"""
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
import numpy as np
from scipy.sparse import isspmatrix
###############################################################################
# Graph laplacian
[docs]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 symmetric 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