"""Laplacian matrix of graphs.
"""
import networkx as nx
from networkx.utils import not_implemented_for
__all__ = [
"laplacian_matrix",
"normalized_laplacian_matrix",
"directed_laplacian_matrix",
"directed_combinatorial_laplacian_matrix",
]
[docs]@not_implemented_for("directed")
def laplacian_matrix(G, nodelist=None, weight="weight"):
"""Returns the Laplacian matrix of G.
The graph Laplacian is the matrix L = D - A, where
A is the adjacency matrix and D is the diagonal matrix of node degrees.
Parameters
----------
G : graph
A NetworkX graph
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().
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
-------
L : SciPy sparse matrix
The Laplacian matrix of G.
Notes
-----
For MultiGraph/MultiDiGraph, the edges weights are summed.
See Also
--------
to_numpy_array
normalized_laplacian_matrix
laplacian_spectrum
"""
import scipy.sparse
if nodelist is None:
nodelist = list(G)
A = nx.to_scipy_sparse_matrix(G, nodelist=nodelist, weight=weight, format="csr")
n, m = A.shape
diags = A.sum(axis=1)
D = scipy.sparse.spdiags(diags.flatten(), [0], m, n, format="csr")
return D - A
[docs]@not_implemented_for("directed")
def normalized_laplacian_matrix(G, nodelist=None, weight="weight"):
r"""Returns the normalized Laplacian matrix of G.
The normalized graph Laplacian is the matrix
.. math::
N = D^{-1/2} L D^{-1/2}
where `L` is the graph Laplacian and `D` is the diagonal matrix of
node degrees.
Parameters
----------
G : graph
A NetworkX graph
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().
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
-------
N : Scipy sparse matrix
The normalized Laplacian matrix of G.
Notes
-----
For MultiGraph/MultiDiGraph, the edges weights are summed.
See to_numpy_array for other options.
If the Graph contains selfloops, D is defined as diag(sum(A,1)), where A is
the adjacency matrix [2]_.
See Also
--------
laplacian_matrix
normalized_laplacian_spectrum
References
----------
.. [1] Fan Chung-Graham, Spectral Graph Theory,
CBMS Regional Conference Series in Mathematics, Number 92, 1997.
.. [2] Steve Butler, Interlacing For Weighted Graphs Using The Normalized
Laplacian, Electronic Journal of Linear Algebra, Volume 16, pp. 90-98,
March 2007.
"""
import numpy as np
import scipy
import scipy.sparse
if nodelist is None:
nodelist = list(G)
A = nx.to_scipy_sparse_matrix(G, nodelist=nodelist, weight=weight, format="csr")
n, m = A.shape
diags = A.sum(axis=1).flatten()
D = scipy.sparse.spdiags(diags, [0], m, n, format="csr")
L = D - A
with scipy.errstate(divide="ignore"):
diags_sqrt = 1.0 / np.sqrt(diags)
diags_sqrt[np.isinf(diags_sqrt)] = 0
DH = scipy.sparse.spdiags(diags_sqrt, [0], m, n, format="csr")
return DH.dot(L.dot(DH))
###############################################################################
# Code based on
# https://bitbucket.org/bedwards/networkx-community/src/370bd69fc02f/networkx/algorithms/community/
[docs]@not_implemented_for("undirected")
@not_implemented_for("multigraph")
def directed_laplacian_matrix(
G, nodelist=None, weight="weight", walk_type=None, alpha=0.95
):
r"""Returns the directed Laplacian matrix of G.
The graph directed Laplacian is the matrix
.. math::
L = I - (\Phi^{1/2} P \Phi^{-1/2} + \Phi^{-1/2} P^T \Phi^{1/2} ) / 2
where `I` is the identity matrix, `P` is the transition matrix of the
graph, and `\Phi` a matrix with the Perron vector of `P` in the diagonal and
zeros elsewhere.
Depending on the value of walk_type, `P` can be the transition matrix
induced by a random walk, a lazy random walk, or a random walk with
teleportation (PageRank).
Parameters
----------
G : DiGraph
A NetworkX graph
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().
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.
walk_type : string or None, optional (default=None)
If None, `P` is selected depending on the properties of the
graph. Otherwise is one of 'random', 'lazy', or 'pagerank'
alpha : real
(1 - alpha) is the teleportation probability used with pagerank
Returns
-------
L : NumPy matrix
Normalized Laplacian of G.
Notes
-----
Only implemented for DiGraphs
See Also
--------
laplacian_matrix
References
----------
.. [1] Fan Chung (2005).
Laplacians and the Cheeger inequality for directed graphs.
Annals of Combinatorics, 9(1), 2005
"""
import numpy as np
from scipy.sparse import spdiags, linalg
P = _transition_matrix(
G, nodelist=nodelist, weight=weight, walk_type=walk_type, alpha=alpha
)
n, m = P.shape
evals, evecs = linalg.eigs(P.T, k=1)
v = evecs.flatten().real
p = v / v.sum()
sqrtp = np.sqrt(p)
Q = spdiags(sqrtp, [0], n, n) * P * spdiags(1.0 / sqrtp, [0], n, n)
I = np.identity(len(G))
return I - (Q + Q.T) / 2.0
[docs]@not_implemented_for("undirected")
@not_implemented_for("multigraph")
def directed_combinatorial_laplacian_matrix(
G, nodelist=None, weight="weight", walk_type=None, alpha=0.95
):
r"""Return the directed combinatorial Laplacian matrix of G.
The graph directed combinatorial Laplacian is the matrix
.. math::
L = \Phi - (\Phi P + P^T \Phi) / 2
where `P` is the transition matrix of the graph and and `\Phi` a matrix
with the Perron vector of `P` in the diagonal and zeros elsewhere.
Depending on the value of walk_type, `P` can be the transition matrix
induced by a random walk, a lazy random walk, or a random walk with
teleportation (PageRank).
Parameters
----------
G : DiGraph
A NetworkX graph
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().
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.
walk_type : string or None, optional (default=None)
If None, `P` is selected depending on the properties of the
graph. Otherwise is one of 'random', 'lazy', or 'pagerank'
alpha : real
(1 - alpha) is the teleportation probability used with pagerank
Returns
-------
L : NumPy matrix
Combinatorial Laplacian of G.
Notes
-----
Only implemented for DiGraphs
See Also
--------
laplacian_matrix
References
----------
.. [1] Fan Chung (2005).
Laplacians and the Cheeger inequality for directed graphs.
Annals of Combinatorics, 9(1), 2005
"""
from scipy.sparse import spdiags, linalg
P = _transition_matrix(
G, nodelist=nodelist, weight=weight, walk_type=walk_type, alpha=alpha
)
n, m = P.shape
evals, evecs = linalg.eigs(P.T, k=1)
v = evecs.flatten().real
p = v / v.sum()
Phi = spdiags(p, [0], n, n)
Phi = Phi.todense()
return Phi - (Phi * P + P.T * Phi) / 2.0
def _transition_matrix(G, nodelist=None, weight="weight", walk_type=None, alpha=0.95):
"""Returns the transition matrix of G.
This is a row stochastic giving the transition probabilities while
performing a random walk on the graph. Depending on the value of walk_type,
P can be the transition matrix induced by a random walk, a lazy random walk,
or a random walk with teleportation (PageRank).
Parameters
----------
G : DiGraph
A NetworkX graph
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().
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.
walk_type : string or None, optional (default=None)
If None, `P` is selected depending on the properties of the
graph. Otherwise is one of 'random', 'lazy', or 'pagerank'
alpha : real
(1 - alpha) is the teleportation probability used with pagerank
Returns
-------
P : NumPy matrix
transition matrix of G.
Raises
------
NetworkXError
If walk_type not specified or alpha not in valid range
"""
import numpy as np
from scipy.sparse import identity, spdiags
if walk_type is None:
if nx.is_strongly_connected(G):
if nx.is_aperiodic(G):
walk_type = "random"
else:
walk_type = "lazy"
else:
walk_type = "pagerank"
M = nx.to_scipy_sparse_matrix(G, nodelist=nodelist, weight=weight, dtype=float)
n, m = M.shape
if walk_type in ["random", "lazy"]:
DI = spdiags(1.0 / np.array(M.sum(axis=1).flat), [0], n, n)
if walk_type == "random":
P = DI * M
else:
I = identity(n)
P = (I + DI * M) / 2.0
elif walk_type == "pagerank":
if not (0 < alpha < 1):
raise nx.NetworkXError("alpha must be between 0 and 1")
# this is using a dense representation
M = M.todense()
# add constant to dangling nodes' row
dangling = np.where(M.sum(axis=1) == 0)
for d in dangling[0]:
M[d] = 1.0 / n
# normalize
M = M / M.sum(axis=1)
P = alpha * M + (1 - alpha) / n
else:
raise nx.NetworkXError("walk_type must be random, lazy, or pagerank")
return P