set to np.float32 for lower memory consumption. solver: string (default='lu') Type of linear solver to use for computing the flow matrix. Options are "full" (uses most memory), "lu" (recommended), and "cg" (uses least memory). Returns ------- nodes : dictionary Dictionary of nodes with betweenness centrality as the value. See Also -------- approximate_current_flow_betweenness_centrality betweenness_centrality edge_betweenness_centrality edge_current_flow_betweenness_centrality Notes ----- Current-flow betweenness can be computed in $O(I(n-1)+mn \log n)$ time [1]_, where $I(n-1)$ is the time needed to compute the inverse Laplacian. For a full matrix this is $O(n^3)$ but using sparse methods you can achieve $O(nm{\sqrt k})$ where $k$ is the Laplacian matrix condition number. The space required is $O(nw)$ where $w$ is the width of the sparse Laplacian matrix. Worse case is $w=n$ for $O(n^2)$. If the edges have a 'weight' attribute they will be used as weights in this algorithm. Unspecified weights are set to 1. References ---------- .. [1] Centrality Measures Based on Current Flow. Ulrik Brandes and Daniel Fleischer, Proc. 22nd Symp. Theoretical Aspects of Computer Science (STACS '05). LNCS 3404, pp. 533-544. Springer-Verlag, 2005. https://doi.org/10.1007/978-3-540-31856-9_44 .. [2] A measure of betweenness centrality based on random walks, M. E. J. Newman, Social Networks 27, 39-54 (2005). rN)rGraph not connected.weightdtypesolver??@csi|] \}}||qSr).0kvorderingre/tmp/pip-target-vg8gfxp4/lib/python/networkx/algorithms/centrality/current_flow_betweenness_subset.py vsz>current_flow_betweenness_centrality_subset..)numpynetworkx.utilsrnx is_connected NetworkXErrornumber_of_nodeslistdictziprange relabel_nodesfromkeysrabsitems)Gsourcestargets normalizedr r rnprnmappingH betweennessrowstssittjnbrrrrr s0P      &(c sddl}t|std|}tt|ttt |} t || } dd| D} t | d} |rA|d|d} nd} t | |||d D]1\}}|D]"}| |}|D]}| |}| |d |||||7<qYqQ| || <qKfd d | DS) a Compute current-flow betweenness centrality for edges using subsets of nodes. Current-flow betweenness centrality uses an electrical current model for information spreading in contrast to betweenness centrality which uses shortest paths. Current-flow betweenness centrality is also known as random-walk betweenness centrality [2]_. Parameters ---------- G : graph A NetworkX graph sources: list of nodes Nodes to use as sources for current targets: list of nodes Nodes to use as sinks for current normalized : bool, optional (default=True) If True the betweenness values are normalized by b=b/(n-1)(n-2) where n is the number of nodes in G. weight : string or None, optional (default=None) Key for edge data used as the edge weight. If None, then use 1 as each edge weight. The weight reflects the capacity or the strength of the edge. dtype: data type (float) Default data type for internal matrices. Set to np.float32 for lower memory consumption. solver: string (default='lu') Type of linear solver to use for computing the flow matrix. Options are "full" (uses most memory), "lu" (recommended), and "cg" (uses least memory). Returns ------- nodes : dict Dictionary of edge tuples with betweenness centrality as the value. See Also -------- betweenness_centrality edge_betweenness_centrality current_flow_betweenness_centrality Notes ----- Current-flow betweenness can be computed in $O(I(n-1)+mn \log n)$ time [1]_, where $I(n-1)$ is the time needed to compute the inverse Laplacian. For a full matrix this is $O(n^3)$ but using sparse methods you can achieve $O(nm{\sqrt k})$ where $k$ is the Laplacian matrix condition number. The space required is $O(nw)$ where $w$ is the width of the sparse Laplacian matrix. Worse case is $w=n$ for $O(n^2)$. If the edges have a 'weight' attribute they will be used as weights in this algorithm. Unspecified weights are set to 1. References ---------- .. [1] Centrality Measures Based on Current Flow. Ulrik Brandes and Daniel Fleischer, Proc. 22nd Symp. Theoretical Aspects of Computer Science (STACS '05). LNCS 3404, pp. 533-544. Springer-Verlag, 2005. https://doi.org/10.1007/978-3-540-31856-9_44 .. [2] A measure of betweenness centrality based on random walks, M. E. J. Newman, Social Networks 27, 39-54 (2005). rNr css$|] \}}tt||fVqdS)N)tuplesorted)rurrrr s"zBedge_current_flow_betweenness_centrality_subset..r rrr rcs&i|]\\}}}||f|qSrr)rr3r4rrrrrs&zCedge_current_flow_betweenness_centrality_subset..)rrrrr r!rr"r#r$r%edgesr&rr'r()r)r*r+r,r r rr-r.r/r0r>r1r9r2er5r6r7r8rrrrys*P     () __doc__networkxr*networkx.algorithms.centrality.flow_matrixrrrr__all__floatrrrrrrs  l