367 lines
10 KiB
Python
367 lines
10 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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"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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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 matrix
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The Laplacian matrix of G.
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Notes
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-----
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For MultiGraph/MultiDiGraph, the edges weights are summed.
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See Also
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--------
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to_numpy_array
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normalized_laplacian_matrix
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laplacian_spectrum
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"""
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import scipy.sparse
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if nodelist is None:
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nodelist = list(G)
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A = nx.to_scipy_sparse_matrix(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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D = scipy.sparse.spdiags(diags.flatten(), [0], m, n, format="csr")
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return D - A
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@not_implemented_for("directed")
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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.
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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 matrix
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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/MultiDiGraph, the edges weights are summed.
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See 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
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import scipy.sparse
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if nodelist is None:
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nodelist = list(G)
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A = nx.to_scipy_sparse_matrix(G, nodelist=nodelist, weight=weight, format="csr")
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n, m = A.shape
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diags = A.sum(axis=1).flatten()
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D = scipy.sparse.spdiags(diags, [0], m, n, format="csr")
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L = D - A
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with scipy.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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DH = scipy.sparse.spdiags(diags_sqrt, [0], m, n, format="csr")
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return DH.dot(L.dot(DH))
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###############################################################################
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# Code based on
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# https://bitbucket.org/bedwards/networkx-community/src/370bd69fc02f/networkx/algorithms/community/
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@not_implemented_for("undirected")
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@not_implemented_for("multigraph")
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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.
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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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from scipy.sparse import spdiags, linalg
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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 = 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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sqrtp = np.sqrt(p)
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Q = spdiags(sqrtp, [0], n, n) * P * spdiags(1.0 / sqrtp, [0], n, n)
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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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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 and `\Phi` a matrix
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with the Perron vector of `P` in the diagonal and zeros elsewhere.
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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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from scipy.sparse import spdiags, linalg
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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 = 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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Phi = spdiags(p, [0], n, n)
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Phi = Phi.todense()
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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 matrix
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transition matrix of G.
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Raises
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------
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NetworkXError
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If walk_type not specified or alpha not in valid range
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"""
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import numpy as np
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from scipy.sparse import identity, spdiags
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if walk_type is None:
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if nx.is_strongly_connected(G):
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if nx.is_aperiodic(G):
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walk_type = "random"
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else:
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walk_type = "lazy"
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else:
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walk_type = "pagerank"
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M = nx.to_scipy_sparse_matrix(G, nodelist=nodelist, weight=weight, dtype=float)
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n, m = M.shape
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if walk_type in ["random", "lazy"]:
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DI = spdiags(1.0 / np.array(M.sum(axis=1).flat), [0], n, n)
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if walk_type == "random":
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P = DI * M
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else:
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I = identity(n)
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P = (I + DI * M) / 2.0
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elif walk_type == "pagerank":
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if not (0 < alpha < 1):
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raise nx.NetworkXError("alpha must be between 0 and 1")
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# this is using a dense representation
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M = M.todense()
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# add constant to dangling nodes' row
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dangling = np.where(M.sum(axis=1) == 0)
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for d in dangling[0]:
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M[d] = 1.0 / n
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# normalize
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M = M / M.sum(axis=1)
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P = alpha * M + (1 - alpha) / n
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else:
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raise nx.NetworkXError("walk_type must be random, lazy, or pagerank")
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return P
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