o 3ήc- @sdZddlZddlmZmZmZmZddlm Z m Z m Z gdZ e de ddde d d d dfd d Ze ddde d fddZe ddde d fddZdS)z-Current-flow betweenness centrality measures.N)CGInverseLaplacianFullInverseLaplacianSuperLUInverseLaplacianflow_matrix_row)not_implemented_forpy_random_statereverse_cuthill_mckee_ordering)#current_flow_betweenness_centrality/approximate_current_flow_betweenness_centrality(edge_current_flow_betweenness_centralitydirectedTfullg?i'c sddl}t|stdtttd} |} tt |t |t t t | } tj| t | |dd} | |} | || |d} t | d}| d | d }| | d |}d }|t|||d || }||krd |d|d}t|d|d |}t |D]R}|t | d \}}}|j| |d}d ||<d||<| |}| D]-}||vrq| |D]!}| |||d }|||||||||7<qqq|rd n|d fdd|DS)aCompute the approximate current-flow betweenness centrality for nodes. Approximates the current-flow betweenness centrality within absolute error of epsilon with high probability [1]_. Parameters ---------- G : graph A NetworkX graph normalized : bool, optional (default=True) If True the betweenness values are normalized by 2/[(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='full') Type of linear solver to use for computing the flow matrix. Options are "full" (uses most memory), "lu" (recommended), and "cg" (uses least memory). epsilon: float Absolute error tolerance. kmax: int Maximum number of sample node pairs to use for approximation. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness`. Returns ------- nodes : dictionary Dictionary of nodes with betweenness centrality as the value. See Also -------- current_flow_betweenness_centrality Notes ----- The running time is $O((1/\epsilon^2)m{\sqrt k} \log n)$ and the space required is $O(m)$ for $n$ nodes and $m$ edges. If the edges have a 'weight' attribute they will be used as weights in this algorithm. Unspecified weights are set to 1. References ---------- .. [1] Ulrik Brandes and Daniel Fleischer: Centrality Measures Based on Current Flow. 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 rNGraph not connected.)rlucg)nodelistweightcsc)dtype?@zNumber random pairs k>kmax (>z) zIncrease kmax or epsiloncsi|] \}}||qSr.0kvfactororderingr^/tmp/pip-target-vg8gfxp4/lib/python/networkx/algorithms/centrality/current_flow_betweenness.py szCapproximate_current_flow_betweenness_centrality..)numpynx is_connected NetworkXErrorrrrnumber_of_nodeslistr relabel_nodesdictziprangelaplacian_matrixasformatastypefromkeysintceillogsamplezerossolvegetabsitems)G normalizedrrsolverepsilonkmaxseednp solvernamenHLC betweennessnbcstarlr msgcstar2k_stpairbpr!nbrwrr"r%r sPM     $     ,r cs8t|s td|}tt|t|ttt |}t |d}t ||||dD]A\}\} } tt| dddt |} t |D]&} || | | | || 7<|| || d| | || 7<qIq/|r||d|d} nd} |D]}t |||d| ||<qfd d |DS) a Compute current-flow betweenness centrality for 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 normalized : bool, optional (default=True) If True the betweenness values are normalized by 2/[(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='full') 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). rrrrr@Nrrrrcsi|] \}}||qSrrrr$rr%r&sz7current_flow_betweenness_centrality..)r(r)r*r+r,rr-r.r/r0r4rargsortfloatr=)r>r?rrr@rFrGrJrowrQrRposirKr!rrYr%r s$ J      *r cs6t|s td|}tt|t|ttt |}dd| D}t |d}|r;|d|d} nd} t ||||dD]J\} } tt| ddd t d |d } t |D]&} || | d | | | | 7<|| || | | | | 7<q`|| | <qEfd d |DS) a( Compute current-flow betweenness centrality for edges. 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 normalized : bool, optional (default=True) If True the betweenness values are normalized by 2/[(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 (default=float) Default data type for internal matrices. Set to np.float32 for lower memory consumption. solver : string (default='full') 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 edge tuples with betweenness centrality as the value. Raises ------ NetworkXError The algorithm does not support DiGraphs. If the input graph is an instance of DiGraph class, NetworkXError is raised. 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. 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