+ jRt^RIt^RIHtRR.t]!R4]!R4]P R444t]!R4]!R4]P R444tR#) z Communicability. N)not_implemented_forcommunicabilitycommunicability_expdirected multigraphc^RIp\P!VRR7pVPP V4wr4WAP V4,VP ,p\V4UUUU u/uF<wrgT\V4UU u/uFwrV \WV,V,4bK up pbK> up ppp#uup piuup pppi)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 matrix exponentiation 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: .. 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 $\lambda_{j}$ [1]_. 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) N)weight) numpynxto_numpy_arraylinalgeighexpT enumeratefloat) GnpAwvecriujvs & h/home/luca/screen-checking/.venv/lib/python3.14/site-packages/networkx/algorithms/communicability_alg.pyrr st !D)A YY^^A FAVVAY#%%/O aL  DA ! E Au_'*+ + EE   E s2C $B:,C :C c d^RIp\V4p\P!W4p^W3R8g&VPP V4p\ \V\\V4444p/pVF9p/Wg&VF,p\WEV,WX,3,4Wg,V&K. K; V#)aHReturns 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` via spectral decomposition 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: .. math:: C(u,v) = (e^A)_{uv}, where ``A`` is the adjacency matrix of `G` [1]_. 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) Ng) scipylistr r r expmdictziprangelenr) rspnodelistrexpAmappingcrrs & rrrRspAwH !&AA3hK 99>>! D3xs8}!567G A ADWZ!789ADG H) __doc__networkxr networkx.utilsr__all__ _dispatchablerrr)rr0s. 3 4Z \"A#!AHZ \"C #!C r)