U ?h@sTdZddlZddlmZddgZededddZededd dZdS) z Communicability. N)not_implemented_forcommunicabilitycommunicability_expZdirectedZ multigraphcCsddl}t|}t||}d||dk<|j|\}}||}tt|t t |}i}|D]} i|| <|D]n} d} || } || } t t |D]8}| |dd|f| |dd|f| ||7} qt | || | <qpq`|S)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_arraylinalgZeighexpdictziprangelenfloat)GnpnodelistAwZvecZexpwmappingcuvspqjrY/var/www/html/venv/lib/python3.8/site-packages/networkx/algorithms/communicability_alg.pyr s$:   6c Csddl}ddl}t|}t||}d||dk<|j|}tt|t t |}i}|D]6}i||<|D]$} t ||||| f||| <qjqZ|S)aReturns 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) rNrr) scipyZ scipy.linalgrr r r Zexpmr rrrr) rspr!rrZexpArrrrrrr r[s7   $)__doc__Znetworkxr Znetworkx.utilsr__all__rrrrrr s  O