+ j)JRt^RIt^RIHt.R Ot]!R4]!R4]P RR/Rl444t]!R4]!R4]P RR/Rl444t]!R4]!R4]P R 444t ]P R 4t R#) z5 Subraph centrality and communicability betweenness. N)not_implemented_fordirected multigraph normalizedFc4^RIp\V4p\P!W4p^WDR8g&VPP V4p\ \VP44pV'dWfP4, p\\W644pV#)aReturns the subgraph centrality for each node of G. Subgraph centrality of a node `n` is the sum of weighted closed walks of all lengths starting and ending at node `n`. The weights decrease with path length. Each closed walk is associated with a connected subgraph ([1]_). Parameters ---------- G: graph normalized : bool If True, normalize the centrality values using the largest eigenvalue of the adjacency matrix so that the centrality values are generally between 0 and 1. Returns ------- nodes:dictionary Dictionary of nodes with subgraph centrality as the value. Raises ------ NetworkXError If the graph is not undirected and simple. See Also -------- subgraph_centrality: Alternative algorithm of the subgraph centrality for each node of G. Notes ----- This version of the algorithm exponentiates the adjacency matrix. The subgraph centrality of a node `u` in G can be found using the matrix exponential of the adjacency matrix of G [1]_, .. math:: SC(u)=(e^A)_{uu} . Examples -------- (Example from [1]_) >>> G = nx.Graph( ... [ ... (1, 2), ... (1, 5), ... (1, 8), ... (2, 3), ... (2, 8), ... (3, 4), ... (3, 6), ... (4, 5), ... (4, 7), ... (5, 6), ... (6, 7), ... (7, 8), ... ] ... ) >>> sc = nx.subgraph_centrality_exp(G) >>> print([f"{node} {sc[node]:0.2f}" for node in sorted(sc)]) ['1 3.90', '2 3.90', '3 3.64', '4 3.71', '5 3.64', '6 3.71', '7 3.64', '8 3.90'] >>> sc = nx.subgraph_centrality(G, normalized=True) >>> print([f"{node} {sc[node]:0.3f}" for node in sorted(sc)]) ['1 0.194', '2 0.194', '3 0.181', '4 0.184', '5 0.181', '6 0.184', '7 0.181', '8 0.194'] References ---------- .. [1] Ernesto Estrada, Juan A. Rodriguez-Velazquez, "Subgraph centrality in complex networks", Physical Review E 71, 056103 (2005). https://arxiv.org/abs/cond-mat/0504730 Ng) scipylistnxto_numpy_arraylinalgexpmmapfloatdiagonalmaxdictzip)GrspnodelistAexpAvaluesscs&$ l/home/luca/screen-checking/.venv/lib/python3.14/site-packages/networkx/algorithms/centrality/subgraph_alg.pysubgraph_centrality_exprsu`AwH !&AA3hK 99>>! D   (F**,& c(# $B Ic ^RIp\V4p\P!W4p^WBP V4&VP P V4wrVVPV4^,pV'd'VPWUP4, 4pMVPV4pWx,p \\V\\V 444p V #)a Returns subgraph centrality for each node in G. Subgraph centrality of a node `n` is the sum of weighted closed walks of all lengths starting and ending at node `n`. The weights decrease with path length. Each closed walk is associated with a connected subgraph ([1]_). Parameters ---------- G: Graph normalized : bool If True, normalize the centrality values using the largest eigenvalue of the adjacency matrix so that the centrality values are generally between 0 and 1. Returns ------- nodes : dictionary Dictionary of nodes with subgraph centrality as the value. Raises ------ NetworkXError If the graph is not undirected and simple. See Also -------- subgraph_centrality_exp: Alternative algorithm of the subgraph centrality for each node of G. Notes ----- This version of the algorithm computes eigenvalues and eigenvectors of the adjacency matrix. Subgraph centrality of a node `u` in G can be found using a spectral decomposition of the adjacency matrix [1]_, .. math:: SC(u)=\sum_{j=1}^{N}(v_{j}^{u})^2 e^{\lambda_{j}}, where `v_j` is an eigenvector of the adjacency matrix `A` of G corresponding to the eigenvalue `\lambda_j`. Examples -------- (Example from [1]_) >>> G = nx.Graph( ... [ ... (1, 2), ... (1, 5), ... (1, 8), ... (2, 3), ... (2, 8), ... (3, 4), ... (3, 6), ... (4, 5), ... (4, 7), ... (5, 6), ... (6, 7), ... (7, 8), ... ] ... ) >>> sc = nx.subgraph_centrality(G) >>> print([f"{node} {sc[node]:0.2f}" for node in sorted(sc)]) ['1 3.90', '2 3.90', '3 3.64', '4 3.71', '5 3.64', '6 3.71', '7 3.64', '8 3.90'] >>> sc = nx.subgraph_centrality(G, normalized=True) >>> print([f"{node} {sc[node]:0.3f}" for node in sorted(sc)]) ['1 0.194', '2 0.194', '3 0.181', '4 0.184', '5 0.181', '6 0.184', '7 0.181', '8 0.194'] References ---------- .. [1] Ernesto Estrada, Juan A. Rodriguez-Velazquez, "Subgraph centrality in complex networks", Physical Review E 71, 056103 (2005). https://arxiv.org/abs/cond-mat/0504730 N)numpyrr r nonzeror eigharrayexprrrr r) rrnprrwvvsquareexpwxgrs &$ rsubgraph_centralityr)osfAwH !&AAjjm 99>>! DAhhqkQGvva%%'k"vvay B c(CrN+ ,B Irc~^RIp^RIp\V4p\V4p\P !W4p^WQP V4&VPPV4p\\V\V444p/pVFp Wy,p WZR3,P4p VRV 3,P4p ^WZR3&^VRV 3&WbPPV4, V, p ^WR3&^V RV 3&WPVPV 44,p \V P44W&WV R3&WRV 3&K \V4pV^8dRRVR, ^,VR, , , pVP!4UUu/uFwppVVV,bK pppV#uuppi)aQReturns subgraph communicability for all pairs of nodes in G. Communicability betweenness measure makes use of the number of walks connecting every pair of nodes as the basis of a betweenness centrality measure. Parameters ---------- G: graph Returns ------- nodes : dictionary Dictionary of nodes with communicability betweenness as the value. Raises ------ NetworkXError If the graph is not undirected and simple. Notes ----- Let `G=(V,E)` be a simple undirected graph with `n` nodes and `m` edges, and `A` denote the adjacency matrix of `G`. Let `G(r)=(V,E(r))` be the graph resulting from removing all edges connected to node `r` but not the node itself. The adjacency matrix for `G(r)` is `A+E(r)`, where `E(r)` has nonzeros only in row and column `r`. The subraph betweenness of a node `r` is [1]_ .. math:: \omega_{r} = \frac{1}{C}\sum_{p}\sum_{q}\frac{G_{prq}}{G_{pq}}, p\neq q, q\neq r, where `G_{prq}=(e^{A}_{pq} - (e^{A+E(r)})_{pq}` is the number of walks involving node r, `G_{pq}=(e^{A})_{pq}` is the number of closed walks starting at node `p` and ending at node `q`, and `C=(n-1)^{2}-(n-1)` is a normalization factor equal to the number of terms in the sum. The resulting `\omega_{r}` takes values between zero and one. The lower bound cannot be attained for a connected graph, and the upper bound is attained in the star graph. References ---------- .. [1] Ernesto Estrada, Desmond J. Higham, Naomichi Hatano, "Communicability Betweenness in Complex Networks" Physica A 388 (2009) 764-774. https://arxiv.org/abs/0905.4102 Examples -------- >>> G = nx.Graph([(0, 1), (1, 2), (1, 5), (5, 4), (2, 4), (2, 3), (4, 3), (3, 6)]) >>> cbc = nx.communicability_betweenness_centrality(G) >>> print([f"{node} {cbc[node]:0.2f}" for node in sorted(cbc)]) ['0 0.03', '1 0.45', '2 0.51', '3 0.45', '4 0.40', '5 0.19', '6 0.03'] N:NNNg?)rrrlenr r rr r rrrangecopydiagrsumitems)rr#rrnrrmappingcbcr%irowcolBorderscalenodevalues& r&communicability_betweenness_centralityr<sHAwH H A !&AAjjm 99>>! D3xq*+G C  J1glln1gllnQ$!Q$ IINN1% % -Q$!Q$ WWRWWQZ  quuw!Q$!Q$" HE qy )US[9:69iikBk{tUtUU]"kB JCsF9cF\\V4P44#)uCReturns the Estrada index of a the graph G. The Estrada Index is a topological index of folding or 3D "compactness" ([1]_). Parameters ---------- G: graph Returns ------- estrada index: float Raises ------ NetworkXError If the graph is not undirected and simple. Notes ----- Let `G=(V,E)` be a simple undirected graph with `n` nodes and let `\lambda_{1}\leq\lambda_{2}\leq\cdots\lambda_{n}` be a non-increasing ordering of the eigenvalues of its adjacency matrix `A`. The Estrada index is ([1]_, [2]_) .. math:: EE(G)=\sum_{j=1}^n e^{\lambda _j}. References ---------- .. [1] E. Estrada, "Characterization of 3D molecular structure", Chem. Phys. Lett. 319, 713 (2000). https://doi.org/10.1016/S0009-2614(00)00158-5 .. [2] José Antonio de la Peñaa, Ivan Gutman, Juan Rada, "Estimating the Estrada index", Linear Algebra and its Applications. 427, 1 (2007). https://doi.org/10.1016/j.laa.2007.06.020 Examples -------- >>> G = nx.Graph([(0, 1), (1, 2), (1, 5), (5, 4), (2, 4), (2, 3), (4, 3), (3, 6)]) >>> ei = nx.estrada_index(G) >>> print(f"{ei:0.5}") 20.55 )r/r)r)rs&r estrada_indexr>;s\ "1%,,. //r)rr)r<r>) __doc__networkxr networkx.utilsr__all__ _dispatchablerr)r<r>rrrEs. Z \"YUY#!YxZ \"__#!_DZ \"a#!aH-0-0r