""" 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 np_random_state __all__ = ["algebraic_connectivity", "fiedler_vector", "spectral_ordering"] 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 surrogate for A Warning: There is no limit on number of iterations. """ def __init__(self, A, M): self._A = A self._M = M def solve(self, B, tol): import numpy as np # Densifying step - can this be kept sparse? B = np.asarray(B) X = np.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): import numpy as np import scipy as sp import scipy.linalg.blas # call as sp.linalg.blas A = self._A M = self._M tol *= sp.linalg.blas.dasum(b) # Initialize. x = np.zeros(b.shape) r = b.copy() z = M(r) rz = sp.linalg.blas.ddot(r, z) p = z.copy() # Iterate. while True: Ap = A(p) alpha = rz / sp.linalg.blas.ddot(p, Ap) x = sp.linalg.blas.daxpy(p, x, a=alpha) r = sp.linalg.blas.daxpy(Ap, r, a=-alpha) if sp.linalg.blas.dasum(r) < tol: return x z = M(r) beta = sp.linalg.blas.ddot(r, z) beta, rz = beta / rz, beta p = sp.linalg.blas.daxpy(p, z, a=beta) 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): import scipy as sp import scipy.sparse.linalg # call as sp.sparse.linalg self._LU = sp.sparse.linalg.splu( A, permc_spec="MMD_AT_PLUS_A", diag_pivot_thresh=0.0, options={"Equil": True, "SymmetricMode": True}, ) def solve(self, B, tol=None): import numpy as np B = np.asarray(B) X = np.ndarray(B.shape, order="F") for j in range(B.shape[1]): X[:, j] = self._LU.solve(B[:, j]) return X 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.""" import numpy as np G = G.subgraph(nodelist) order = reverse_cuthill_mckee_ordering(G) n = len(nodelist) index = dict(zip(nodelist, range(n))) x = np.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 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' 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. """ import numpy as np import scipy as sp import scipy.linalg # call as sp.linalg import scipy.linalg.blas # call as sp.linalg.blas import scipy.sparse # call as sp.sparse n = X.shape[0] if normalized: # Form the normalized Laplacian matrix and determine the eigenvector of # its nullspace. e = np.sqrt(L.diagonal()) # TODO: rm csr_array wrapper when spdiags array creation becomes available D = sp.sparse.csr_array(sp.sparse.spdiags(1 / e, 0, n, n, format="csr")) L = D @ L @ D e *= 1.0 / np.linalg.norm(e, 2) if normalized: def project(X): """Make X orthogonal to the nullspace of L.""" X = np.asarray(X) for j in range(X.shape[1]): X[:, j] -= (X[:, j] @ e) * e else: def project(X): """Make X orthogonal to the nullspace of L.""" X = np.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_lu": # Convert A to CSC to suppress SparseEfficiencyWarning. A = sp.sparse.csc_array(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 = _LUSolver(A) else: raise nx.NetworkXError(f"Unknown linear system solver: {method}") # Initialize. Lnorm = abs(L).sum(axis=1).flatten().max() project(X) W = np.ndarray(X.shape, order="F") while True: # Orthonormalize X. X = np.linalg.qr(X)[0] # Compute iteration matrix H. W[:, :] = L @ X H = X.T @ W sigma, Y = sp.linalg.eigh(H, overwrite_a=True) # Compute the Ritz vectors. X = X @ Y # Test for convergence exploiting the fact that L * X == W * Y. res = sp.linalg.blas.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 = (sp.linalg.inv(W.T @ X) @ W.T).T # Preserves Fortran storage order. project(X) return sigma, np.asarray(X) def _get_fiedler_func(method): """Returns a function that solves the Fiedler eigenvalue problem.""" import numpy as np if method == "tracemin": # old style keyword `. 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. 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 @np_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 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_lu' LU factorization =============== ======================================== seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`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. See Also -------- laplacian_matrix """ import numpy as np 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 np.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 @np_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_lu' LU factorization =============== ======================================== seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`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