657 lines
21 KiB
Python
657 lines
21 KiB
Python
"""
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Algebraic connectivity and Fiedler vectors of undirected graphs.
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"""
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from functools import partial
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import networkx as nx
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from networkx.utils import (
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not_implemented_for,
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np_random_state,
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reverse_cuthill_mckee_ordering,
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)
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__all__ = [
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"algebraic_connectivity",
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"fiedler_vector",
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"spectral_ordering",
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"spectral_bisection",
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]
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class _PCGSolver:
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"""Preconditioned conjugate gradient method.
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To solve Ax = b:
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M = A.diagonal() # or some other preconditioner
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solver = _PCGSolver(lambda x: A * x, lambda x: M * x)
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x = solver.solve(b)
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The inputs A and M are functions which compute
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matrix multiplication on the argument.
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A - multiply by the matrix A in Ax=b
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M - multiply by M, the preconditioner surrogate for A
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Warning: There is no limit on number of iterations.
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"""
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def __init__(self, A, M):
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self._A = A
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self._M = M
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def solve(self, B, tol):
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import numpy as np
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# Densifying step - can this be kept sparse?
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B = np.asarray(B)
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X = np.ndarray(B.shape, order="F")
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for j in range(B.shape[1]):
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X[:, j] = self._solve(B[:, j], tol)
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return X
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def _solve(self, b, tol):
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import numpy as np
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import scipy as sp
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A = self._A
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M = self._M
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tol *= sp.linalg.blas.dasum(b)
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# Initialize.
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x = np.zeros(b.shape)
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r = b.copy()
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z = M(r)
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rz = sp.linalg.blas.ddot(r, z)
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p = z.copy()
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# Iterate.
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while True:
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Ap = A(p)
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alpha = rz / sp.linalg.blas.ddot(p, Ap)
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x = sp.linalg.blas.daxpy(p, x, a=alpha)
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r = sp.linalg.blas.daxpy(Ap, r, a=-alpha)
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if sp.linalg.blas.dasum(r) < tol:
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return x
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z = M(r)
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beta = sp.linalg.blas.ddot(r, z)
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beta, rz = beta / rz, beta
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p = sp.linalg.blas.daxpy(p, z, a=beta)
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class _LUSolver:
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"""LU factorization.
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To solve Ax = b:
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solver = _LUSolver(A)
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x = solver.solve(b)
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optional argument `tol` on solve method is ignored but included
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to match _PCGsolver API.
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"""
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def __init__(self, A):
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import scipy as sp
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self._LU = sp.sparse.linalg.splu(
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A,
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permc_spec="MMD_AT_PLUS_A",
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diag_pivot_thresh=0.0,
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options={"Equil": True, "SymmetricMode": True},
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)
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def solve(self, B, tol=None):
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import numpy as np
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B = np.asarray(B)
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X = np.ndarray(B.shape, order="F")
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for j in range(B.shape[1]):
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X[:, j] = self._LU.solve(B[:, j])
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return X
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def _preprocess_graph(G, weight):
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"""Compute edge weights and eliminate zero-weight edges."""
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if G.is_directed():
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H = nx.MultiGraph()
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H.add_nodes_from(G)
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H.add_weighted_edges_from(
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((u, v, e.get(weight, 1.0)) for u, v, e in G.edges(data=True) if u != v),
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weight=weight,
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)
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G = H
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if not G.is_multigraph():
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edges = (
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(u, v, abs(e.get(weight, 1.0))) for u, v, e in G.edges(data=True) if u != v
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)
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else:
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edges = (
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(u, v, sum(abs(e.get(weight, 1.0)) for e in G[u][v].values()))
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for u, v in G.edges()
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if u != v
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)
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H = nx.Graph()
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H.add_nodes_from(G)
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H.add_weighted_edges_from((u, v, e) for u, v, e in edges if e != 0)
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return H
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def _rcm_estimate(G, nodelist):
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"""Estimate the Fiedler vector using the reverse Cuthill-McKee ordering."""
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import numpy as np
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G = G.subgraph(nodelist)
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order = reverse_cuthill_mckee_ordering(G)
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n = len(nodelist)
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index = dict(zip(nodelist, range(n)))
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x = np.ndarray(n, dtype=float)
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for i, u in enumerate(order):
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x[index[u]] = i
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x -= (n - 1) / 2.0
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return x
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def _tracemin_fiedler(L, X, normalized, tol, method):
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"""Compute the Fiedler vector of L using the TraceMIN-Fiedler algorithm.
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The Fiedler vector of a connected undirected graph is the eigenvector
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corresponding to the second smallest eigenvalue of the Laplacian matrix
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of the graph. This function starts with the Laplacian L, not the Graph.
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Parameters
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----------
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L : Laplacian of a possibly weighted or normalized, but undirected graph
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X : Initial guess for a solution. Usually a matrix of random numbers.
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This function allows more than one column in X to identify more than
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one eigenvector if desired.
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normalized : bool
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Whether the normalized Laplacian matrix is used.
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tol : float
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Tolerance of relative residual in eigenvalue computation.
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Warning: There is no limit on number of iterations.
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method : string
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Should be 'tracemin_pcg' or 'tracemin_lu'.
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Otherwise exception is raised.
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Returns
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-------
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sigma, X : Two NumPy arrays of floats.
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The lowest eigenvalues and corresponding eigenvectors of L.
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The size of input X determines the size of these outputs.
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As this is for Fiedler vectors, the zero eigenvalue (and
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constant eigenvector) are avoided.
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"""
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import numpy as np
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import scipy as sp
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n = X.shape[0]
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if normalized:
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# Form the normalized Laplacian matrix and determine the eigenvector of
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# its nullspace.
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e = np.sqrt(L.diagonal())
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# TODO: rm csr_array wrapper when spdiags array creation becomes available
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D = sp.sparse.csr_array(sp.sparse.spdiags(1 / e, 0, n, n, format="csr"))
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L = D @ L @ D
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e *= 1.0 / np.linalg.norm(e, 2)
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if normalized:
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def project(X):
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"""Make X orthogonal to the nullspace of L."""
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X = np.asarray(X)
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for j in range(X.shape[1]):
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X[:, j] -= (X[:, j] @ e) * e
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else:
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def project(X):
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"""Make X orthogonal to the nullspace of L."""
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X = np.asarray(X)
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for j in range(X.shape[1]):
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X[:, j] -= X[:, j].sum() / n
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if method == "tracemin_pcg":
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D = L.diagonal().astype(float)
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solver = _PCGSolver(lambda x: L @ x, lambda x: D * x)
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elif method == "tracemin_lu":
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# Convert A to CSC to suppress SparseEfficiencyWarning.
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A = sp.sparse.csc_array(L, dtype=float, copy=True)
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# Force A to be nonsingular. Since A is the Laplacian matrix of a
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# connected graph, its rank deficiency is one, and thus one diagonal
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# element needs to modified. Changing to infinity forces a zero in the
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# corresponding element in the solution.
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i = (A.indptr[1:] - A.indptr[:-1]).argmax()
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A[i, i] = float("inf")
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solver = _LUSolver(A)
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else:
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raise nx.NetworkXError(f"Unknown linear system solver: {method}")
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# Initialize.
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Lnorm = abs(L).sum(axis=1).flatten().max()
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project(X)
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W = np.ndarray(X.shape, order="F")
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while True:
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# Orthonormalize X.
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X = np.linalg.qr(X)[0]
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# Compute iteration matrix H.
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W[:, :] = L @ X
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H = X.T @ W
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sigma, Y = sp.linalg.eigh(H, overwrite_a=True)
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# Compute the Ritz vectors.
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X = X @ Y
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# Test for convergence exploiting the fact that L * X == W * Y.
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res = sp.linalg.blas.dasum(W @ Y[:, 0] - sigma[0] * X[:, 0]) / Lnorm
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if res < tol:
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break
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# Compute X = L \ X / (X' * (L \ X)).
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# L \ X can have an arbitrary projection on the nullspace of L,
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# which will be eliminated.
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W[:, :] = solver.solve(X, tol)
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X = (sp.linalg.inv(W.T @ X) @ W.T).T # Preserves Fortran storage order.
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project(X)
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return sigma, np.asarray(X)
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def _get_fiedler_func(method):
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"""Returns a function that solves the Fiedler eigenvalue problem."""
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import numpy as np
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if method == "tracemin": # old style keyword <v2.1
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method = "tracemin_pcg"
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if method in ("tracemin_pcg", "tracemin_lu"):
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def find_fiedler(L, x, normalized, tol, seed):
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q = 1 if method == "tracemin_pcg" else min(4, L.shape[0] - 1)
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X = np.asarray(seed.normal(size=(q, L.shape[0]))).T
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sigma, X = _tracemin_fiedler(L, X, normalized, tol, method)
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return sigma[0], X[:, 0]
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elif method == "lanczos" or method == "lobpcg":
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def find_fiedler(L, x, normalized, tol, seed):
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import scipy as sp
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L = sp.sparse.csc_array(L, dtype=float)
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n = L.shape[0]
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if normalized:
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# TODO: rm csc_array wrapping when spdiags array becomes available
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D = sp.sparse.csc_array(
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sp.sparse.spdiags(
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1.0 / np.sqrt(L.diagonal()), [0], n, n, format="csc"
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)
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)
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L = D @ L @ D
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if method == "lanczos" or n < 10:
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# Avoid LOBPCG when n < 10 due to
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# https://github.com/scipy/scipy/issues/3592
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# https://github.com/scipy/scipy/pull/3594
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sigma, X = sp.sparse.linalg.eigsh(
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L, 2, which="SM", tol=tol, return_eigenvectors=True
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)
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return sigma[1], X[:, 1]
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else:
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X = np.asarray(np.atleast_2d(x).T)
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# TODO: rm csr_array wrapping when spdiags array becomes available
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M = sp.sparse.csr_array(sp.sparse.spdiags(1.0 / L.diagonal(), 0, n, n))
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Y = np.ones(n)
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if normalized:
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Y /= D.diagonal()
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sigma, X = sp.sparse.linalg.lobpcg(
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L, X, M=M, Y=np.atleast_2d(Y).T, tol=tol, maxiter=n, largest=False
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)
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return sigma[0], X[:, 0]
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else:
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raise nx.NetworkXError(f"unknown method {method!r}.")
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return find_fiedler
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@not_implemented_for("directed")
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@np_random_state(5)
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@nx._dispatch(edge_attrs="weight")
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def algebraic_connectivity(
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G, weight="weight", normalized=False, tol=1e-8, method="tracemin_pcg", seed=None
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):
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r"""Returns the algebraic connectivity of an undirected graph.
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The algebraic connectivity of a connected undirected graph is the second
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smallest eigenvalue of its Laplacian matrix.
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Parameters
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----------
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G : NetworkX graph
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An undirected graph.
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weight : object, optional (default: None)
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The data key used to determine the weight of each edge. If None, then
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each edge has unit weight.
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normalized : bool, optional (default: False)
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Whether the normalized Laplacian matrix is used.
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tol : float, optional (default: 1e-8)
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Tolerance of relative residual in eigenvalue computation.
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method : string, optional (default: 'tracemin_pcg')
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Method of eigenvalue computation. It must be one of the tracemin
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options shown below (TraceMIN), 'lanczos' (Lanczos iteration)
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or 'lobpcg' (LOBPCG).
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The TraceMIN algorithm uses a linear system solver. The following
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values allow specifying the solver to be used.
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=============== ========================================
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Value Solver
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=============== ========================================
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'tracemin_pcg' Preconditioned conjugate gradient method
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'tracemin_lu' LU factorization
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=============== ========================================
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seed : integer, random_state, or None (default)
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Indicator of random number generation state.
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See :ref:`Randomness<randomness>`.
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Returns
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-------
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algebraic_connectivity : float
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Algebraic connectivity.
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Raises
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------
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NetworkXNotImplemented
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If G is directed.
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NetworkXError
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If G has less than two nodes.
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Notes
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-----
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Edge weights are interpreted by their absolute values. For MultiGraph's,
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weights of parallel edges are summed. Zero-weighted edges are ignored.
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See Also
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--------
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laplacian_matrix
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Examples
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--------
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For undirected graphs algebraic connectivity can tell us if a graph is connected or not
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`G` is connected iff ``algebraic_connectivity(G) > 0``:
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>>> G = nx.complete_graph(5)
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>>> nx.algebraic_connectivity(G) > 0
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True
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>>> G.add_node(10) # G is no longer connected
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>>> nx.algebraic_connectivity(G) > 0
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False
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"""
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if len(G) < 2:
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raise nx.NetworkXError("graph has less than two nodes.")
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G = _preprocess_graph(G, weight)
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if not nx.is_connected(G):
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return 0.0
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L = nx.laplacian_matrix(G)
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if L.shape[0] == 2:
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return 2.0 * L[0, 0] if not normalized else 2.0
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find_fiedler = _get_fiedler_func(method)
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x = None if method != "lobpcg" else _rcm_estimate(G, G)
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sigma, fiedler = find_fiedler(L, x, normalized, tol, seed)
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return sigma
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@not_implemented_for("directed")
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@np_random_state(5)
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@nx._dispatch(edge_attrs="weight")
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def fiedler_vector(
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G, weight="weight", normalized=False, tol=1e-8, method="tracemin_pcg", seed=None
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):
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"""Returns the Fiedler vector of a connected undirected graph.
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The Fiedler vector of a connected undirected graph is the eigenvector
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corresponding to the second smallest eigenvalue of the Laplacian matrix
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of the graph.
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Parameters
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----------
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G : NetworkX graph
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An undirected graph.
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weight : object, optional (default: None)
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The data key used to determine the weight of each edge. If None, then
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each edge has unit weight.
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normalized : bool, optional (default: False)
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Whether the normalized Laplacian matrix is used.
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tol : float, optional (default: 1e-8)
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Tolerance of relative residual in eigenvalue computation.
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method : string, optional (default: 'tracemin_pcg')
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Method of eigenvalue computation. It must be one of the tracemin
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options shown below (TraceMIN), 'lanczos' (Lanczos iteration)
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or 'lobpcg' (LOBPCG).
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The TraceMIN algorithm uses a linear system solver. The following
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values allow specifying the solver to be used.
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|
=============== ========================================
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Value Solver
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=============== ========================================
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'tracemin_pcg' Preconditioned conjugate gradient method
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'tracemin_lu' LU factorization
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=============== ========================================
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seed : integer, random_state, or None (default)
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Indicator of random number generation state.
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See :ref:`Randomness<randomness>`.
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Returns
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-------
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fiedler_vector : NumPy array of floats.
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Fiedler vector.
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Raises
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------
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NetworkXNotImplemented
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If G is directed.
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NetworkXError
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If G has less than two nodes or is not connected.
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Notes
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-----
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Edge weights are interpreted by their absolute values. For MultiGraph's,
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weights of parallel edges are summed. Zero-weighted edges are ignored.
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See Also
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--------
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laplacian_matrix
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Examples
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--------
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Given a connected graph the signs of the values in the Fiedler vector can be
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used to partition the graph into two components.
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>>> G = nx.barbell_graph(5, 0)
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>>> nx.fiedler_vector(G, normalized=True, seed=1)
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array([-0.32864129, -0.32864129, -0.32864129, -0.32864129, -0.26072899,
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0.26072899, 0.32864129, 0.32864129, 0.32864129, 0.32864129])
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The connected components are the two 5-node cliques of the barbell graph.
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"""
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import numpy as np
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if len(G) < 2:
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raise nx.NetworkXError("graph has less than two nodes.")
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G = _preprocess_graph(G, weight)
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if not nx.is_connected(G):
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raise nx.NetworkXError("graph is not connected.")
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if len(G) == 2:
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return np.array([1.0, -1.0])
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find_fiedler = _get_fiedler_func(method)
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L = nx.laplacian_matrix(G)
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x = None if method != "lobpcg" else _rcm_estimate(G, G)
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sigma, fiedler = find_fiedler(L, x, normalized, tol, seed)
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return fiedler
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@np_random_state(5)
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@nx._dispatch(edge_attrs="weight")
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def spectral_ordering(
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G, weight="weight", normalized=False, tol=1e-8, method="tracemin_pcg", seed=None
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):
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"""Compute the spectral_ordering of a graph.
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The spectral ordering of a graph is an ordering of its nodes where nodes
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in the same weakly connected components appear contiguous and ordered by
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their corresponding elements in the Fiedler vector of the component.
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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_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.
|
|
|
|
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
|
|
|
|
|
|
@nx._dispatch(edge_attrs="weight")
|
|
def spectral_bisection(
|
|
G, weight="weight", normalized=False, tol=1e-8, method="tracemin_pcg", seed=None
|
|
):
|
|
"""Bisect the graph using the Fiedler vector.
|
|
|
|
This method uses the Fiedler vector to bisect a graph.
|
|
The partition is defined by the nodes which are associated with
|
|
either positive or negative values in the vector.
|
|
|
|
Parameters
|
|
----------
|
|
G : NetworkX Graph
|
|
|
|
weight : str, optional (default: weight)
|
|
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_lu' LU factorization
|
|
=============== ========================================
|
|
|
|
seed : integer, random_state, or None (default)
|
|
Indicator of random number generation state.
|
|
See :ref:`Randomness<randomness>`.
|
|
|
|
Returns
|
|
-------
|
|
bisection : tuple of sets
|
|
Sets with the bisection of nodes
|
|
|
|
Examples
|
|
--------
|
|
>>> G = nx.barbell_graph(3, 0)
|
|
>>> nx.spectral_bisection(G)
|
|
({0, 1, 2}, {3, 4, 5})
|
|
|
|
References
|
|
----------
|
|
.. [1] M. E. J Newman 'Networks: An Introduction', pages 364-370
|
|
Oxford University Press 2011.
|
|
"""
|
|
import numpy as np
|
|
|
|
v = nx.fiedler_vector(G, weight, normalized, tol, method, seed)
|
|
nodes = np.array(list(G))
|
|
pos_vals = v >= 0
|
|
|
|
return set(nodes[~pos_vals]), set(nodes[pos_vals])
|