129 lines
4 KiB
Python
129 lines
4 KiB
Python
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"""
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Laplacian of a compressed-sparse graph
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"""
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# Authors: Aric Hagberg <hagberg@lanl.gov>
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# Gael Varoquaux <gael.varoquaux@normalesup.org>
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# Jake Vanderplas <vanderplas@astro.washington.edu>
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# License: BSD
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from __future__ import division, print_function, absolute_import
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import numpy as np
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from scipy.sparse import isspmatrix
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###############################################################################
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# Graph laplacian
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def laplacian(csgraph, normed=False, return_diag=False, use_out_degree=False):
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"""
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Return the Laplacian matrix of a directed graph.
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Parameters
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----------
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csgraph : array_like or sparse matrix, 2 dimensions
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compressed-sparse graph, with shape (N, N).
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normed : bool, optional
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If True, then compute normalized Laplacian.
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return_diag : bool, optional
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If True, then also return an array related to vertex degrees.
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use_out_degree : bool, optional
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If True, then use out-degree instead of in-degree.
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This distinction matters only if the graph is asymmetric.
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Default: False.
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Returns
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-------
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lap : ndarray or sparse matrix
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The N x N laplacian matrix of csgraph. It will be a numpy array (dense)
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if the input was dense, or a sparse matrix otherwise.
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diag : ndarray, optional
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The length-N diagonal of the Laplacian matrix.
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For the normalized Laplacian, this is the array of square roots
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of vertex degrees or 1 if the degree is zero.
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Notes
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-----
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The Laplacian matrix of a graph is sometimes referred to as the
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"Kirchoff matrix" or the "admittance matrix", and is useful in many
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parts of spectral graph theory. In particular, the eigen-decomposition
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of the laplacian matrix can give insight into many properties of the graph.
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Examples
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--------
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>>> from scipy.sparse import csgraph
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>>> G = np.arange(5) * np.arange(5)[:, np.newaxis]
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>>> G
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array([[ 0, 0, 0, 0, 0],
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[ 0, 1, 2, 3, 4],
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[ 0, 2, 4, 6, 8],
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[ 0, 3, 6, 9, 12],
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[ 0, 4, 8, 12, 16]])
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>>> csgraph.laplacian(G, normed=False)
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array([[ 0, 0, 0, 0, 0],
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[ 0, 9, -2, -3, -4],
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[ 0, -2, 16, -6, -8],
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[ 0, -3, -6, 21, -12],
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[ 0, -4, -8, -12, 24]])
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"""
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if csgraph.ndim != 2 or csgraph.shape[0] != csgraph.shape[1]:
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raise ValueError('csgraph must be a square matrix or array')
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if normed and (np.issubdtype(csgraph.dtype, np.signedinteger)
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or np.issubdtype(csgraph.dtype, np.uint)):
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csgraph = csgraph.astype(float)
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create_lap = _laplacian_sparse if isspmatrix(csgraph) else _laplacian_dense
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degree_axis = 1 if use_out_degree else 0
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lap, d = create_lap(csgraph, normed=normed, axis=degree_axis)
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if return_diag:
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return lap, d
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return lap
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def _setdiag_dense(A, d):
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A.flat[::len(d)+1] = d
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def _laplacian_sparse(graph, normed=False, axis=0):
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if graph.format in ('lil', 'dok'):
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m = graph.tocoo()
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needs_copy = False
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else:
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m = graph
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needs_copy = True
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w = m.sum(axis=axis).getA1() - m.diagonal()
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if normed:
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m = m.tocoo(copy=needs_copy)
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isolated_node_mask = (w == 0)
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w = np.where(isolated_node_mask, 1, np.sqrt(w))
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m.data /= w[m.row]
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m.data /= w[m.col]
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m.data *= -1
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m.setdiag(1 - isolated_node_mask)
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else:
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if m.format == 'dia':
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m = m.copy()
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else:
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m = m.tocoo(copy=needs_copy)
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m.data *= -1
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m.setdiag(w)
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return m, w
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def _laplacian_dense(graph, normed=False, axis=0):
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m = np.array(graph)
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np.fill_diagonal(m, 0)
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w = m.sum(axis=axis)
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if normed:
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isolated_node_mask = (w == 0)
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w = np.where(isolated_node_mask, 1, np.sqrt(w))
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m /= w
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m /= w[:, np.newaxis]
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m *= -1
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_setdiag_dense(m, 1 - isolated_node_mask)
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else:
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m *= -1
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_setdiag_dense(m, w)
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return m, w
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