Old engine for Continuous Time Bayesian Networks. Superseded by reCTBN. 🐍
https://github.com/madlabunimib/PyCTBN
You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
600 lines
19 KiB
600 lines
19 KiB
"""
|
|
Algebraic connectivity and Fiedler vectors of undirected graphs.
|
|
"""
|
|
from functools import partial
|
|
import networkx as nx
|
|
from networkx.utils import not_implemented_for
|
|
from networkx.utils import reverse_cuthill_mckee_ordering
|
|
from networkx.utils import random_state
|
|
|
|
try:
|
|
from numpy import array, asarray, dot, ndarray, ones, sqrt, zeros, atleast_2d
|
|
from numpy.linalg import norm, qr
|
|
from scipy.linalg import eigh, inv
|
|
from scipy.sparse import csc_matrix, spdiags
|
|
from scipy.sparse.linalg import eigsh, lobpcg
|
|
|
|
__all__ = ["algebraic_connectivity", "fiedler_vector", "spectral_ordering"]
|
|
except ImportError:
|
|
__all__ = []
|
|
|
|
try:
|
|
from scipy.linalg.blas import dasum, daxpy, ddot
|
|
except ImportError:
|
|
if __all__:
|
|
# Make sure the imports succeeded.
|
|
# Use minimal replacements if BLAS is unavailable from SciPy.
|
|
dasum = partial(norm, ord=1)
|
|
ddot = dot
|
|
|
|
def daxpy(x, y, a):
|
|
y += a * x
|
|
return y
|
|
|
|
|
|
class _PCGSolver:
|
|
"""Preconditioned conjugate gradient method.
|
|
|
|
To solve Ax = b:
|
|
M = A.diagonal() # or some other preconditioner
|
|
solver = _PCGSolver(lambda x: A * x, lambda x: M * x)
|
|
x = solver.solve(b)
|
|
|
|
The inputs A and M are functions which compute
|
|
matrix multiplication on the argument.
|
|
A - multiply by the matrix A in Ax=b
|
|
M - multiply by M, the preconditioner surragate for A
|
|
|
|
Warning: There is no limit on number of iterations.
|
|
"""
|
|
|
|
def __init__(self, A, M):
|
|
self._A = A
|
|
self._M = M or (lambda x: x.copy())
|
|
|
|
def solve(self, B, tol):
|
|
B = asarray(B)
|
|
X = ndarray(B.shape, order="F")
|
|
for j in range(B.shape[1]):
|
|
X[:, j] = self._solve(B[:, j], tol)
|
|
return X
|
|
|
|
def _solve(self, b, tol):
|
|
A = self._A
|
|
M = self._M
|
|
tol *= dasum(b)
|
|
# Initialize.
|
|
x = zeros(b.shape)
|
|
r = b.copy()
|
|
z = M(r)
|
|
rz = ddot(r, z)
|
|
p = z.copy()
|
|
# Iterate.
|
|
while True:
|
|
Ap = A(p)
|
|
alpha = rz / ddot(p, Ap)
|
|
x = daxpy(p, x, a=alpha)
|
|
r = daxpy(Ap, r, a=-alpha)
|
|
if dasum(r) < tol:
|
|
return x
|
|
z = M(r)
|
|
beta = ddot(r, z)
|
|
beta, rz = beta / rz, beta
|
|
p = daxpy(p, z, a=beta)
|
|
|
|
|
|
class _CholeskySolver:
|
|
"""Cholesky factorization.
|
|
|
|
To solve Ax = b:
|
|
solver = _CholeskySolver(A)
|
|
x = solver.solve(b)
|
|
|
|
optional argument `tol` on solve method is ignored but included
|
|
to match _PCGsolver API.
|
|
"""
|
|
|
|
def __init__(self, A):
|
|
if not self._cholesky:
|
|
raise nx.NetworkXError("Cholesky solver unavailable.")
|
|
self._chol = self._cholesky(A)
|
|
|
|
def solve(self, B, tol=None):
|
|
return self._chol(B)
|
|
|
|
try:
|
|
from scikits.sparse.cholmod import cholesky
|
|
|
|
_cholesky = cholesky
|
|
except ImportError:
|
|
_cholesky = None
|
|
|
|
|
|
class _LUSolver:
|
|
"""LU factorization.
|
|
|
|
To solve Ax = b:
|
|
solver = _LUSolver(A)
|
|
x = solver.solve(b)
|
|
|
|
optional argument `tol` on solve method is ignored but included
|
|
to match _PCGsolver API.
|
|
"""
|
|
|
|
def __init__(self, A):
|
|
if not self._splu:
|
|
raise nx.NetworkXError("LU solver unavailable.")
|
|
self._LU = self._splu(A)
|
|
|
|
def solve(self, B, tol=None):
|
|
B = asarray(B)
|
|
X = ndarray(B.shape, order="F")
|
|
for j in range(B.shape[1]):
|
|
X[:, j] = self._LU.solve(B[:, j])
|
|
return X
|
|
|
|
try:
|
|
from scipy.sparse.linalg import splu
|
|
|
|
_splu = partial(
|
|
splu,
|
|
permc_spec="MMD_AT_PLUS_A",
|
|
diag_pivot_thresh=0.0,
|
|
options={"Equil": True, "SymmetricMode": True},
|
|
)
|
|
except ImportError:
|
|
_splu = None
|
|
|
|
|
|
def _preprocess_graph(G, weight):
|
|
"""Compute edge weights and eliminate zero-weight edges.
|
|
"""
|
|
if G.is_directed():
|
|
H = nx.MultiGraph()
|
|
H.add_nodes_from(G)
|
|
H.add_weighted_edges_from(
|
|
((u, v, e.get(weight, 1.0)) for u, v, e in G.edges(data=True) if u != v),
|
|
weight=weight,
|
|
)
|
|
G = H
|
|
if not G.is_multigraph():
|
|
edges = (
|
|
(u, v, abs(e.get(weight, 1.0))) for u, v, e in G.edges(data=True) if u != v
|
|
)
|
|
else:
|
|
edges = (
|
|
(u, v, sum(abs(e.get(weight, 1.0)) for e in G[u][v].values()))
|
|
for u, v in G.edges()
|
|
if u != v
|
|
)
|
|
H = nx.Graph()
|
|
H.add_nodes_from(G)
|
|
H.add_weighted_edges_from((u, v, e) for u, v, e in edges if e != 0)
|
|
return H
|
|
|
|
|
|
def _rcm_estimate(G, nodelist):
|
|
"""Estimate the Fiedler vector using the reverse Cuthill-McKee ordering.
|
|
"""
|
|
G = G.subgraph(nodelist)
|
|
order = reverse_cuthill_mckee_ordering(G)
|
|
n = len(nodelist)
|
|
index = dict(zip(nodelist, range(n)))
|
|
x = ndarray(n, dtype=float)
|
|
for i, u in enumerate(order):
|
|
x[index[u]] = i
|
|
x -= (n - 1) / 2.0
|
|
return x
|
|
|
|
|
|
def _tracemin_fiedler(L, X, normalized, tol, method):
|
|
"""Compute the Fiedler vector of L using the TraceMIN-Fiedler algorithm.
|
|
|
|
The Fiedler vector of a connected undirected graph is the eigenvector
|
|
corresponding to the second smallest eigenvalue of the Laplacian matrix of
|
|
of the graph. This function starts with the Laplacian L, not the Graph.
|
|
|
|
Parameters
|
|
----------
|
|
L : Laplacian of a possibly weighted or normalized, but undirected graph
|
|
|
|
X : Initial guess for a solution. Usually a matrix of random numbers.
|
|
This function allows more than one column in X to identify more than
|
|
one eigenvector if desired.
|
|
|
|
normalized : bool
|
|
Whether the normalized Laplacian matrix is used.
|
|
|
|
tol : float
|
|
Tolerance of relative residual in eigenvalue computation.
|
|
Warning: There is no limit on number of iterations.
|
|
|
|
method : string
|
|
Should be 'tracemin_pcg', 'tracemin_chol' or 'tracemin_lu'.
|
|
Otherwise exception is raised.
|
|
|
|
Returns
|
|
-------
|
|
sigma, X : Two NumPy arrays of floats.
|
|
The lowest eigenvalues and corresponding eigenvectors of L.
|
|
The size of input X determines the size of these outputs.
|
|
As this is for Fiedler vectors, the zero eigenvalue (and
|
|
constant eigenvector) are avoided.
|
|
"""
|
|
n = X.shape[0]
|
|
|
|
if normalized:
|
|
# Form the normalized Laplacian matrix and determine the eigenvector of
|
|
# its nullspace.
|
|
e = sqrt(L.diagonal())
|
|
D = spdiags(1.0 / e, [0], n, n, format="csr")
|
|
L = D * L * D
|
|
e *= 1.0 / norm(e, 2)
|
|
|
|
if normalized:
|
|
|
|
def project(X):
|
|
"""Make X orthogonal to the nullspace of L.
|
|
"""
|
|
X = asarray(X)
|
|
for j in range(X.shape[1]):
|
|
X[:, j] -= dot(X[:, j], e) * e
|
|
|
|
else:
|
|
|
|
def project(X):
|
|
"""Make X orthogonal to the nullspace of L.
|
|
"""
|
|
X = asarray(X)
|
|
for j in range(X.shape[1]):
|
|
X[:, j] -= X[:, j].sum() / n
|
|
|
|
if method == "tracemin_pcg":
|
|
D = L.diagonal().astype(float)
|
|
solver = _PCGSolver(lambda x: L * x, lambda x: D * x)
|
|
elif method == "tracemin_chol" or method == "tracemin_lu":
|
|
# Convert A to CSC to suppress SparseEfficiencyWarning.
|
|
A = csc_matrix(L, dtype=float, copy=True)
|
|
# Force A to be nonsingular. Since A is the Laplacian matrix of a
|
|
# connected graph, its rank deficiency is one, and thus one diagonal
|
|
# element needs to modified. Changing to infinity forces a zero in the
|
|
# corresponding element in the solution.
|
|
i = (A.indptr[1:] - A.indptr[:-1]).argmax()
|
|
A[i, i] = float("inf")
|
|
if method == "tracemin_chol":
|
|
solver = _CholeskySolver(A)
|
|
else:
|
|
solver = _LUSolver(A)
|
|
else:
|
|
raise nx.NetworkXError("Unknown linear system solver: " + method)
|
|
|
|
# Initialize.
|
|
Lnorm = abs(L).sum(axis=1).flatten().max()
|
|
project(X)
|
|
W = ndarray(X.shape, order="F")
|
|
|
|
while True:
|
|
# Orthonormalize X.
|
|
X = qr(X)[0]
|
|
# Compute iteration matrix H.
|
|
W[:, :] = L @ X
|
|
H = X.T @ W
|
|
sigma, Y = eigh(H, overwrite_a=True)
|
|
# Compute the Ritz vectors.
|
|
X = X @ Y
|
|
# Test for convergence exploiting the fact that L * X == W * Y.
|
|
res = dasum(W @ Y[:, 0] - sigma[0] * X[:, 0]) / Lnorm
|
|
if res < tol:
|
|
break
|
|
# Compute X = L \ X / (X' * (L \ X)).
|
|
# L \ X can have an arbitrary projection on the nullspace of L,
|
|
# which will be eliminated.
|
|
W[:, :] = solver.solve(X, tol)
|
|
X = (inv(W.T @ X) @ W.T).T # Preserves Fortran storage order.
|
|
project(X)
|
|
|
|
return sigma, asarray(X)
|
|
|
|
|
|
def _get_fiedler_func(method):
|
|
"""Returns a function that solves the Fiedler eigenvalue problem.
|
|
"""
|
|
if method == "tracemin": # old style keyword <v2.1
|
|
method = "tracemin_pcg"
|
|
if method in ("tracemin_pcg", "tracemin_chol", "tracemin_lu"):
|
|
|
|
def find_fiedler(L, x, normalized, tol, seed):
|
|
q = 1 if method == "tracemin_pcg" else min(4, L.shape[0] - 1)
|
|
X = asarray(seed.normal(size=(q, L.shape[0]))).T
|
|
sigma, X = _tracemin_fiedler(L, X, normalized, tol, method)
|
|
return sigma[0], X[:, 0]
|
|
|
|
elif method == "lanczos" or method == "lobpcg":
|
|
|
|
def find_fiedler(L, x, normalized, tol, seed):
|
|
L = csc_matrix(L, dtype=float)
|
|
n = L.shape[0]
|
|
if normalized:
|
|
D = spdiags(1.0 / sqrt(L.diagonal()), [0], n, n, format="csc")
|
|
L = D * L * D
|
|
if method == "lanczos" or n < 10:
|
|
# Avoid LOBPCG when n < 10 due to
|
|
# https://github.com/scipy/scipy/issues/3592
|
|
# https://github.com/scipy/scipy/pull/3594
|
|
sigma, X = eigsh(L, 2, which="SM", tol=tol, return_eigenvectors=True)
|
|
return sigma[1], X[:, 1]
|
|
else:
|
|
X = asarray(atleast_2d(x).T)
|
|
M = spdiags(1.0 / L.diagonal(), [0], n, n)
|
|
Y = ones(n)
|
|
if normalized:
|
|
Y /= D.diagonal()
|
|
sigma, X = lobpcg(
|
|
L, X, M=M, Y=atleast_2d(Y).T, tol=tol, maxiter=n, largest=False
|
|
)
|
|
return sigma[0], X[:, 0]
|
|
|
|
else:
|
|
raise nx.NetworkXError(f"unknown method '{method}'.")
|
|
|
|
return find_fiedler
|
|
|
|
|
|
@random_state(5)
|
|
@not_implemented_for("directed")
|
|
def algebraic_connectivity(
|
|
G, weight="weight", normalized=False, tol=1e-8, method="tracemin_pcg", seed=None
|
|
):
|
|
"""Returns the algebraic connectivity of an undirected graph.
|
|
|
|
The algebraic connectivity of a connected undirected graph is the second
|
|
smallest eigenvalue of its Laplacian matrix.
|
|
|
|
Parameters
|
|
----------
|
|
G : NetworkX graph
|
|
An undirected graph.
|
|
|
|
weight : object, optional (default: None)
|
|
The data key used to determine the weight of each edge. If None, then
|
|
each edge has unit weight.
|
|
|
|
normalized : bool, optional (default: False)
|
|
Whether the normalized Laplacian matrix is used.
|
|
|
|
tol : float, optional (default: 1e-8)
|
|
Tolerance of relative residual in eigenvalue computation.
|
|
|
|
method : string, optional (default: 'tracemin_pcg')
|
|
Method of eigenvalue computation. It must be one of the tracemin
|
|
options shown below (TraceMIN), 'lanczos' (Lanczos iteration)
|
|
or 'lobpcg' (LOBPCG).
|
|
|
|
The TraceMIN algorithm uses a linear system solver. The following
|
|
values allow specifying the solver to be used.
|
|
|
|
=============== ========================================
|
|
Value Solver
|
|
=============== ========================================
|
|
'tracemin_pcg' Preconditioned conjugate gradient method
|
|
'tracemin_chol' Cholesky factorization
|
|
'tracemin_lu' LU factorization
|
|
=============== ========================================
|
|
|
|
seed : integer, random_state, or None (default)
|
|
Indicator of random number generation state.
|
|
See :ref:`Randomness<randomness>`.
|
|
|
|
Returns
|
|
-------
|
|
algebraic_connectivity : float
|
|
Algebraic connectivity.
|
|
|
|
Raises
|
|
------
|
|
NetworkXNotImplemented
|
|
If G is directed.
|
|
|
|
NetworkXError
|
|
If G has less than two nodes.
|
|
|
|
Notes
|
|
-----
|
|
Edge weights are interpreted by their absolute values. For MultiGraph's,
|
|
weights of parallel edges are summed. Zero-weighted edges are ignored.
|
|
|
|
To use Cholesky factorization in the TraceMIN algorithm, the
|
|
:samp:`scikits.sparse` package must be installed.
|
|
|
|
See Also
|
|
--------
|
|
laplacian_matrix
|
|
"""
|
|
if len(G) < 2:
|
|
raise nx.NetworkXError("graph has less than two nodes.")
|
|
G = _preprocess_graph(G, weight)
|
|
if not nx.is_connected(G):
|
|
return 0.0
|
|
|
|
L = nx.laplacian_matrix(G)
|
|
if L.shape[0] == 2:
|
|
return 2.0 * L[0, 0] if not normalized else 2.0
|
|
|
|
find_fiedler = _get_fiedler_func(method)
|
|
x = None if method != "lobpcg" else _rcm_estimate(G, G)
|
|
sigma, fiedler = find_fiedler(L, x, normalized, tol, seed)
|
|
return sigma
|
|
|
|
|
|
@random_state(5)
|
|
@not_implemented_for("directed")
|
|
def fiedler_vector(
|
|
G, weight="weight", normalized=False, tol=1e-8, method="tracemin_pcg", seed=None
|
|
):
|
|
"""Returns the Fiedler vector of a connected undirected graph.
|
|
|
|
The Fiedler vector of a connected undirected graph is the eigenvector
|
|
corresponding to the second smallest eigenvalue of the Laplacian matrix of
|
|
of the graph.
|
|
|
|
Parameters
|
|
----------
|
|
G : NetworkX graph
|
|
An undirected graph.
|
|
|
|
weight : object, optional (default: None)
|
|
The data key used to determine the weight of each edge. If None, then
|
|
each edge has unit weight.
|
|
|
|
normalized : bool, optional (default: False)
|
|
Whether the normalized Laplacian matrix is used.
|
|
|
|
tol : float, optional (default: 1e-8)
|
|
Tolerance of relative residual in eigenvalue computation.
|
|
|
|
method : string, optional (default: 'tracemin_pcg')
|
|
Method of eigenvalue computation. It must be one of the tracemin
|
|
options shown below (TraceMIN), 'lanczos' (Lanczos iteration)
|
|
or 'lobpcg' (LOBPCG).
|
|
|
|
The TraceMIN algorithm uses a linear system solver. The following
|
|
values allow specifying the solver to be used.
|
|
|
|
=============== ========================================
|
|
Value Solver
|
|
=============== ========================================
|
|
'tracemin_pcg' Preconditioned conjugate gradient method
|
|
'tracemin_chol' Cholesky factorization
|
|
'tracemin_lu' LU factorization
|
|
=============== ========================================
|
|
|
|
seed : integer, random_state, or None (default)
|
|
Indicator of random number generation state.
|
|
See :ref:`Randomness<randomness>`.
|
|
|
|
Returns
|
|
-------
|
|
fiedler_vector : NumPy array of floats.
|
|
Fiedler vector.
|
|
|
|
Raises
|
|
------
|
|
NetworkXNotImplemented
|
|
If G is directed.
|
|
|
|
NetworkXError
|
|
If G has less than two nodes or is not connected.
|
|
|
|
Notes
|
|
-----
|
|
Edge weights are interpreted by their absolute values. For MultiGraph's,
|
|
weights of parallel edges are summed. Zero-weighted edges are ignored.
|
|
|
|
To use Cholesky factorization in the TraceMIN algorithm, the
|
|
:samp:`scikits.sparse` package must be installed.
|
|
|
|
See Also
|
|
--------
|
|
laplacian_matrix
|
|
"""
|
|
if len(G) < 2:
|
|
raise nx.NetworkXError("graph has less than two nodes.")
|
|
G = _preprocess_graph(G, weight)
|
|
if not nx.is_connected(G):
|
|
raise nx.NetworkXError("graph is not connected.")
|
|
|
|
if len(G) == 2:
|
|
return array([1.0, -1.0])
|
|
|
|
find_fiedler = _get_fiedler_func(method)
|
|
L = nx.laplacian_matrix(G)
|
|
x = None if method != "lobpcg" else _rcm_estimate(G, G)
|
|
sigma, fiedler = find_fiedler(L, x, normalized, tol, seed)
|
|
return fiedler
|
|
|
|
|
|
@random_state(5)
|
|
def spectral_ordering(
|
|
G, weight="weight", normalized=False, tol=1e-8, method="tracemin_pcg", seed=None
|
|
):
|
|
"""Compute the spectral_ordering of a graph.
|
|
|
|
The spectral ordering of a graph is an ordering of its nodes where nodes
|
|
in the same weakly connected components appear contiguous and ordered by
|
|
their corresponding elements in the Fiedler vector of the component.
|
|
|
|
Parameters
|
|
----------
|
|
G : NetworkX graph
|
|
A graph.
|
|
|
|
weight : object, optional (default: None)
|
|
The data key used to determine the weight of each edge. If None, then
|
|
each edge has unit weight.
|
|
|
|
normalized : bool, optional (default: False)
|
|
Whether the normalized Laplacian matrix is used.
|
|
|
|
tol : float, optional (default: 1e-8)
|
|
Tolerance of relative residual in eigenvalue computation.
|
|
|
|
method : string, optional (default: 'tracemin_pcg')
|
|
Method of eigenvalue computation. It must be one of the tracemin
|
|
options shown below (TraceMIN), 'lanczos' (Lanczos iteration)
|
|
or 'lobpcg' (LOBPCG).
|
|
|
|
The TraceMIN algorithm uses a linear system solver. The following
|
|
values allow specifying the solver to be used.
|
|
|
|
=============== ========================================
|
|
Value Solver
|
|
=============== ========================================
|
|
'tracemin_pcg' Preconditioned conjugate gradient method
|
|
'tracemin_chol' Cholesky factorization
|
|
'tracemin_lu' LU factorization
|
|
=============== ========================================
|
|
|
|
seed : integer, random_state, or None (default)
|
|
Indicator of random number generation state.
|
|
See :ref:`Randomness<randomness>`.
|
|
|
|
Returns
|
|
-------
|
|
spectral_ordering : NumPy array of floats.
|
|
Spectral ordering of nodes.
|
|
|
|
Raises
|
|
------
|
|
NetworkXError
|
|
If G is empty.
|
|
|
|
Notes
|
|
-----
|
|
Edge weights are interpreted by their absolute values. For MultiGraph's,
|
|
weights of parallel edges are summed. Zero-weighted edges are ignored.
|
|
|
|
To use Cholesky factorization in the TraceMIN algorithm, the
|
|
:samp:`scikits.sparse` package must be installed.
|
|
|
|
See Also
|
|
--------
|
|
laplacian_matrix
|
|
"""
|
|
if len(G) == 0:
|
|
raise nx.NetworkXError("graph is empty.")
|
|
G = _preprocess_graph(G, weight)
|
|
|
|
find_fiedler = _get_fiedler_func(method)
|
|
order = []
|
|
for component in nx.connected_components(G):
|
|
size = len(component)
|
|
if size > 2:
|
|
L = nx.laplacian_matrix(G, component)
|
|
x = None if method != "lobpcg" else _rcm_estimate(G, component)
|
|
sigma, fiedler = find_fiedler(L, x, normalized, tol, seed)
|
|
sort_info = zip(fiedler, range(size), component)
|
|
order.extend(u for x, c, u in sorted(sort_info))
|
|
else:
|
|
order.extend(component)
|
|
|
|
return order
|
|
|