"""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. 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
