"""Laplacian matrix of graphs.
All calculations here are done using the out-degree. For Laplacians using
in-degree, use `G.reverse(copy=False)` instead of `G` and take the transpose.
The `laplacian_matrix` function provides an unnormalized matrix,
while `normalized_laplacian_matrix`, `directed_laplacian_matrix`,
and `directed_combinatorial_laplacian_matrix` are all normalized.
"""
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]
@nx._dispatchable(edge_attrs="weight")
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 array
The Laplacian matrix of G.
Notes
-----
For MultiGraph, the edges weights are summed.
This returns an unnormalized matrix. For a normalized output,
use `normalized_laplacian_matrix`, `directed_laplacian_matrix`,
or `directed_combinatorial_laplacian_matrix`.
This calculation uses the out-degree of the graph `G`. To use the
in-degree for calculations instead, use `G.reverse(copy=False)` and
take the transpose.
See Also
--------
:func:`~networkx.convert_matrix.to_numpy_array`
normalized_laplacian_matrix
directed_laplacian_matrix
directed_combinatorial_laplacian_matrix
:func:`~networkx.linalg.spectrum.laplacian_spectrum`
Examples
--------
For graphs with multiple connected components, L is permutation-similar
to a block diagonal matrix where each block is the respective Laplacian
matrix for each component.
>>> G = nx.Graph([(1, 2), (2, 3), (4, 5)])
>>> print(nx.laplacian_matrix(G).toarray())
[[ 1 -1 0 0 0]
[-1 2 -1 0 0]
[ 0 -1 1 0 0]
[ 0 0 0 1 -1]
[ 0 0 0 -1 1]]
>>> edges = [
... (1, 2),
... (2, 1),
... (2, 4),
... (4, 3),
... (3, 4),
... ]
>>> DiG = nx.DiGraph(edges)
>>> print(nx.laplacian_matrix(DiG).toarray())
[[ 1 -1 0 0]
[-1 2 -1 0]
[ 0 0 1 -1]
[ 0 0 -1 1]]
Notice that node 4 is represented by the third column and row. This is because
by default the row/column order is the order of `G.nodes` (i.e. the node added
order -- in the edgelist, 4 first appears in (2, 4), before node 3 in edge (4, 3).)
To control the node order of the matrix, use the `nodelist` argument.
>>> print(nx.laplacian_matrix(DiG, nodelist=[1, 2, 3, 4]).toarray())
[[ 1 -1 0 0]
[-1 2 0 -1]
[ 0 0 1 -1]
[ 0 0 -1 1]]
This calculation uses the out-degree of the graph `G`. To use the
in-degree for calculations instead, use `G.reverse(copy=False)` and
take the transpose.
>>> print(nx.laplacian_matrix(DiG.reverse(copy=False)).toarray().T)
[[ 1 -1 0 0]
[-1 1 -1 0]
[ 0 0 2 -1]
[ 0 0 -1 1]]
References
----------
.. [1] Langville, Amy N., and Carl D. Meyer. Google’s PageRank and Beyond:
The Science of Search Engine Rankings. Princeton University Press, 2006.
"""
import scipy as sp
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"))
return D - A
[docs]
@nx._dispatchable(edge_attrs="weight")
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 [1]_.
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 array
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
the adjacency matrix [2]_.
This calculation uses the out-degree of the graph `G`. To use the
in-degree for calculations instead, use `G.reverse(copy=False)` and
take the transpose.
For an unnormalized output, use `laplacian_matrix`.
Examples
--------
>>> import numpy as np
>>> edges = [
... (1, 2),
... (2, 1),
... (2, 4),
... (4, 3),
... (3, 4),
... ]
>>> DiG = nx.DiGraph(edges)
>>> print(nx.normalized_laplacian_matrix(DiG).toarray())
[[ 1. -0.70710678 0. 0. ]
[-0.70710678 1. -0.70710678 0. ]
[ 0. 0. 1. -1. ]
[ 0. 0. -1. 1. ]]
Notice that node 4 is represented by the third column and row. This is because
by default the row/column order is the order of `G.nodes` (i.e. the node added
order -- in the edgelist, 4 first appears in (2, 4), before node 3 in edge (4, 3).)
To control the node order of the matrix, use the `nodelist` argument.
>>> print(nx.normalized_laplacian_matrix(DiG, nodelist=[1, 2, 3, 4]).toarray())
[[ 1. -0.70710678 0. 0. ]
[-0.70710678 1. 0. -0.70710678]
[ 0. 0. 1. -1. ]
[ 0. 0. -1. 1. ]]
>>> G = nx.Graph(edges)
>>> print(nx.normalized_laplacian_matrix(G).toarray())
[[ 1. -0.70710678 0. 0. ]
[-0.70710678 1. -0.5 0. ]
[ 0. -0.5 1. -0.70710678]
[ 0. 0. -0.70710678 1. ]]
See Also
--------
laplacian_matrix
normalized_laplacian_spectrum
directed_laplacian_matrix
directed_combinatorial_laplacian_matrix
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.
.. [3] Langville, Amy N., and Carl D. Meyer. Google’s PageRank and Beyond:
The Science of Search Engine Rankings. Princeton University Press, 2006.
"""
import numpy as np
import scipy as sp
if nodelist is None:
nodelist = list(G)
A = nx.to_scipy_sparse_array(G, nodelist=nodelist, weight=weight, format="csr")
n, _ = 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, n, n, format="csr"))
L = D - A
with np.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, n, n, format="csr"))
return DH @ (L @ DH)
[docs]
@nx._dispatchable(edge_attrs="weight")
def total_spanning_tree_weight(G, weight=None, root=None):
"""
Returns the total weight of all spanning trees of `G`.
Kirchoff's Tree Matrix Theorem [1]_, [2]_ 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.
For unweighted graphs, the total weight equals the number of spanning trees in `G`.
For directed graphs, the total weight follows by summing over all directed
spanning trees in `G` that start in the `root` node [3]_.
.. deprecated:: 3.3
``total_spanning_tree_weight`` is deprecated and will be removed in v3.5.
Use ``nx.number_of_spanning_trees(G)`` instead.
Parameters
----------
G : NetworkX Graph
weight : string or None, optional (default=None)
The key for the edge attribute holding the edge weight.
If None, then each edge has weight 1.
root : node (only required for directed graphs)
A node in the directed graph `G`.
Returns
-------
total_weight : float
Undirected graphs:
The sum of the total multiplicative weights for all spanning trees in `G`.
Directed graphs:
The sum of the total multiplicative weights for all spanning trees of `G`,
rooted at node `root`.
Raises
------
NetworkXPointlessConcept
If `G` does not contain any nodes.
NetworkXError
If the graph `G` is not (weakly) connected,
or if `G` is directed and the root node is not specified or not in G.
Examples
--------
>>> G = nx.complete_graph(5)
>>> round(nx.total_spanning_tree_weight(G))
125
>>> G = nx.Graph()
>>> G.add_edge(1, 2, weight=2)
>>> G.add_edge(1, 3, weight=1)
>>> G.add_edge(2, 3, weight=1)
>>> round(nx.total_spanning_tree_weight(G, "weight"))
5
Notes
-----
Self-loops are excluded. Multi-edges are contracted in one edge
equal to the sum of the weights.
References
----------
.. [1] Wikipedia
"Kirchhoff's theorem."
https://en.wikipedia.org/wiki/Kirchhoff%27s_theorem
.. [2] Kirchhoff, G. R.
Über die Auflösung der Gleichungen, auf welche man
bei der Untersuchung der linearen Vertheilung
Galvanischer Ströme geführt wird
Annalen der Physik und Chemie, vol. 72, pp. 497-508, 1847.
.. [3] Margoliash, J.
"Matrix-Tree Theorem for Directed Graphs"
https://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Margoliash.pdf
"""
import warnings
warnings.warn(
(
"\n\ntotal_spanning_tree_weight is deprecated and will be removed in v3.5.\n"
"Use `nx.number_of_spanning_trees(G)` instead."
),
category=DeprecationWarning,
stacklevel=3,
)
return nx.number_of_spanning_trees(G, weight=weight, root=root)
###############################################################################
# Code based on work from https://github.com/bjedwards
[docs]
@not_implemented_for("undirected")
@not_implemented_for("multigraph")
@nx._dispatchable(edge_attrs="weight")
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 - \frac{1}{2} \left (\Phi^{1/2} P \Phi^{-1/2} + \Phi^{-1/2} P^T \Phi^{1/2} \right )
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 [1]_.
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)
One of ``"random"``, ``"lazy"``, or ``"pagerank"``. If ``walk_type=None``
(the default), then a value is selected according to the properties of `G`:
- ``walk_type="random"`` if `G` is strongly connected and aperiodic
- ``walk_type="lazy"`` if `G` is strongly connected but not aperiodic
- ``walk_type="pagerank"`` for all other cases.
alpha : real
(1 - alpha) is the teleportation probability used with pagerank
Returns
-------
L : NumPy matrix
Normalized Laplacian of G.
Notes
-----
Only implemented for DiGraphs
The result is always a symmetric matrix.
This calculation uses the out-degree of the graph `G`. To use the
in-degree for calculations instead, use `G.reverse(copy=False)` and
take the transpose.
See Also
--------
laplacian_matrix
normalized_laplacian_matrix
directed_combinatorial_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
import scipy as sp
# 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()
# p>=0 by Perron-Frobenius Thm. Use abs() to fix roundoff across zero gh-6865
sqrtp = np.sqrt(np.abs(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))
return I - (Q + Q.T) / 2.0
@not_implemented_for("undirected")
@not_implemented_for("multigraph")
@nx._dispatchable(edge_attrs="weight")
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 - \frac{1}{2} \left (\Phi P + P^T \Phi \right)
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 [1]_.
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)
One of ``"random"``, ``"lazy"``, or ``"pagerank"``. If ``walk_type=None``
(the default), then a value is selected according to the properties of `G`:
- ``walk_type="random"`` if `G` is strongly connected and aperiodic
- ``walk_type="lazy"`` if `G` is strongly connected but not aperiodic
- ``walk_type="pagerank"`` for all other cases.
alpha : real
(1 - alpha) is the teleportation probability used with pagerank
Returns
-------
L : NumPy matrix
Combinatorial Laplacian of G.
Notes
-----
Only implemented for DiGraphs
The result is always a symmetric matrix.
This calculation uses the out-degree of the graph `G`. To use the
in-degree for calculations instead, use `G.reverse(copy=False)` and
take the transpose.
See Also
--------
laplacian_matrix
normalized_laplacian_matrix
directed_laplacian_matrix
References
----------
.. [1] Fan Chung (2005).
Laplacians and the Cheeger inequality for directed graphs.
Annals of Combinatorics, 9(1), 2005
"""
import scipy as sp
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()
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)
One of ``"random"``, ``"lazy"``, or ``"pagerank"``. If ``walk_type=None``
(the default), then a value is selected according to the properties of `G`:
- ``walk_type="random"`` if `G` is strongly connected and aperiodic
- ``walk_type="lazy"`` if `G` is strongly connected but not aperiodic
- ``walk_type="pagerank"`` for all other cases.
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
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
