U +þjg£ ã@s<dZddlZddlmZdgZedƒedƒddd„ƒƒZdS) z5Bethe Hessian or deformed Laplacian matrix of graphs.éN)Únot_implemented_forÚbethe_hessian_matrixZdirectedZ multigraphc CsÒddl}ddl}|dkr t|ƒ}|dkr\tdd„t |¡Dƒƒtdd„t |¡Dƒƒd}tj||dd}|j\}}|j  |jj |jdd d||dd ¡}|j  |jj ||dd ¡} |d d| |||S) uìReturns the Bethe Hessian matrix of G. The Bethe Hessian is a family of matrices parametrized by r, defined as H(r) = (r^2 - 1) I - r A + D where A is the adjacency matrix, D is the diagonal matrix of node degrees, and I is the identify matrix. It is equal to the graph laplacian when the regularizer r = 1. The default choice of regularizer should be the ratio [2]_ .. math:: r_m = \left(\sum k_i \right)^{-1}\left(\sum k_i^2 \right) - 1 Parameters ---------- G : Graph A NetworkX graph r : float Regularizer parameter nodelist : list, optional The rows and columns are ordered according to the nodes in nodelist. If nodelist is None, then the ordering is produced by ``G.nodes()``. Returns ------- H : scipy.sparse.csr_array The Bethe Hessian matrix of `G`, with parameter `r`. Examples -------- >>> k = [3, 2, 2, 1, 0] >>> G = nx.havel_hakimi_graph(k) >>> H = nx.bethe_hessian_matrix(G) >>> H.toarray() array([[ 3.5625, -1.25 , -1.25 , -1.25 , 0. ], [-1.25 , 2.5625, -1.25 , 0. , 0. ], [-1.25 , -1.25 , 2.5625, 0. , 0. ], [-1.25 , 0. , 0. , 1.5625, 0. ], [ 0. , 0. , 0. , 0. , 0.5625]]) See Also -------- bethe_hessian_spectrum adjacency_matrix laplacian_matrix References ---------- .. [1] A. Saade, F. Krzakala and L. Zdeborová "Spectral Clustering of Graphs with the Bethe Hessian", Advances in Neural Information Processing Systems, 2014. .. [2] C. M. Le, E. Levina "Estimating the number of communities in networks by spectral methods" arXiv:1507.00827, 2015. rNcss|]\}}|dVqdS)éN©©Ú.0ÚvÚdrrúF/tmp/pip-unpacked-wheel-9meu_2xh/networkx/linalg/bethehessianmatrix.pyÚ Gsz'bethe_hessian_matrix..css|]\}}|VqdS)Nrrrrr r GséZcsr)ÚnodelistÚformat)Zaxis)rr) ÚscipyZ scipy.sparseÚlistÚsumÚnxZdegreeZto_scipy_sparse_arrayÚshapeÚsparseZ csr_arrayZspdiagsZeye) ÚGÚrr ÚsprÚAÚnÚmÚDÚIrrr rs94 &)NN)Ú__doc__ZnetworkxrZnetworkx.utilsrÚ__all__rrrrr Ús