3 d@s<dZddlZddlmZdgZededdddZdS) z5Bethe Hessian or deformed Laplacian matrix of graphs.N)not_implemented_forbethe_hessian_matrixZdirectedZ multigraphc Csddl}|dkrt|}|dkrTtddtj|Dtddtj|Dd}tj||dd}|j\}}|jdd }|jj|j dg||dd }|jj ||dd } |d d| |||S) uReturns 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 : Numpy matrix The Bethe Hessian matrix of G, with paramter r. Examples -------- >>> k = [3, 2, 2, 1, 0] >>> G = nx.havel_hakimi_graph(k) >>> H = nx.modularity_matrix(G) See Also -------- bethe_hessian_spectrum to_numpy_array 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. Lee, E. Levina "Estimating the number of communities in networks by spectral methods" arXiv:1507.00827, 2015. rNcSsg|]\}}|dqS)).0vdrrF/tmp/pip-build-7vycvbft/networkx/networkx/linalg/bethehessianmatrix.py Fsz(bethe_hessian_matrix..cSsg|] \}}|qSrr)rrrrrr r FsZcsr)nodelistformat)Zaxis)r r) Z scipy.sparselistsumnxZdegreeZto_scipy_sparse_matrixshapesparseZspdiagsflattenZeye) Grr ZscipyAnmZdiagsDIrrr rs8.  )NN)__doc__ZnetworkxrZnetworkx.utilsr__all__rrrrr s