bethe_hessian_matrix(G, r=None, nodelist=None)
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
$$r_m = \left(\sum k_i \right)^{-1}\left(\sum k_i^2 \right) - 1$$A NetworkX graph
Regularizer parameter
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 the Bethe Hessian matrix of G.
>>> k = [3, 2, 2, 1, 0]See :
... 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]])
The following pages refer to to this document either explicitly or contain code examples using this.
networkx.linalg.spectrum.bethe_hessian_spectrum
networkx.linalg.bethehessianmatrix.bethe_hessian_matrix
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