hSrSSKrSSKJr SS/r\"S5\"S5\R S555r\"S5\"S5\R S 555rg) z Communicability. N)not_implemented_forcommunicabilitycommunicability_expdirected multigraphc SSKn[U5n[R"X5nSX3S:g'URR U5upEUR U5n[[U[[U5555n0nUHen 0X'UHXn Sn Xyn Xzn [[U55H#nXSS2U4U USS2U4U -Xn-- n M% [U 5XU 'MZ Mg U$)aReturns communicability between all pairs of nodes in G. The communicability between pairs of nodes in G is the sum of walks of different lengths starting at node u and ending at node v. Parameters ---------- G: graph Returns ------- comm: dictionary of dictionaries Dictionary of dictionaries keyed by nodes with communicability as the value. Raises ------ NetworkXError If the graph is not undirected and simple. See Also -------- communicability_exp: Communicability between all pairs of nodes in G using spectral decomposition. communicability_betweenness_centrality: Communicability betweenness centrality for each node in G. Notes ----- This algorithm uses a spectral decomposition of the adjacency matrix. Let G=(V,E) be a simple undirected graph. Using the connection between the powers of the adjacency matrix and the number of walks in the graph, the communicability between nodes `u` and `v` based on the graph spectrum is [1]_ .. math:: C(u,v)=\sum_{j=1}^{n}\phi_{j}(u)\phi_{j}(v)e^{\lambda_{j}}, where `\phi_{j}(u)` is the `u\rm{th}` element of the `j\rm{th}` orthonormal eigenvector of the adjacency matrix associated with the eigenvalue `\lambda_{j}`. References ---------- .. [1] Ernesto Estrada, Naomichi Hatano, "Communicability in complex networks", Phys. Rev. E 77, 036111 (2008). https://arxiv.org/abs/0707.0756 Examples -------- >>> G = nx.Graph([(0, 1), (1, 2), (1, 5), (5, 4), (2, 4), (2, 3), (4, 3), (3, 6)]) >>> c = nx.communicability(G) rN) numpylistnxto_numpy_arraylinalgeighexpdictziprangelenfloat)GnpnodelistAwvecexpwmappingcuvspqjs Y/var/www/html/env/lib/python3.13/site-packages/networkx/algorithms/communicability_alg.pyrr svAwH !&AA3hK YY^^A FA 66!9D3xs8}!567G A AA A A3x=)AYq\C1IaL047::*AhADG  Hc DSSKn[U5n[R"X5nSX3S:g'URR U5n[ [U[[U5555n0nUH)n0Xg'UHn[XEUXX45XgU'M M+ U$)a Returns communicability between all pairs of nodes in G. Communicability between pair of node (u,v) of node in G is the sum of walks of different lengths starting at node u and ending at node v. Parameters ---------- G: graph Returns ------- comm: dictionary of dictionaries Dictionary of dictionaries keyed by nodes with communicability as the value. Raises ------ NetworkXError If the graph is not undirected and simple. See Also -------- communicability: Communicability between pairs of nodes in G. communicability_betweenness_centrality: Communicability betweenness centrality for each node in G. Notes ----- This algorithm uses matrix exponentiation of the adjacency matrix. Let G=(V,E) be a simple undirected graph. Using the connection between the powers of the adjacency matrix and the number of walks in the graph, the communicability between nodes u and v is [1]_, .. math:: C(u,v) = (e^A)_{uv}, where `A` is the adjacency matrix of G. References ---------- .. [1] Ernesto Estrada, Naomichi Hatano, "Communicability in complex networks", Phys. Rev. E 77, 036111 (2008). https://arxiv.org/abs/0707.0756 Examples -------- >>> G = nx.Graph([(0, 1), (1, 2), (1, 5), (5, 4), (2, 4), (2, 3), (4, 3), (3, 6)]) >>> c = nx.communicability_exp(G) rNr r ) scipyr r rrexpmrrrrr) rsprrexpArrr r!s r&rr]spAwH !&AA3hK 99>>! D3xs8}!567G A ADWZ!789ADG Hr') __doc__networkxr networkx.utilsr__all__ _dispatchablerrr'r&r3s. 3 4Z \"L #!L ^Z \"C #!C r'