VhH. VdZddlZddlmZmZmZmZddlm Z m Z m Z gdZ e de dejd d ded d d dfdZe dejd d ded fdZe dejd d ded fdZy)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_centralitydirectedweight) edge_attrsTfullg?i'cddl}tj|stjdtt t d} |j} tt|} tj|tt| t| } tj| t| |jd} | j!|} | || |}tj#| d}| d z | d z z}| | d z z|z }d }|t%|j'||z d z|j)| zz}||kDrd |d|d}tj|d|d |zz }t|D]}|j+t| d x\}}}|j-| |}d ||<d||<|j/|}| D]]}||vr| |D]N}| ||j1|d }||xxt3||j5||||z z|zz cc<P_|rd }n|d z }|j7Dcic]\}}| |||zc}}Scc}}w)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)nodelistrcsc)dtype?@zNumber random pairs k>kmax (>z) zIncrease kmax or epsilon)numpynx is_connected NetworkXErrorrrrnumber_of_nodeslistr relabel_nodesdictziprangelaplacian_matrixasformatastypefromkeysintceillogsamplezerossolvegetfloatabsitems) G normalizedrrsolverepsilonkmaxseednp solvernamenorderingHLC betweennessnbcstarlkmsgcstar2k_stpairbpvnbrwfactors g/home/dcms/DCMS/lib/python3.12/site-packages/networkx/algorithms/centrality/current_flow_betweenness.pyr r s~\ ??1 566$% J A2156H DXuQx!89:A Aa@II%PA A 61E*A--3'K c'a#g B QK" E A CQ.:; <??q1uoG 1X Mkk%(A..1t HHQeH $!! GGAJ MADyt MaDIMM&#.A%BFF1Q4!C&=,A(AG(K"LL M M Mc0;0A0A0C D1HQKV # DD DsI(c tj|stjd|j}t t |}tj |tt|t|}tj|d}t||||D]\} \} } tt| jdddt|} t|D]R} || xx| | | z | j| zz cc<|| xx|| z dz | | z | j| zz cc<T|r |dz |dz z}nd}|jDcic]\}}||||z dz|z c}}Scc}}w) 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:Nrrrr)r!r"r#r$r%r r&r'r(r)r-rargsortitemr7)r8r9rrr:NrArBrErowrMrNposirFr@rPs rVr r stV ??1 566 A2156H DXuQx!89:A--3'K&quVTA Va3s{{}TrT*E!H56q AA Nq3q6zSXXa[8 8N Nq1uqy3q61SXXa[@ @N AA #g!c' " 8C8I8I8K L1HQK!a%3+ + LL Ls&Fc `tj|stjd|j}t t |}tj |tt|t|}d|jD}tj|d} |r |dz |dz z} nd} t||||D]\} } tt| jdddtd |d z} t|D]R}| | xx|d z| |z | j|zz cc<| | xx||z | |z | j|zz cc<T| | xx| zcc<| jDcic]\\}}}||||f|c}}}Scc}}}w) 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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