ó ú£Õ\c@s[dZddlZddlTdjddgƒZddd d d d gZed ƒedƒedƒd„ƒƒƒZ edƒedƒedƒd„ƒƒƒZ ed ƒedƒedƒe d„ƒƒƒZ d„Z edd ƒedƒedƒd„ƒƒƒZed ƒedƒedƒd„ƒƒƒZedƒd„ƒZd„ZdS(s* Communicability and centrality measures. iÿÿÿÿN(t*s sAric Hagberg (hagberg@lanl.gov)s$Franck Kalala (franckkalala@yahoo.frtcommunicability_centrality_exptcommunicability_centralityt&communicability_betweenness_centralitytcommunicabilitytcommunicability_expt estrada_indextscipytdirectedt multigraphcCstddl}|jƒ}tj||ƒ}d||dk<|jj|ƒ}tt|tt |j ƒƒƒƒ}|S(sÐReturn the communicability centrality for each node of G Communicability centrality, also called subgraph centrality, of a node `n` is the sum of closed walks of all lengths starting and ending at node `n`. Parameters ---------- G: graph Returns ------- nodes:dictionary Dictionary of nodes with communicability centrality as the value. Raises ------ NetworkXError If the graph is not undirected and simple. See Also -------- communicability: Communicability between all pairs of nodes in G. communicability_centrality: Communicability centrality for each node of G. Notes ----- This version of the algorithm exponentiates the adjacency matrix. The communicability centrality of a node `u` in G can be found using the matrix exponential of the adjacency matrix of G [1]_ [2]_, .. math:: SC(u)=(e^A)_{uu} . References ---------- .. [1] Ernesto Estrada, Juan A. Rodriguez-Velazquez, "Subgraph centrality in complex networks", Physical Review E 71, 056103 (2005). http://arxiv.org/abs/cond-mat/0504730 .. [2] Ernesto Estrada, Naomichi Hatano, "Communicability in complex networks", Phys. Rev. E 77, 036111 (2008). http://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)]) >>> sc = nx.communicability_centrality_exp(G) iÿÿÿÿNig( t scipy.linalgtnodestnxtto_numpy_matrixtlinalgtexpmtdicttziptmaptfloattdiagonal(tGRtnodelisttAtexpAtsc((sj/Users/dxp/prism/prism-games/prism-examples/smgs/car/networkx/algorithms/centrality/communicability_alg.pyRs:  $tnumpyc Cs´ddl}ddl}|jƒ}tj||ƒ}d||dk<|jj|ƒ\}}|j|ƒd}|j|ƒ}|j ||ƒ}t t |t t |ƒƒƒ} | S(sReturn communicability centrality for each node in G. Communicability centrality, also called subgraph centrality, of a node `n` is the sum of closed walks of all lengths starting and ending at node `n`. Parameters ---------- G: graph Returns ------- nodes: dictionary Dictionary of nodes with communicability centrality as the value. Raises ------ NetworkXError If the graph is not undirected and simple. See Also -------- communicability: Communicability between all pairs of nodes in G. communicability_centrality: Communicability centrality for each node of G. Notes ----- This version of the algorithm computes eigenvalues and eigenvectors of the adjacency matrix. Communicability centrality of a node `u` in G can be found using a spectral decomposition of the adjacency matrix [1]_ [2]_, .. 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 corresponding to the eigenvalue `\lambda_j`. Examples -------- >>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)]) >>> sc = nx.communicability_centrality(G) References ---------- .. [1] Ernesto Estrada, Juan A. Rodriguez-Velazquez, "Subgraph centrality in complex networks", Physical Review E 71, 056103 (2005). http://arxiv.org/abs/cond-mat/0504730 .. [2] Ernesto Estrada, Naomichi Hatano, "Communicability in complex networks", Phys. Rev. E 77, 036111 (2008). http://arxiv.org/abs/0707.0756 iÿÿÿÿNigi(Rt numpy.linalgR R R RteightarraytexptdotRRRR( RRRRtwtvtvsquaretexpwtxgR((sj/Users/dxp/prism/prism-games/prism-examples/smgs/car/networkx/algorithms/centrality/communicability_alg.pyRZs=   cCs¾ddl}ddl}|jƒ}t|ƒ}tj||ƒ}d||dk<|jj|ƒ}tt |t |ƒƒƒ}i}x |D]} || } || dd…fj ƒ} |dd…| fj ƒ} d|| dd…f>> 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) iÿÿÿÿNigit normalized(RR R tlenR R RRRRtrangetcopytdiagRtsumt_rescale(RR%RRtnRRtmappingRR!titrowtcoltB((sj/Users/dxp/prism/prism-games/prism-examples/smgs/car/networkx/algorithms/centrality/communicability_alg.pyR¥s0K      cCs~|tkrJt|ƒ}|dkr-d}qJd|dd|d}n|dk rzx!|D]}||c|9>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)]) >>> c = nx.communicability(G) iÿÿÿÿNigi( RR R R R RRRRRR'R&R(RRRRRR tvecR#R-RtuR!tstptqtj((sj/Users/dxp/prism/prism-games/prism-examples/smgs/car/networkx/algorithms/centrality/communicability_alg.pyRs&=   !     Nc CsÉddl}|jƒ}tj||ƒ}d||dk<|jj|ƒ}tt|tt |ƒƒƒƒ}i}xO|D]G}i||>> G = nx.Graph([(0,1),(1,2),(1,5),(5,4),(2,4),(2,3),(4,3),(3,6)]) >>> c = nx.communicability_exp(G) iÿÿÿÿNig( R R R R RRRRR'R&R( RRRRRR-RR7R!((sj/Users/dxp/prism/prism-games/prism-examples/smgs/car/networkx/algorithms/centrality/communicability_alg.pyRos<  !   .cCstt|ƒjƒƒS(s^Return the Estrada index of a the graph G. Parameters ---------- G: graph Returns ------- estrada index: float Raises ------ NetworkXError If the graph is not undirected and simple. See also -------- estrada_index_exp 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 .. 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). 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) (R*Rtvalues(R((sj/Users/dxp/prism/prism-games/prism-examples/smgs/car/networkx/algorithms/centrality/communicability_alg.pyRºs)cCs`ddlm}yddl}Wn|dƒ‚nXyddl}Wn|dƒ‚nXdS(Niÿÿÿÿ(tSkipTestsNumPy not availablesSciPy not available(tnoseR=RR(tmoduleR=RR((sj/Users/dxp/prism/prism-games/prism-examples/smgs/car/networkx/algorithms/centrality/communicability_alg.pyt setup_moduleæs(t__doc__tnetworkxR tnetworkx.utilstjoint __author__t__all__trequiretnot_implemented_forRRR2RR+RRRR@(((sj/Users/dxp/prism/prism-games/prism-examples/smgs/car/networkx/algorithms/centrality/communicability_alg.pyts:       B  I   f   Q  I,