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"""Laplacian matrix of graphs.
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All calculations here are done using the out-degree. For Laplacians using
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in-degree, use `G.reverse(copy=False)` instead of `G` and take the transpose.
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The `laplacian_matrix` function provides an unnormalized matrix,
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while `normalized_laplacian_matrix`, `directed_laplacian_matrix`,
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and `directed_combinatorial_laplacian_matrix` are all normalized.
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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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@nx._dispatchable(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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This returns an unnormalized matrix. For a normalized output,
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use `normalized_laplacian_matrix`, `directed_laplacian_matrix`,
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or `directed_combinatorial_laplacian_matrix`.
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This calculation uses the out-degree of the graph `G`. To use the
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in-degree for calculations instead, use `G.reverse(copy=False)` and
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take the transpose.
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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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directed_laplacian_matrix
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directed_combinatorial_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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>>> edges = [
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... (1, 2),
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... (2, 1),
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... (2, 4),
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... (4, 3),
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... (3, 4),
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... ]
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>>> DiG = nx.DiGraph(edges)
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>>> print(nx.laplacian_matrix(DiG).toarray())
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[[ 1 -1 0 0]
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[-1 2 -1 0]
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[ 0 0 1 -1]
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[ 0 0 -1 1]]
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Notice that node 4 is represented by the third column and row. This is because
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by default the row/column order is the order of `G.nodes` (i.e. the node added
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order -- in the edgelist, 4 first appears in (2, 4), before node 3 in edge (4, 3).)
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To control the node order of the matrix, use the `nodelist` argument.
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>>> print(nx.laplacian_matrix(DiG, nodelist=[1, 2, 3, 4]).toarray())
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[[ 1 -1 0 0]
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[-1 2 0 -1]
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[ 0 0 1 -1]
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[ 0 0 -1 1]]
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This calculation uses the out-degree of the graph `G`. To use the
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in-degree for calculations instead, use `G.reverse(copy=False)` and
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take the transpose.
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>>> print(nx.laplacian_matrix(DiG.reverse(copy=False)).toarray().T)
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[[ 1 -1 0 0]
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[-1 1 -1 0]
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[ 0 0 2 -1]
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[ 0 0 -1 1]]
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References
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----------
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.. [1] Langville, Amy N., and Carl D. Meyer. Google’s PageRank and Beyond:
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The Science of Search Engine Rankings. Princeton University Press, 2006.
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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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@nx._dispatchable(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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This calculation uses the out-degree of the graph `G`. To use the
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in-degree for calculations instead, use `G.reverse(copy=False)` and
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take the transpose.
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For an unnormalized output, use `laplacian_matrix`.
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Examples
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--------
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>>> import numpy as np
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>>> edges = [
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... (1, 2),
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... (2, 1),
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... (2, 4),
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... (4, 3),
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... (3, 4),
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... ]
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>>> DiG = nx.DiGraph(edges)
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>>> print(nx.normalized_laplacian_matrix(DiG).toarray())
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[[ 1. -0.70710678 0. 0. ]
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[-0.70710678 1. -0.70710678 0. ]
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[ 0. 0. 1. -1. ]
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[ 0. 0. -1. 1. ]]
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Notice that node 4 is represented by the third column and row. This is because
|
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|
|
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by default the row/column order is the order of `G.nodes` (i.e. the node added
|
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order -- in the edgelist, 4 first appears in (2, 4), before node 3 in edge (4, 3).)
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To control the node order of the matrix, use the `nodelist` argument.
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>>> print(nx.normalized_laplacian_matrix(DiG, nodelist=[1, 2, 3, 4]).toarray())
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[[ 1. -0.70710678 0. 0. ]
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[-0.70710678 1. 0. -0.70710678]
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[ 0. 0. 1. -1. ]
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[ 0. 0. -1. 1. ]]
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>>> G = nx.Graph(edges)
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>>> print(nx.normalized_laplacian_matrix(G).toarray())
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[[ 1. -0.70710678 0. 0. ]
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[-0.70710678 1. -0.5 0. ]
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[ 0. -0.5 1. -0.70710678]
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[ 0. 0. -0.70710678 1. ]]
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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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directed_laplacian_matrix
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directed_combinatorial_laplacian_matrix
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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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.. [3] Langville, Amy N., and Carl D. Meyer. Google’s PageRank and Beyond:
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The Science of Search Engine Rankings. Princeton University Press, 2006.
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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, _ = 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, n, 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, n, n, format="csr"))
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return DH @ (L @ DH)
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@nx._dispatchable(edge_attrs="weight")
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def total_spanning_tree_weight(G, weight=None, root=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 [1]_, [2]_ states that the determinant of any
|
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cofactor of the Laplacian matrix of a graph is the number of spanning trees
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in the graph. For a weighted Laplacian matrix, it is the sum across all
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spanning trees of the multiplicative weight of each tree. That is, the
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weight of each tree is the product of its edge weights.
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For unweighted graphs, the total weight equals the number of spanning trees in `G`.
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For directed graphs, the total weight follows by summing over all directed
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spanning trees in `G` that start in the `root` node [3]_.
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|
.. deprecated:: 3.3
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|
``total_spanning_tree_weight`` is deprecated and will be removed in v3.5.
|
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|
|
Use ``nx.number_of_spanning_trees(G)`` instead.
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|
|
Parameters
|
|
|
|
|
|
----------
|
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|
|
G : NetworkX Graph
|
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|
|
weight : string or None, optional (default=None)
|
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|
|
|
|
The key for the edge attribute holding the edge weight.
|
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|
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If None, then each edge has weight 1.
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root : node (only required for directed graphs)
|
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|
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A node in the directed graph `G`.
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Returns
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|
|
-------
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|
|
total_weight : float
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|
|
Undirected graphs:
|
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|
|
The sum of the total multiplicative weights for all spanning trees in `G`.
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Directed graphs:
|
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|
|
The sum of the total multiplicative weights for all spanning trees of `G`,
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|
|
rooted at node `root`.
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Raises
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|
------
|
|
|
|
|
|
NetworkXPointlessConcept
|
|
|
|
|
|
If `G` does not contain any nodes.
|
|
|
|
|
|
|
|
|
|
|
|
NetworkXError
|
|
|
|
|
|
If the graph `G` is not (weakly) connected,
|
|
|
|
|
|
or if `G` is directed and the root node is not specified or not in G.
|
|
|
|
|
|
|
|
|
|
|
|
Examples
|
|
|
|
|
|
--------
|
|
|
|
|
|
>>> G = nx.complete_graph(5)
|
|
|
|
|
|
>>> round(nx.total_spanning_tree_weight(G))
|
|
|
|
|
|
125
|
|
|
|
|
|
|
|
|
|
|
|
>>> G = nx.Graph()
|
|
|
|
|
|
>>> G.add_edge(1, 2, weight=2)
|
|
|
|
|
|
>>> G.add_edge(1, 3, weight=1)
|
|
|
|
|
|
>>> G.add_edge(2, 3, weight=1)
|
|
|
|
|
|
>>> round(nx.total_spanning_tree_weight(G, "weight"))
|
|
|
|
|
|
5
|
|
|
|
|
|
|
|
|
|
|
|
Notes
|
|
|
|
|
|
-----
|
|
|
|
|
|
Self-loops are excluded. Multi-edges are contracted in one edge
|
|
|
|
|
|
equal to the sum of the weights.
|
|
|
|
|
|
|
|
|
|
|
|
References
|
|
|
|
|
|
----------
|
|
|
|
|
|
.. [1] Wikipedia
|
|
|
|
|
|
"Kirchhoff's theorem."
|
|
|
|
|
|
https://en.wikipedia.org/wiki/Kirchhoff%27s_theorem
|
|
|
|
|
|
.. [2] Kirchhoff, G. R.
|
|
|
|
|
|
Über die Auflösung der Gleichungen, auf welche man
|
|
|
|
|
|
bei der Untersuchung der linearen Vertheilung
|
|
|
|
|
|
Galvanischer Ströme geführt wird
|
|
|
|
|
|
Annalen der Physik und Chemie, vol. 72, pp. 497-508, 1847.
|
|
|
|
|
|
.. [3] Margoliash, J.
|
|
|
|
|
|
"Matrix-Tree Theorem for Directed Graphs"
|
|
|
|
|
|
https://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Margoliash.pdf
|
|
|
|
|
|
"""
|
|
|
|
|
|
import warnings
|
|
|
|
|
|
|
|
|
|
|
|
warnings.warn(
|
|
|
|
|
|
(
|
|
|
|
|
|
"\n\ntotal_spanning_tree_weight is deprecated and will be removed in v3.5.\n"
|
|
|
|
|
|
"Use `nx.number_of_spanning_trees(G)` instead."
|
|
|
|
|
|
),
|
|
|
|
|
|
category=DeprecationWarning,
|
|
|
|
|
|
stacklevel=3,
|
|
|
|
|
|
)
|
|
|
|
|
|
|
|
|
|
|
|
return nx.number_of_spanning_trees(G, weight=weight, root=root)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
###############################################################################
|
|
|
|
|
|
# Code based on work from https://github.com/bjedwards
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
@not_implemented_for("undirected")
|
|
|
|
|
|
@not_implemented_for("multigraph")
|
|
|
|
|
|
@nx._dispatchable(edge_attrs="weight")
|
|
|
|
|
|
def directed_laplacian_matrix(
|
|
|
|
|
|
G, nodelist=None, weight="weight", walk_type=None, alpha=0.95
|
|
|
|
|
|
):
|
|
|
|
|
|
r"""Returns the directed Laplacian matrix of G.
|
|
|
|
|
|
|
|
|
|
|
|
The graph directed Laplacian is the matrix
|
|
|
|
|
|
|
|
|
|
|
|
.. math::
|
|
|
|
|
|
|
|
|
|
|
|
L = I - \frac{1}{2} \left (\Phi^{1/2} P \Phi^{-1/2} + \Phi^{-1/2} P^T \Phi^{1/2} \right )
|
|
|
|
|
|
|
|
|
|
|
|
where `I` is the identity matrix, `P` is the transition matrix of the
|
|
|
|
|
|
graph, and `\Phi` a matrix with the Perron vector of `P` in the diagonal and
|
|
|
|
|
|
zeros elsewhere [1]_.
|
|
|
|
|
|
|
|
|
|
|
|
Depending on the value of walk_type, `P` can be the transition matrix
|
|
|
|
|
|
induced by a random walk, a lazy random walk, or a random walk with
|
|
|
|
|
|
teleportation (PageRank).
|
|
|
|
|
|
|
|
|
|
|
|
Parameters
|
|
|
|
|
|
----------
|
|
|
|
|
|
G : DiGraph
|
|
|
|
|
|
A NetworkX graph
|
|
|
|
|
|
|
|
|
|
|
|
nodelist : list, optional
|
|
|
|
|
|
The rows and columns are ordered according to the nodes in nodelist.
|
|
|
|
|
|
If nodelist is None, then the ordering is produced by G.nodes().
|
|
|
|
|
|
|
|
|
|
|
|
weight : string or None, optional (default='weight')
|
|
|
|
|
|
The edge data key used to compute each value in the matrix.
|
|
|
|
|
|
If None, then each edge has weight 1.
|
|
|
|
|
|
|
|
|
|
|
|
walk_type : string or None, optional (default=None)
|
|
|
|
|
|
One of ``"random"``, ``"lazy"``, or ``"pagerank"``. If ``walk_type=None``
|
|
|
|
|
|
(the default), then a value is selected according to the properties of `G`:
|
|
|
|
|
|
- ``walk_type="random"`` if `G` is strongly connected and aperiodic
|
|
|
|
|
|
- ``walk_type="lazy"`` if `G` is strongly connected but not aperiodic
|
|
|
|
|
|
- ``walk_type="pagerank"`` for all other cases.
|
|
|
|
|
|
|
|
|
|
|
|
alpha : real
|
|
|
|
|
|
(1 - alpha) is the teleportation probability used with pagerank
|
|
|
|
|
|
|
|
|
|
|
|
Returns
|
|
|
|
|
|
-------
|
|
|
|
|
|
L : NumPy matrix
|
|
|
|
|
|
Normalized Laplacian of G.
|
|
|
|
|
|
|
|
|
|
|
|
Notes
|
|
|
|
|
|
-----
|
|
|
|
|
|
Only implemented for DiGraphs
|
|
|
|
|
|
|
|
|
|
|
|
The result is always a symmetric matrix.
|
|
|
|
|
|
|
|
|
|
|
|
This calculation uses the out-degree of the graph `G`. To use the
|
|
|
|
|
|
in-degree for calculations instead, use `G.reverse(copy=False)` and
|
|
|
|
|
|
take the transpose.
|
|
|
|
|
|
|
|
|
|
|
|
See Also
|
|
|
|
|
|
--------
|
|
|
|
|
|
laplacian_matrix
|
|
|
|
|
|
normalized_laplacian_matrix
|
|
|
|
|
|
directed_combinatorial_laplacian_matrix
|
|
|
|
|
|
|
|
|
|
|
|
References
|
|
|
|
|
|
----------
|
|
|
|
|
|
.. [1] Fan Chung (2005).
|
|
|
|
|
|
Laplacians and the Cheeger inequality for directed graphs.
|
|
|
|
|
|
Annals of Combinatorics, 9(1), 2005
|
|
|
|
|
|
"""
|
|
|
|
|
|
import numpy as np
|
|
|
|
|
|
import scipy as sp
|
|
|
|
|
|
|
|
|
|
|
|
# NOTE: P has type ndarray if walk_type=="pagerank", else csr_array
|
|
|
|
|
|
P = _transition_matrix(
|
|
|
|
|
|
G, nodelist=nodelist, weight=weight, walk_type=walk_type, alpha=alpha
|
|
|
|
|
|
)
|
|
|
|
|
|
|
|
|
|
|
|
n, m = P.shape
|
|
|
|
|
|
|
|
|
|
|
|
evals, evecs = sp.sparse.linalg.eigs(P.T, k=1)
|
|
|
|
|
|
v = evecs.flatten().real
|
|
|
|
|
|
p = v / v.sum()
|
|
|
|
|
|
# p>=0 by Perron-Frobenius Thm. Use abs() to fix roundoff across zero gh-6865
|
|
|
|
|
|
sqrtp = np.sqrt(np.abs(p))
|
|
|
|
|
|
Q = (
|
|
|
|
|
|
# TODO: rm csr_array wrapper when spdiags creates arrays
|
|
|
|
|
|
sp.sparse.csr_array(sp.sparse.spdiags(sqrtp, 0, n, n))
|
|
|
|
|
|
@ P
|
|
|
|
|
|
# TODO: rm csr_array wrapper when spdiags creates arrays
|
|
|
|
|
|
@ sp.sparse.csr_array(sp.sparse.spdiags(1.0 / sqrtp, 0, n, n))
|
|
|
|
|
|
)
|
|
|
|
|
|
# NOTE: This could be sparsified for the non-pagerank cases
|
|
|
|
|
|
I = np.identity(len(G))
|
|
|
|
|
|
|
|
|
|
|
|
return I - (Q + Q.T) / 2.0
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
@not_implemented_for("undirected")
|
|
|
|
|
|
@not_implemented_for("multigraph")
|
|
|
|
|
|
@nx._dispatchable(edge_attrs="weight")
|
|
|
|
|
|
def directed_combinatorial_laplacian_matrix(
|
|
|
|
|
|
G, nodelist=None, weight="weight", walk_type=None, alpha=0.95
|
|
|
|
|
|
):
|
|
|
|
|
|
r"""Return the directed combinatorial Laplacian matrix of G.
|
|
|
|
|
|
|
|
|
|
|
|
The graph directed combinatorial Laplacian is the matrix
|
|
|
|
|
|
|
|
|
|
|
|
.. math::
|
|
|
|
|
|
|
|
|
|
|
|
L = \Phi - \frac{1}{2} \left (\Phi P + P^T \Phi \right)
|
|
|
|
|
|
|
|
|
|
|
|
where `P` is the transition matrix of the graph and `\Phi` a matrix
|
|
|
|
|
|
with the Perron vector of `P` in the diagonal and zeros elsewhere [1]_.
|
|
|
|
|
|
|
|
|
|
|
|
Depending on the value of walk_type, `P` can be the transition matrix
|
|
|
|
|
|
induced by a random walk, a lazy random walk, or a random walk with
|
|
|
|
|
|
teleportation (PageRank).
|
|
|
|
|
|
|
|
|
|
|
|
Parameters
|
|
|
|
|
|
----------
|
|
|
|
|
|
G : DiGraph
|
|
|
|
|
|
A NetworkX graph
|
|
|
|
|
|
|
|
|
|
|
|
nodelist : list, optional
|
|
|
|
|
|
The rows and columns are ordered according to the nodes in nodelist.
|
|
|
|
|
|
If nodelist is None, then the ordering is produced by G.nodes().
|
|
|
|
|
|
|
|
|
|
|
|
weight : string or None, optional (default='weight')
|
|
|
|
|
|
The edge data key used to compute each value in the matrix.
|
|
|
|
|
|
If None, then each edge has weight 1.
|
|
|
|
|
|
|
|
|
|
|
|
walk_type : string or None, optional (default=None)
|
|
|
|
|
|
One of ``"random"``, ``"lazy"``, or ``"pagerank"``. If ``walk_type=None``
|
|
|
|
|
|
(the default), then a value is selected according to the properties of `G`:
|
|
|
|
|
|
- ``walk_type="random"`` if `G` is strongly connected and aperiodic
|
|
|
|
|
|
- ``walk_type="lazy"`` if `G` is strongly connected but not aperiodic
|
|
|
|
|
|
- ``walk_type="pagerank"`` for all other cases.
|
|
|
|
|
|
|
|
|
|
|
|
alpha : real
|
|
|
|
|
|
(1 - alpha) is the teleportation probability used with pagerank
|
|
|
|
|
|
|
|
|
|
|
|
Returns
|
|
|
|
|
|
-------
|
|
|
|
|
|
L : NumPy matrix
|
|
|
|
|
|
Combinatorial Laplacian of G.
|
|
|
|
|
|
|
|
|
|
|
|
Notes
|
|
|
|
|
|
-----
|
|
|
|
|
|
Only implemented for DiGraphs
|
|
|
|
|
|
|
|
|
|
|
|
The result is always a symmetric matrix.
|
|
|
|
|
|
|
|
|
|
|
|
This calculation uses the out-degree of the graph `G`. To use the
|
|
|
|
|
|
in-degree for calculations instead, use `G.reverse(copy=False)` and
|
|
|
|
|
|
take the transpose.
|
|
|
|
|
|
|
|
|
|
|
|
See Also
|
|
|
|
|
|
--------
|
|
|
|
|
|
laplacian_matrix
|
|
|
|
|
|
normalized_laplacian_matrix
|
|
|
|
|
|
directed_laplacian_matrix
|
|
|
|
|
|
|
|
|
|
|
|
References
|
|
|
|
|
|
----------
|
|
|
|
|
|
.. [1] Fan Chung (2005).
|
|
|
|
|
|
Laplacians and the Cheeger inequality for directed graphs.
|
|
|
|
|
|
Annals of Combinatorics, 9(1), 2005
|
|
|
|
|
|
"""
|
|
|
|
|
|
import scipy as sp
|
|
|
|
|
|
|
|
|
|
|
|
P = _transition_matrix(
|
|
|
|
|
|
G, nodelist=nodelist, weight=weight, walk_type=walk_type, alpha=alpha
|
|
|
|
|
|
)
|
|
|
|
|
|
|
|
|
|
|
|
n, m = P.shape
|
|
|
|
|
|
|
|
|
|
|
|
evals, evecs = sp.sparse.linalg.eigs(P.T, k=1)
|
|
|
|
|
|
v = evecs.flatten().real
|
|
|
|
|
|
p = v / v.sum()
|
|
|
|
|
|
# NOTE: could be improved by not densifying
|
|
|
|
|
|
# TODO: Rm csr_array wrapper when spdiags array creation becomes available
|
|
|
|
|
|
Phi = sp.sparse.csr_array(sp.sparse.spdiags(p, 0, n, n)).toarray()
|
|
|
|
|
|
|
|
|
|
|
|
return Phi - (Phi @ P + P.T @ Phi) / 2.0
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
def _transition_matrix(G, nodelist=None, weight="weight", walk_type=None, alpha=0.95):
|
|
|
|
|
|
"""Returns the transition matrix of G.
|
|
|
|
|
|
|
|
|
|
|
|
This is a row stochastic giving the transition probabilities while
|
|
|
|
|
|
performing a random walk on the graph. Depending on the value of walk_type,
|
|
|
|
|
|
P can be the transition matrix induced by a random walk, a lazy random walk,
|
|
|
|
|
|
or a random walk with teleportation (PageRank).
|
|
|
|
|
|
|
|
|
|
|
|
Parameters
|
|
|
|
|
|
----------
|
|
|
|
|
|
G : DiGraph
|
|
|
|
|
|
A NetworkX graph
|
|
|
|
|
|
|
|
|
|
|
|
nodelist : list, optional
|
|
|
|
|
|
The rows and columns are ordered according to the nodes in nodelist.
|
|
|
|
|
|
If nodelist is None, then the ordering is produced by G.nodes().
|
|
|
|
|
|
|
|
|
|
|
|
weight : string or None, optional (default='weight')
|
|
|
|
|
|
The edge data key used to compute each value in the matrix.
|
|
|
|
|
|
If None, then each edge has weight 1.
|
|
|
|
|
|
|
|
|
|
|
|
walk_type : string or None, optional (default=None)
|
|
|
|
|
|
One of ``"random"``, ``"lazy"``, or ``"pagerank"``. If ``walk_type=None``
|
|
|
|
|
|
(the default), then a value is selected according to the properties of `G`:
|
|
|
|
|
|
- ``walk_type="random"`` if `G` is strongly connected and aperiodic
|
|
|
|
|
|
- ``walk_type="lazy"`` if `G` is strongly connected but not aperiodic
|
|
|
|
|
|
- ``walk_type="pagerank"`` for all other cases.
|
|
|
|
|
|
|
|
|
|
|
|
alpha : real
|
|
|
|
|
|
(1 - alpha) is the teleportation probability used with pagerank
|
|
|
|
|
|
|
|
|
|
|
|
Returns
|
|
|
|
|
|
-------
|
|
|
|
|
|
P : numpy.ndarray
|
|
|
|
|
|
transition matrix of G.
|
|
|
|
|
|
|
|
|
|
|
|
Raises
|
|
|
|
|
|
------
|
|
|
|
|
|
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"
|
|
|
|
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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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A = nx.to_scipy_sparse_array(G, nodelist=nodelist, weight=weight, dtype=float)
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n, m = A.shape
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if walk_type in ["random", "lazy"]:
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# TODO: Rm csr_array wrapper when spdiags array creation becomes available
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DI = sp.sparse.csr_array(sp.sparse.spdiags(1.0 / A.sum(axis=1), 0, n, n))
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if walk_type == "random":
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P = DI @ A
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else:
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# TODO: Rm csr_array wrapper when identity array creation becomes available
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I = sp.sparse.csr_array(sp.sparse.identity(n))
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P = (I + DI @ A) / 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. NOTE: This should be sparsified!
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A = A.toarray()
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# add constant to dangling nodes' row
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A[A.sum(axis=1) == 0, :] = 1 / n
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# normalize
|
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A = A / A.sum(axis=1)[np.newaxis, :].T
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P = alpha * A + (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
|
