Warning

This documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.

# Source code for networkx.linalg.algebraicconnectivity

# -*- coding: utf-8 -*-
"""
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 re import compile

try:
from numpy import (array, asmatrix, asarray, dot, matrix, ndarray, ones,
reshape, sqrt, zeros)
from numpy.linalg import norm, qr
from numpy.random import normal
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

_tracemin_method = compile('^tracemin(?:_(.*))?\$')

class _PCGSolver(object):
"""

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):
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(object):
"""Cholesky factorization.
"""

def __init__(self, A):
if not self._cholesky:
raise nx.NetworkXError('Cholesky solver unavailable.')
self._chol = self._cholesky(A)

def solve(self, B):
return self._chol(B)

try:
from scikits.sparse.cholmod import cholesky
_cholesky = cholesky
except ImportError:
_cholesky = None

class _LUSolver(object):
"""LU factorization.
"""

def __init__(self, A):
if not self._splu:
raise nx.NetworkXError('LU solver unavailable.')
self._LU = self._splu(A)

def solve(self, B):
B = asarray(B)
X = ndarray(B.shape, order='F')
for j in range(B.shape):
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.,
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()
for u, v, e in G.edges_iter(data=True)
if u != v), weight=weight)
G = H
if not G.is_multigraph():
edges = ((u, v, abs(e.get(weight, 1.)))
for u, v, e in G.edges_iter(data=True) if u != v)
else:
edges = ((u, v, sum(abs(e.get(weight, 1.)) for e in G[u][v].values()))
for u, v in G.edges_iter() if u != v)
H = nx.Graph()
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.
return x

def _tracemin_fiedler(L, X, normalized, tol, method):
"""Compute the Fiedler vector of L using the TraceMIN-Fiedler algorithm.
"""
n = X.shape

if normalized:
# Form the normalized Laplacian matrix and determine the eigenvector of
# its nullspace.
e = sqrt(L.diagonal())
D = spdiags(1. / e, , n, n, format='csr')
L = D * L * D
e *= 1. / norm(e, 2)

if not normalized:
def project(X):
"""Make X orthogonal to the nullspace of L.
"""
X = asarray(X)
for j in range(X.shape):
X[:, j] -= X[:, j].sum() / n
else:
def project(X):
"""Make X orthogonal to the nullspace of L.
"""
X = asarray(X)
for j in range(X.shape):
X[:, j] -= dot(X[:, j], e) * e

if method is None:
method = 'pcg'
if method == 'pcg':
# See comments below for the semantics of P and D.
def P(x):
x -= asarray(x * X * X.T)[0, :]
if not normalized:
x -= x.sum() / n
else:
x = daxpy(e, x, a=-ddot(x, e))
return x
solver = _PCGSolver(lambda x: P(L * P(x)), lambda x: D * x)
elif method == 'chol' or method == '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')
solver = (_CholeskySolver if method == 'chol' else _LUSolver)(A)
else:
raise nx.NetworkXError('unknown linear system solver.')

# Initialize.
Lnorm = abs(L).sum(axis=1).flatten().max()
project(X)
W = asmatrix(ndarray(X.shape, order='F'))

while True:
# Orthonormalize X.
X = qr(X)
# Compute interation matrix H.
W[:, :] = L * X
H = X.T * W
sigma, Y = eigh(H, overwrite_a=True)
# Compute the Ritz vectors.
X *= Y
# Test for convergence exploiting the fact that L * X == W * Y.
res = dasum(W * asmatrix(Y)[:, 0] - sigma * X[:, 0]) / Lnorm
if res < tol:
break
# Depending on the linear solver to be used, two mathematically
# equivalent formulations are used.
if method == 'pcg':
# Compute X = X - (P * L * P) \ (P * L * X) where
# P = I - [e X] * [e X]' is a projection onto the orthogonal
# complement of [e X].
W *= Y  # L * X == W * Y
W -= (W.T * X * X.T).T
project(W)
# Compute the diagonal of P * L * P as a Jacobi preconditioner.
D = L.diagonal()
D += 2. * (asarray(X) * asarray(W)).sum(axis=1)
D += (asarray(X) * asarray(X * (W.T * X))).sum(axis=1)
D[D < tol * Lnorm] = 1.
D = 1. / D
# Since TraceMIN is globally convergent, the relative residual can
# be loose.
X -= solver.solve(W, 0.1)
else:
# 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)
project(W)
X = (inv(W.T * X) * W.T).T  # Preserves Fortran storage order.

return sigma, asarray(X)

def _get_fiedler_func(method):
"""Return a function that solves the Fiedler eigenvalue problem.
"""
match = _tracemin_method.match(method)
if match:
method = match.group(1)
def find_fiedler(L, x, normalized, tol):
q = 2 if method == 'pcg' else min(4, L.shape - 1)
X = asmatrix(normal(size=(q, L.shape))).T
sigma, X = _tracemin_fiedler(L, X, normalized, tol, method)
return sigma, X[:, 0]
elif method == 'lanczos' or method == 'lobpcg':
def find_fiedler(L, x, normalized, tol):
L = csc_matrix(L, dtype=float)
n = L.shape
if normalized:
D = spdiags(1. / sqrt(L.diagonal()), , 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, X[:, 1]
else:
X = asarray(asmatrix(x).T)
M = spdiags(1. / L.diagonal(), , n, n)
Y = ones(n)
if normalized:
Y /= D.diagonal()
sigma, X = lobpcg(L, X, M=M, Y=asmatrix(Y).T, tol=tol,
maxiter=n, largest=False)
return sigma, X[:, 0]
else:
raise nx.NetworkXError("unknown method '%s'." % method)

return find_fiedler

@not_implemented_for('directed')
[docs]def algebraic_connectivity(G, weight='weight', normalized=False, tol=1e-8,
method='tracemin'):
"""Return 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
The data key used to determine the weight of each edge. If None, then
each edge has unit weight. Default value: None.

normalized : bool, optional
Whether the normalized Laplacian matrix is used. Default value: False.

tol : float, optional
Tolerance of relative residual in eigenvalue computation. Default
value: 1e-8.

method : string, optional
Method of eigenvalue computation. It should be one of 'tracemin'
(TraceMIN), 'lanczos' (Lanczos iteration) and 'lobpcg' (LOBPCG).
Default value: 'tracemin'.

The TraceMIN algorithm uses a linear system solver. The following
values allow specifying the solver to be used.

=============== ========================================
Value           Solver
=============== ========================================
'tracemin_chol' Cholesky factorization
'tracemin_lu'   LU factorization
=============== ========================================

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.

--------
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.

L = nx.laplacian_matrix(G)
if L.shape == 2:
return 2. * L[0, 0] if not normalized else 2.

find_fiedler = _get_fiedler_func(method)
x = None if method != 'lobpcg' else _rcm_estimate(G, G)
return find_fiedler(L, x, normalized, tol)

@not_implemented_for('directed')
[docs]def fiedler_vector(G, weight='weight', normalized=False, tol=1e-8,
method='tracemin'):
"""Return 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
The data key used to determine the weight of each edge. If None, then
each edge has unit weight. Default value: None.

normalized : bool, optional
Whether the normalized Laplacian matrix is used. Default value: False.

tol : float, optional
Tolerance of relative residual in eigenvalue computation. Default
value: 1e-8.

method : string, optional
Method of eigenvalue computation. It should be one of 'tracemin'
(TraceMIN), 'lanczos' (Lanczos iteration) and 'lobpcg' (LOBPCG).
Default value: 'tracemin'.

The TraceMIN algorithm uses a linear system solver. The following
values allow specifying the solver to be used.

=============== ========================================
Value           Solver
=============== ========================================
'tracemin_chol' Cholesky factorization
'tracemin_lu'   LU factorization
=============== ========================================

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.

--------
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., -1.])

find_fiedler = _get_fiedler_func(method)
L = nx.laplacian_matrix(G)
x = None if method != 'lobpcg' else _rcm_estimate(G, G)
return find_fiedler(L, x, normalized, tol)

[docs]def spectral_ordering(G, weight='weight', normalized=False, tol=1e-8,
method='tracemin'):
"""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
The data key used to determine the weight of each edge. If None, then
each edge has unit weight. Default value: None.

normalized : bool, optional
Whether the normalized Laplacian matrix is used. Default value: False.

tol : float, optional
Tolerance of relative residual in eigenvalue computation. Default
value: 1e-8.

method : string, optional
Method of eigenvalue computation. It should be one of 'tracemin'
(TraceMIN), 'lanczos' (Lanczos iteration) and 'lobpcg' (LOBPCG).
Default value: 'tracemin'.

The TraceMIN algorithm uses a linear system solver. The following
values allow specifying the solver to be used.

=============== ========================================
Value           Solver
=============== ========================================
'tracemin_chol' Cholesky factorization
'tracemin_lu'   LU factorization
=============== ========================================

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.

--------
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)
fiedler = find_fiedler(L, x, normalized, tol)
order.extend(
u for x, c, u in sorted(zip(fiedler, range(size), component)))
else:
order.extend(component)

return order

# fixture for nose tests
def setup_module(module):
from nose import SkipTest
try:
import numpy
import scipy.sparse
except ImportError:
raise SkipTest('SciPy not available.')