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") 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 : 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. Zdeborov√° "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. """ import scipy as sp import scipy.sparse # call as sp.sparse 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_matrix(G, nodelist=nodelist, format="csr") n, m = A.shape diags = A.sum(axis=1) D = sp.sparse.spdiags(diags.flatten(), [0], m, n, format="csr") I = sp.sparse.eye(m, n, format="csr") return (r ** 2 - 1) * I - r * A + D