You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
129 lines
4.0 KiB
Python
129 lines
4.0 KiB
Python
6 years ago
|
"""
|
||
|
|
Laplacian of a compressed-sparse graph
|
||
|
|
"""
|
||
|
|
|
||
|
|
# Authors: Aric Hagberg <hagberg@lanl.gov>
|
||
|
|
# Gael Varoquaux <gael.varoquaux@normalesup.org>
|
||
|
|
# Jake Vanderplas <vanderplas@astro.washington.edu>
|
||
|
|
# License: BSD
|
||
|
|
|
||
|
|
from __future__ import division, print_function, absolute_import
|
||
|
|
|
||
|
|
import numpy as np
|
||
|
|
from scipy.sparse import isspmatrix
|
||
|
|
|
||
|
|
|
||
|
|
###############################################################################
|
||
|
|
# Graph laplacian
|
||
|
|
def laplacian(csgraph, normed=False, return_diag=False, use_out_degree=False):
|
||
|
|
"""
|
||
|
|
Return the Laplacian matrix of a directed graph.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
csgraph : array_like or sparse matrix, 2 dimensions
|
||
|
|
compressed-sparse graph, with shape (N, N).
|
||
|
|
normed : bool, optional
|
||
|
|
If True, then compute normalized Laplacian.
|
||
|
|
return_diag : bool, optional
|
||
|
|
If True, then also return an array related to vertex degrees.
|
||
|
|
use_out_degree : bool, optional
|
||
|
|
If True, then use out-degree instead of in-degree.
|
||
|
|
This distinction matters only if the graph is asymmetric.
|
||
|
|
Default: False.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
lap : ndarray or sparse matrix
|
||
|
|
The N x N laplacian matrix of csgraph. It will be a numpy array (dense)
|
||
|
|
if the input was dense, or a sparse matrix otherwise.
|
||
|
|
diag : ndarray, optional
|
||
|
|
The length-N diagonal of the Laplacian matrix.
|
||
|
|
For the normalized Laplacian, this is the array of square roots
|
||
|
|
of vertex degrees or 1 if the degree is zero.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
The Laplacian matrix of a graph is sometimes referred to as the
|
||
|
|
"Kirchoff matrix" or the "admittance matrix", and is useful in many
|
||
|
|
parts of spectral graph theory. In particular, the eigen-decomposition
|
||
|
|
of the laplacian matrix can give insight into many properties of the graph.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from scipy.sparse import csgraph
|
||
|
|
>>> G = np.arange(5) * np.arange(5)[:, np.newaxis]
|
||
|
|
>>> G
|
||
|
|
array([[ 0, 0, 0, 0, 0],
|
||
|
|
[ 0, 1, 2, 3, 4],
|
||
|
|
[ 0, 2, 4, 6, 8],
|
||
|
|
[ 0, 3, 6, 9, 12],
|
||
|
|
[ 0, 4, 8, 12, 16]])
|
||
|
|
>>> csgraph.laplacian(G, normed=False)
|
||
|
|
array([[ 0, 0, 0, 0, 0],
|
||
|
|
[ 0, 9, -2, -3, -4],
|
||
|
|
[ 0, -2, 16, -6, -8],
|
||
|
|
[ 0, -3, -6, 21, -12],
|
||
|
|
[ 0, -4, -8, -12, 24]])
|
||
|
|
"""
|
||
|
|
if csgraph.ndim != 2 or csgraph.shape[0] != csgraph.shape[1]:
|
||
|
|
raise ValueError('csgraph must be a square matrix or array')
|
||
|
|
|
||
|
|
if normed and (np.issubdtype(csgraph.dtype, np.signedinteger)
|
||
|
|
or np.issubdtype(csgraph.dtype, np.uint)):
|
||
|
|
csgraph = csgraph.astype(float)
|
||
|
|
|
||
|
|
create_lap = _laplacian_sparse if isspmatrix(csgraph) else _laplacian_dense
|
||
|
|
degree_axis = 1 if use_out_degree else 0
|
||
|
|
lap, d = create_lap(csgraph, normed=normed, axis=degree_axis)
|
||
|
|
if return_diag:
|
||
|
|
return lap, d
|
||
|
|
return lap
|
||
|
|
|
||
|
|
|
||
|
|
def _setdiag_dense(A, d):
|
||
|
|
A.flat[::len(d)+1] = d
|
||
|
|
|
||
|
|
|
||
|
|
def _laplacian_sparse(graph, normed=False, axis=0):
|
||
|
|
if graph.format in ('lil', 'dok'):
|
||
|
|
m = graph.tocoo()
|
||
|
|
needs_copy = False
|
||
|
|
else:
|
||
|
|
m = graph
|
||
|
|
needs_copy = True
|
||
|
|
w = m.sum(axis=axis).getA1() - m.diagonal()
|
||
|
|
if normed:
|
||
|
|
m = m.tocoo(copy=needs_copy)
|
||
|
|
isolated_node_mask = (w == 0)
|
||
|
|
w = np.where(isolated_node_mask, 1, np.sqrt(w))
|
||
|
|
m.data /= w[m.row]
|
||
|
|
m.data /= w[m.col]
|
||
|
|
m.data *= -1
|
||
|
|
m.setdiag(1 - isolated_node_mask)
|
||
|
|
else:
|
||
|
|
if m.format == 'dia':
|
||
|
|
m = m.copy()
|
||
|
|
else:
|
||
|
|
m = m.tocoo(copy=needs_copy)
|
||
|
|
m.data *= -1
|
||
|
|
m.setdiag(w)
|
||
|
|
return m, w
|
||
|
|
|
||
|
|
|
||
|
|
def _laplacian_dense(graph, normed=False, axis=0):
|
||
|
|
m = np.array(graph)
|
||
|
|
np.fill_diagonal(m, 0)
|
||
|
|
w = m.sum(axis=axis)
|
||
|
|
if normed:
|
||
|
|
isolated_node_mask = (w == 0)
|
||
|
|
w = np.where(isolated_node_mask, 1, np.sqrt(w))
|
||
|
|
m /= w
|
||
|
|
m /= w[:, np.newaxis]
|
||
|
|
m *= -1
|
||
|
|
_setdiag_dense(m, 1 - isolated_node_mask)
|
||
|
|
else:
|
||
|
|
m *= -1
|
||
|
|
_setdiag_dense(m, w)
|
||
|
|
return m, w
|