/
_laplacian.py
128 lines (111 loc) · 3.97 KB
/
_laplacian.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
"""
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, int)
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