Source code for networkx.linalg.bethehessianmatrix

"""Bethe Hessian or deformed Laplacian matrix of graphs."""
import networkx as nx
from networkx.utils import not_implemented_for

__all__ = ["bethe_hessian_matrix"]


[docs] @not_implemented_for("directed") @not_implemented_for("multigraph") @nx._dispatchable 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. Zdeborov√° "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