"""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. 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.
"""
import scipy.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 = scipy.sparse.spdiags(diags.flatten(), [0], m, n, format="csr")
I = scipy.sparse.eye(m, n, format="csr")
return (r ** 2 - 1) * I - r * A + D