429 lines
13 KiB
Python
429 lines
13 KiB
Python
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"""Laplacian matrix of graphs.
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"""
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import networkx as nx
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from networkx.utils import not_implemented_for
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__all__ = [
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"laplacian_matrix",
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"normalized_laplacian_matrix",
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"total_spanning_tree_weight",
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"directed_laplacian_matrix",
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"directed_combinatorial_laplacian_matrix",
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]
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@not_implemented_for("directed")
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@nx._dispatch(edge_attrs="weight")
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def laplacian_matrix(G, nodelist=None, weight="weight"):
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"""Returns the Laplacian matrix of G.
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The graph Laplacian is the matrix L = D - A, where
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A is the adjacency matrix and D is the diagonal matrix of node degrees.
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Parameters
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----------
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G : graph
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A NetworkX graph
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nodelist : list, optional
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The rows and columns are ordered according to the nodes in nodelist.
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If nodelist is None, then the ordering is produced by G.nodes().
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weight : string or None, optional (default='weight')
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The edge data key used to compute each value in the matrix.
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If None, then each edge has weight 1.
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Returns
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-------
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L : SciPy sparse array
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The Laplacian matrix of G.
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Notes
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-----
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For MultiGraph, the edges weights are summed.
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See Also
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--------
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:func:`~networkx.convert_matrix.to_numpy_array`
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normalized_laplacian_matrix
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:func:`~networkx.linalg.spectrum.laplacian_spectrum`
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Examples
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--------
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For graphs with multiple connected components, L is permutation-similar
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to a block diagonal matrix where each block is the respective Laplacian
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matrix for each component.
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>>> G = nx.Graph([(1, 2), (2, 3), (4, 5)])
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>>> print(nx.laplacian_matrix(G).toarray())
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[[ 1 -1 0 0 0]
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[-1 2 -1 0 0]
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[ 0 -1 1 0 0]
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[ 0 0 0 1 -1]
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[ 0 0 0 -1 1]]
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"""
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import scipy as sp
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if nodelist is None:
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nodelist = list(G)
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A = nx.to_scipy_sparse_array(G, nodelist=nodelist, weight=weight, format="csr")
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n, m = A.shape
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# TODO: rm csr_array wrapper when spdiags can produce arrays
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D = sp.sparse.csr_array(sp.sparse.spdiags(A.sum(axis=1), 0, m, n, format="csr"))
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return D - A
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@not_implemented_for("directed")
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@nx._dispatch(edge_attrs="weight")
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def normalized_laplacian_matrix(G, nodelist=None, weight="weight"):
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r"""Returns the normalized Laplacian matrix of G.
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The normalized graph Laplacian is the matrix
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.. math::
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N = D^{-1/2} L D^{-1/2}
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where `L` is the graph Laplacian and `D` is the diagonal matrix of
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node degrees [1]_.
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Parameters
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----------
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G : graph
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A NetworkX graph
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nodelist : list, optional
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The rows and columns are ordered according to the nodes in nodelist.
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If nodelist is None, then the ordering is produced by G.nodes().
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weight : string or None, optional (default='weight')
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The edge data key used to compute each value in the matrix.
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If None, then each edge has weight 1.
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Returns
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-------
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N : SciPy sparse array
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The normalized Laplacian matrix of G.
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Notes
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-----
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For MultiGraph, the edges weights are summed.
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See :func:`to_numpy_array` for other options.
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If the Graph contains selfloops, D is defined as ``diag(sum(A, 1))``, where A is
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the adjacency matrix [2]_.
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See Also
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--------
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laplacian_matrix
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normalized_laplacian_spectrum
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References
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----------
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.. [1] Fan Chung-Graham, Spectral Graph Theory,
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CBMS Regional Conference Series in Mathematics, Number 92, 1997.
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.. [2] Steve Butler, Interlacing For Weighted Graphs Using The Normalized
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Laplacian, Electronic Journal of Linear Algebra, Volume 16, pp. 90-98,
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March 2007.
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"""
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import numpy as np
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import scipy as sp
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if nodelist is None:
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nodelist = list(G)
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A = nx.to_scipy_sparse_array(G, nodelist=nodelist, weight=weight, format="csr")
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n, m = A.shape
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diags = A.sum(axis=1)
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# TODO: rm csr_array wrapper when spdiags can produce arrays
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D = sp.sparse.csr_array(sp.sparse.spdiags(diags, 0, m, n, format="csr"))
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L = D - A
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with np.errstate(divide="ignore"):
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diags_sqrt = 1.0 / np.sqrt(diags)
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diags_sqrt[np.isinf(diags_sqrt)] = 0
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# TODO: rm csr_array wrapper when spdiags can produce arrays
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DH = sp.sparse.csr_array(sp.sparse.spdiags(diags_sqrt, 0, m, n, format="csr"))
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return DH @ (L @ DH)
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@nx._dispatch(edge_attrs="weight")
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def total_spanning_tree_weight(G, weight=None):
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"""
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Returns the total weight of all spanning trees of `G`.
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Kirchoff's Tree Matrix Theorem states that the determinant of any cofactor of the
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Laplacian matrix of a graph is the number of spanning trees in the graph. For a
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weighted Laplacian matrix, it is the sum across all spanning trees of the
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multiplicative weight of each tree. That is, the weight of each tree is the
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product of its edge weights.
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Parameters
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----------
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G : NetworkX Graph
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The graph to use Kirchhoff's theorem on.
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weight : string or None
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The key for the edge attribute holding the edge weight. If `None`, then
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each edge is assumed to have a weight of 1 and this function returns the
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total number of spanning trees in `G`.
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Returns
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-------
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float
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The sum of the total multiplicative weights for all spanning trees in `G`
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"""
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import numpy as np
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G_laplacian = nx.laplacian_matrix(G, weight=weight).toarray()
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# Determinant ignoring first row and column
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return abs(np.linalg.det(G_laplacian[1:, 1:]))
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###############################################################################
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# Code based on work from https://github.com/bjedwards
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@not_implemented_for("undirected")
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@not_implemented_for("multigraph")
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@nx._dispatch(edge_attrs="weight")
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def directed_laplacian_matrix(
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G, nodelist=None, weight="weight", walk_type=None, alpha=0.95
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):
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r"""Returns the directed Laplacian matrix of G.
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The graph directed Laplacian is the matrix
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.. math::
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L = I - (\Phi^{1/2} P \Phi^{-1/2} + \Phi^{-1/2} P^T \Phi^{1/2} ) / 2
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where `I` is the identity matrix, `P` is the transition matrix of the
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graph, and `\Phi` a matrix with the Perron vector of `P` in the diagonal and
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zeros elsewhere [1]_.
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Depending on the value of walk_type, `P` can be the transition matrix
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induced by a random walk, a lazy random walk, or a random walk with
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teleportation (PageRank).
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Parameters
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----------
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G : DiGraph
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A NetworkX graph
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nodelist : list, optional
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The rows and columns are ordered according to the nodes in nodelist.
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If nodelist is None, then the ordering is produced by G.nodes().
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weight : string or None, optional (default='weight')
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The edge data key used to compute each value in the matrix.
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If None, then each edge has weight 1.
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walk_type : string or None, optional (default=None)
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If None, `P` is selected depending on the properties of the
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graph. Otherwise is one of 'random', 'lazy', or 'pagerank'
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alpha : real
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(1 - alpha) is the teleportation probability used with pagerank
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Returns
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-------
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L : NumPy matrix
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Normalized Laplacian of G.
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Notes
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-----
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Only implemented for DiGraphs
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See Also
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--------
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laplacian_matrix
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References
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----------
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.. [1] Fan Chung (2005).
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Laplacians and the Cheeger inequality for directed graphs.
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Annals of Combinatorics, 9(1), 2005
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"""
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import numpy as np
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import scipy as sp
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# NOTE: P has type ndarray if walk_type=="pagerank", else csr_array
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P = _transition_matrix(
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G, nodelist=nodelist, weight=weight, walk_type=walk_type, alpha=alpha
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)
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n, m = P.shape
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evals, evecs = sp.sparse.linalg.eigs(P.T, k=1)
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v = evecs.flatten().real
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p = v / v.sum()
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# p>=0 by Perron-Frobenius Thm. Use abs() to fix roundoff across zero gh-6865
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sqrtp = np.sqrt(np.abs(p))
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Q = (
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# TODO: rm csr_array wrapper when spdiags creates arrays
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sp.sparse.csr_array(sp.sparse.spdiags(sqrtp, 0, n, n))
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@ P
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# TODO: rm csr_array wrapper when spdiags creates arrays
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@ sp.sparse.csr_array(sp.sparse.spdiags(1.0 / sqrtp, 0, n, n))
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)
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# NOTE: This could be sparsified for the non-pagerank cases
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I = np.identity(len(G))
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return I - (Q + Q.T) / 2.0
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@not_implemented_for("undirected")
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@not_implemented_for("multigraph")
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@nx._dispatch(edge_attrs="weight")
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def directed_combinatorial_laplacian_matrix(
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G, nodelist=None, weight="weight", walk_type=None, alpha=0.95
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):
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r"""Return the directed combinatorial Laplacian matrix of G.
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The graph directed combinatorial Laplacian is the matrix
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.. math::
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L = \Phi - (\Phi P + P^T \Phi) / 2
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where `P` is the transition matrix of the graph and `\Phi` a matrix
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with the Perron vector of `P` in the diagonal and zeros elsewhere [1]_.
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Depending on the value of walk_type, `P` can be the transition matrix
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induced by a random walk, a lazy random walk, or a random walk with
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teleportation (PageRank).
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Parameters
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----------
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G : DiGraph
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A NetworkX graph
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nodelist : list, optional
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The rows and columns are ordered according to the nodes in nodelist.
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If nodelist is None, then the ordering is produced by G.nodes().
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weight : string or None, optional (default='weight')
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The edge data key used to compute each value in the matrix.
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If None, then each edge has weight 1.
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walk_type : string or None, optional (default=None)
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If None, `P` is selected depending on the properties of the
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graph. Otherwise is one of 'random', 'lazy', or 'pagerank'
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alpha : real
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(1 - alpha) is the teleportation probability used with pagerank
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Returns
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-------
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L : NumPy matrix
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Combinatorial Laplacian of G.
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Notes
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-----
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Only implemented for DiGraphs
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See Also
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--------
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laplacian_matrix
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References
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----------
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.. [1] Fan Chung (2005).
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Laplacians and the Cheeger inequality for directed graphs.
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Annals of Combinatorics, 9(1), 2005
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"""
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import scipy as sp
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P = _transition_matrix(
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G, nodelist=nodelist, weight=weight, walk_type=walk_type, alpha=alpha
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)
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n, m = P.shape
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evals, evecs = sp.sparse.linalg.eigs(P.T, k=1)
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v = evecs.flatten().real
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p = v / v.sum()
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# NOTE: could be improved by not densifying
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# TODO: Rm csr_array wrapper when spdiags array creation becomes available
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Phi = sp.sparse.csr_array(sp.sparse.spdiags(p, 0, n, n)).toarray()
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return Phi - (Phi @ P + P.T @ Phi) / 2.0
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def _transition_matrix(G, nodelist=None, weight="weight", walk_type=None, alpha=0.95):
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"""Returns the transition matrix of G.
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This is a row stochastic giving the transition probabilities while
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performing a random walk on the graph. Depending on the value of walk_type,
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P can be the transition matrix induced by a random walk, a lazy random walk,
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or a random walk with teleportation (PageRank).
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Parameters
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----------
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G : DiGraph
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A NetworkX graph
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nodelist : list, optional
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The rows and columns are ordered according to the nodes in nodelist.
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If nodelist is None, then the ordering is produced by G.nodes().
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weight : string or None, optional (default='weight')
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The edge data key used to compute each value in the matrix.
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If None, then each edge has weight 1.
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walk_type : string or None, optional (default=None)
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If None, `P` is selected depending on the properties of the
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graph. Otherwise is one of 'random', 'lazy', or 'pagerank'
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alpha : real
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(1 - alpha) is the teleportation probability used with pagerank
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Returns
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-------
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P : numpy.ndarray
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transition matrix of G.
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Raises
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|
------
|
||
|
|
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
|