# Source code for networkx.linalg.laplacianmatrix

```"""Laplacian matrix of graphs.
"""
import networkx as nx
from networkx.utils import not_implemented_for

__all__ = [
"laplacian_matrix",
"normalized_laplacian_matrix",
"total_spanning_tree_weight",
"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, the edges weights are summed.

--------
to_numpy_array
normalized_laplacian_matrix
laplacian_spectrum
"""
import scipy as sp
import scipy.sparse  # call as sp.sparse

if nodelist is None:
nodelist = list(G)
A = nx.to_scipy_sparse_array(G, nodelist=nodelist, weight=weight, format="csr")
n, m = A.shape
# TODO: rm csr_array wrapper when spdiags can produce arrays
D = sp.sparse.csr_array(sp.sparse.spdiags(A.sum(axis=1), 0, m, n, format="csr"))
import warnings

warnings.warn(
"laplacian_matrix will return a scipy.sparse array instead of a matrix in Networkx 3.0.",
FutureWarning,
stacklevel=2,
)
# TODO: rm sp.sparse.csr_matrix in version 3.0
return sp.sparse.csr_matrix(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, the edges weights are summed.
See :func:`to_numpy_array` for other options.

If the Graph contains selfloops, D is defined as ``diag(sum(A, 1))``, where A is

--------
laplacian_matrix
normalized_laplacian_spectrum

References
----------
..  Fan Chung-Graham, Spectral Graph Theory,
CBMS Regional Conference Series in Mathematics, Number 92, 1997.
..  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 as sp
import scipy.sparse  # call as sp.sparse

if nodelist is None:
nodelist = list(G)
A = nx.to_scipy_sparse_array(G, nodelist=nodelist, weight=weight, format="csr")
n, m = A.shape
diags = A.sum(axis=1)
# TODO: rm csr_array wrapper when spdiags can produce arrays
D = sp.sparse.csr_array(sp.sparse.spdiags(diags, 0, m, n, format="csr"))
L = D - A
with sp.errstate(divide="ignore"):
diags_sqrt = 1.0 / np.sqrt(diags)
diags_sqrt[np.isinf(diags_sqrt)] = 0
# TODO: rm csr_array wrapper when spdiags can produce arrays
DH = sp.sparse.csr_array(sp.sparse.spdiags(diags_sqrt, 0, m, n, format="csr"))
import warnings

warnings.warn(
"normalized_laplacian_matrix will return a scipy.sparse array instead of a matrix in Networkx 3.0.",
FutureWarning,
stacklevel=2,
)
# TODO: rm csr_matrix wrapper for NX 3.0
return sp.sparse.csr_matrix(DH @ (L @ DH))

def total_spanning_tree_weight(G, weight=None):
"""
Returns the total weight of all spanning trees of `G`.

Kirchoff's Tree Matrix Theorem states that the determinant of any cofactor of the
Laplacian matrix of a graph is the number of spanning trees in the graph. For a
weighted Laplacian matrix, it is the sum across all spanning trees of the
multiplicative weight of each tree. That is, the weight of each tree is the
product of its edge weights.

Parameters
----------
G : NetworkX Graph
The graph to use Kirchhoff's theorem on.

weight : string or None
The key for the edge attribute holding the edge weight. If `None`, then
each edge is assumed to have a weight of 1 and this function returns the
total number of spanning trees in `G`.

Returns
-------
float
The sum of the total multiplicative weights for all spanning trees in `G`
"""
import numpy as np

G_laplacian = nx.laplacian_matrix(G, weight=weight).toarray()
# Determinant ignoring first row and column
return abs(np.linalg.det(G_laplacian[1:, 1:]))

###############################################################################
# Code based on work from https://github.com/bjedwards

[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

--------
laplacian_matrix

References
----------
..  Fan Chung (2005).
Laplacians and the Cheeger inequality for directed graphs.
Annals of Combinatorics, 9(1), 2005
"""
import numpy as np
import scipy as sp
import scipy.sparse  # call as sp.sparse
import scipy.sparse.linalg  # call as sp.sparse.linalg

# NOTE: P has type ndarray if walk_type=="pagerank", else csr_array
P = _transition_matrix(
G, nodelist=nodelist, weight=weight, walk_type=walk_type, alpha=alpha
)

n, m = P.shape

evals, evecs = sp.sparse.linalg.eigs(P.T, k=1)
v = evecs.flatten().real
p = v / v.sum()
sqrtp = np.sqrt(p)
Q = (
# TODO: rm csr_array wrapper when spdiags creates arrays
sp.sparse.csr_array(sp.sparse.spdiags(sqrtp, 0, n, n))
[docs]        @ P
# TODO: rm csr_array wrapper when spdiags creates arrays
@ sp.sparse.csr_array(sp.sparse.spdiags(1.0 / sqrtp, 0, n, n))
)
# NOTE: This could be sparsified for the non-pagerank cases
I = np.identity(len(G))

import warnings

warnings.warn(
"directed_laplacian_matrix will return a numpy array instead of a matrix in NetworkX 3.0",
FutureWarning,
stacklevel=2,
)
# TODO: rm np.asmatrix for networkx 3.0
return np.asmatrix(I - (Q + Q.T) / 2.0)

@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 `\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

--------
laplacian_matrix

References
----------
..  Fan Chung (2005).
Laplacians and the Cheeger inequality for directed graphs.
Annals of Combinatorics, 9(1), 2005
"""
import scipy as sp
import scipy.sparse  # call as sp.sparse
import scipy.sparse.linalg  # call as sp.sparse.linalg

P = _transition_matrix(
G, nodelist=nodelist, weight=weight, walk_type=walk_type, alpha=alpha
)

n, m = P.shape

evals, evecs = sp.sparse.linalg.eigs(P.T, k=1)
v = evecs.flatten().real
p = v / v.sum()
# NOTE: could be improved by not densifying
# TODO: Rm csr_array wrapper when spdiags array creation becomes available
Phi = sp.sparse.csr_array(sp.sparse.spdiags(p, 0, n, n)).toarray()

import warnings

warnings.warn(
"directed_combinatorial_laplacian_matrix will return a numpy array instead of a matrix in NetworkX 3.0",
FutureWarning,
stacklevel=2,
)
# TODO: Rm np.asmatrix for networkx 3.0
import numpy as np

return np.asmatrix(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.ndarray
transition matrix of G.

Raises
------
NetworkXError
If walk_type not specified or alpha not in valid range
"""
import numpy as np
import scipy as sp
import scipy.sparse  # call as sp.sparse

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"

A = nx.to_scipy_sparse_array(G, nodelist=nodelist, weight=weight, dtype=float)
n, m = A.shape
if walk_type in ["random", "lazy"]:
# TODO: Rm csr_array wrapper when spdiags array creation becomes available
DI = sp.sparse.csr_array(sp.sparse.spdiags(1.0 / A.sum(axis=1), 0, n, n))
if walk_type == "random":
P = DI @ A
else:
# TODO: Rm csr_array wrapper when identity array creation becomes available
I = sp.sparse.csr_array(sp.sparse.identity(n))
P = (I + DI @ A) / 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. NOTE: This should be sparsified!
A = A.toarray()
# add constant to dangling nodes' row
A[A.sum(axis=1) == 0, :] = 1 / n
# normalize
A = A / A.sum(axis=1)[np.newaxis, :].T
P = alpha * A + (1 - alpha) / n
else:
raise nx.NetworkXError("walk_type must be random, lazy, or pagerank")

return P
```