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| """Bethe Hessian or deformed Laplacian matrix of graphs.""" | |
| import networkx as nx | |
| from networkx.utils import not_implemented_for | |
| __all__ = ["bethe_hessian_matrix"] | |
| def bethe_hessian_matrix(G, r=None, nodelist=None): | |
| r"""Returns 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. | |
| """ | |
| import scipy as sp | |
| if nodelist is None: | |
| nodelist = list(G) | |
| if r is None: | |
| r = sum(d**2 for v, d in nx.degree(G)) / sum(d for v, d in nx.degree(G)) - 1 | |
| A = nx.to_scipy_sparse_array(G, nodelist=nodelist, format="csr") | |
| n, m = A.shape | |
| # TODO: Rm csr_array wrapper when spdiags array creation becomes available | |
| D = sp.sparse.csr_array(sp.sparse.spdiags(A.sum(axis=1), 0, m, n, format="csr")) | |
| # TODO: Rm csr_array wrapper when eye array creation becomes available | |
| I = sp.sparse.csr_array(sp.sparse.eye(m, n, format="csr")) | |
| return (r**2 - 1) * I - r * A + D | |