"""
Eigenvalue spectrum of graphs.
"""
# Copyright (C) 2004-2019 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
import networkx as nx
__author__ = "\n".join(['Aric Hagberg <aric.hagberg@gmail.com>',
'Pieter Swart (swart@lanl.gov)',
'Dan Schult(dschult@colgate.edu)',
'Jean-Gabriel Young (jean.gabriel.young@gmail.com)'])
__all__ = ['laplacian_spectrum', 'adjacency_spectrum', 'modularity_spectrum',
'normalized_laplacian_spectrum', 'bethe_hessian_spectrum']
[docs]def laplacian_spectrum(G, weight='weight'):
"""Returns eigenvalues of the Laplacian of G
Parameters
----------
G : graph
A NetworkX graph
weight : string or None, optional (default='weight')
The edge data key used to compute each value in the matrix.
If None, then each edge has weight 1.
Returns
-------
evals : NumPy array
Eigenvalues
Notes
-----
For MultiGraph/MultiDiGraph, the edges weights are summed.
See to_numpy_matrix for other options.
See Also
--------
laplacian_matrix
"""
from scipy.linalg import eigvalsh
return eigvalsh(nx.laplacian_matrix(G, weight=weight).todense())
[docs]def normalized_laplacian_spectrum(G, weight='weight'):
"""Return eigenvalues of the normalized Laplacian of G
Parameters
----------
G : graph
A NetworkX graph
weight : string or None, optional (default='weight')
The edge data key used to compute each value in the matrix.
If None, then each edge has weight 1.
Returns
-------
evals : NumPy array
Eigenvalues
Notes
-----
For MultiGraph/MultiDiGraph, the edges weights are summed.
See to_numpy_matrix for other options.
See Also
--------
normalized_laplacian_matrix
"""
from scipy.linalg import eigvalsh
return eigvalsh(nx.normalized_laplacian_matrix(G, weight=weight).todense())
[docs]def adjacency_spectrum(G, weight='weight'):
"""Returns eigenvalues of the adjacency matrix of G.
Parameters
----------
G : graph
A NetworkX graph
weight : string or None, optional (default='weight')
The edge data key used to compute each value in the matrix.
If None, then each edge has weight 1.
Returns
-------
evals : NumPy array
Eigenvalues
Notes
-----
For MultiGraph/MultiDiGraph, the edges weights are summed.
See to_numpy_matrix for other options.
See Also
--------
adjacency_matrix
"""
from scipy.linalg import eigvals
return eigvals(nx.adjacency_matrix(G, weight=weight).todense())
[docs]def modularity_spectrum(G):
"""Returns eigenvalues of the modularity matrix of G.
Parameters
----------
G : Graph
A NetworkX Graph or DiGraph
Returns
-------
evals : NumPy array
Eigenvalues
See Also
--------
modularity_matrix
References
----------
.. [1] M. E. J. Newman, "Modularity and community structure in networks",
Proc. Natl. Acad. Sci. USA, vol. 103, pp. 8577-8582, 2006.
"""
from scipy.linalg import eigvals
if G.is_directed():
return eigvals(nx.directed_modularity_matrix(G))
else:
return eigvals(nx.modularity_matrix(G))
[docs]def bethe_hessian_spectrum(G, r=None):
"""Returns eigenvalues of the Bethe Hessian matrix of G.
Parameters
----------
G : Graph
A NetworkX Graph or DiGraph
r : float
Regularizer parameter
Returns
-------
evals : NumPy array
Eigenvalues
See Also
--------
bethe_hessian_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.
"""
from scipy.linalg import eigvalsh
return eigvalsh(nx.bethe_hessian_matrix(G, r).todense())
# fixture for pytest
def setup_module(module):
import pytest
scipy.linalg = pytest.importorskip('scipy.linalg')