563 lines
18 KiB
Python
563 lines
18 KiB
Python
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"""
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Laplacian of a compressed-sparse graph
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"""
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import numpy as np
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from scipy.sparse import issparse
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from scipy.sparse.linalg import LinearOperator
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from scipy.sparse._sputils import convert_pydata_sparse_to_scipy, is_pydata_spmatrix
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###############################################################################
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# Graph laplacian
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def laplacian(
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csgraph,
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normed=False,
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return_diag=False,
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use_out_degree=False,
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*,
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copy=True,
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form="array",
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dtype=None,
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symmetrized=False,
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):
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"""
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Return the Laplacian 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 symmetrically normalized Laplacian.
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Default: False.
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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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Default: False.
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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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copy: bool, optional
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If False, then change `csgraph` in place if possible,
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avoiding doubling the memory use.
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Default: True, for backward compatibility.
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form: 'array', or 'function', or 'lo'
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Determines the format of the output Laplacian:
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* 'array' is a numpy array;
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* 'function' is a pointer to evaluating the Laplacian-vector
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or Laplacian-matrix product;
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* 'lo' results in the format of the `LinearOperator`.
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Choosing 'function' or 'lo' always avoids doubling
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the memory use, ignoring `copy` value.
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Default: 'array', for backward compatibility.
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dtype: None or one of numeric numpy dtypes, optional
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The dtype of the output. If ``dtype=None``, the dtype of the
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output matches the dtype of the input csgraph, except for
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the case ``normed=True`` and integer-like csgraph, where
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the output dtype is 'float' allowing accurate normalization,
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but dramatically increasing the memory use.
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Default: None, for backward compatibility.
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symmetrized: bool, optional
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If True, then the output Laplacian is symmetric/Hermitian.
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The symmetrization is done by ``csgraph + csgraph.T.conj``
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without dividing by 2 to preserve integer dtypes if possible
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prior to the construction of the Laplacian.
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The symmetrization will increase the memory footprint of
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sparse matrices unless the sparsity pattern is symmetric or
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`form` is 'function' or 'lo'.
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Default: False, for backward compatibility.
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Returns
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-------
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lap : ndarray, or sparse matrix, or `LinearOperator`
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The N x N Laplacian of csgraph. It will be a NumPy array (dense)
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if the input was dense, or a sparse matrix otherwise, or
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the format of a function or `LinearOperator` if
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`form` equals 'function' or 'lo', respectively.
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diag : ndarray, optional
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The length-N main 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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"Kirchhoff matrix" or just the "Laplacian", and is useful in many
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parts of spectral graph theory.
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In particular, the eigen-decomposition of the Laplacian can give
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insight into many properties of the graph, e.g.,
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is commonly used for spectral data embedding and clustering.
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The constructed Laplacian doubles the memory use if ``copy=True`` and
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``form="array"`` which is the default.
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Choosing ``copy=False`` has no effect unless ``form="array"``
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or the matrix is sparse in the ``coo`` format, or dense array, except
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for the integer input with ``normed=True`` that forces the float output.
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Sparse input is reformatted into ``coo`` if ``form="array"``,
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which is the default.
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If the input adjacency matrix is not symmetric, the Laplacian is
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also non-symmetric unless ``symmetrized=True`` is used.
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Diagonal entries of the input adjacency matrix are ignored and
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replaced with zeros for the purpose of normalization where ``normed=True``.
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The normalization uses the inverse square roots of row-sums of the input
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adjacency matrix, and thus may fail if the row-sums contain
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negative or complex with a non-zero imaginary part values.
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The normalization is symmetric, making the normalized Laplacian also
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symmetric if the input csgraph was symmetric.
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References
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----------
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.. [1] Laplacian matrix. https://en.wikipedia.org/wiki/Laplacian_matrix
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Examples
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--------
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>>> import numpy as np
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>>> from scipy.sparse import csgraph
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Our first illustration is the symmetric graph
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>>> G = np.arange(4) * np.arange(4)[:, np.newaxis]
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>>> G
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array([[0, 0, 0, 0],
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[0, 1, 2, 3],
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[0, 2, 4, 6],
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[0, 3, 6, 9]])
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and its symmetric Laplacian matrix
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>>> csgraph.laplacian(G)
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array([[ 0, 0, 0, 0],
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[ 0, 5, -2, -3],
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[ 0, -2, 8, -6],
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[ 0, -3, -6, 9]])
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The non-symmetric graph
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>>> G = np.arange(9).reshape(3, 3)
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>>> G
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array([[0, 1, 2],
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[3, 4, 5],
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[6, 7, 8]])
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has different row- and column sums, resulting in two varieties
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of the Laplacian matrix, using an in-degree, which is the default
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>>> L_in_degree = csgraph.laplacian(G)
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>>> L_in_degree
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array([[ 9, -1, -2],
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[-3, 8, -5],
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[-6, -7, 7]])
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or alternatively an out-degree
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>>> L_out_degree = csgraph.laplacian(G, use_out_degree=True)
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>>> L_out_degree
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array([[ 3, -1, -2],
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[-3, 8, -5],
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[-6, -7, 13]])
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Constructing a symmetric Laplacian matrix, one can add the two as
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>>> L_in_degree + L_out_degree.T
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array([[ 12, -4, -8],
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[ -4, 16, -12],
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[ -8, -12, 20]])
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or use the ``symmetrized=True`` option
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>>> csgraph.laplacian(G, symmetrized=True)
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array([[ 12, -4, -8],
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[ -4, 16, -12],
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[ -8, -12, 20]])
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that is equivalent to symmetrizing the original graph
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>>> csgraph.laplacian(G + G.T)
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array([[ 12, -4, -8],
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[ -4, 16, -12],
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[ -8, -12, 20]])
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The goal of normalization is to make the non-zero diagonal entries
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of the Laplacian matrix to be all unit, also scaling off-diagonal
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entries correspondingly. The normalization can be done manually, e.g.,
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>>> G = np.array([[0, 1, 1], [1, 0, 1], [1, 1, 0]])
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>>> L, d = csgraph.laplacian(G, return_diag=True)
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>>> L
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array([[ 2, -1, -1],
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[-1, 2, -1],
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[-1, -1, 2]])
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>>> d
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array([2, 2, 2])
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>>> scaling = np.sqrt(d)
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>>> scaling
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array([1.41421356, 1.41421356, 1.41421356])
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>>> (1/scaling)*L*(1/scaling)
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array([[ 1. , -0.5, -0.5],
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[-0.5, 1. , -0.5],
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[-0.5, -0.5, 1. ]])
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Or using ``normed=True`` option
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>>> L, d = csgraph.laplacian(G, return_diag=True, normed=True)
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>>> L
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array([[ 1. , -0.5, -0.5],
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[-0.5, 1. , -0.5],
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[-0.5, -0.5, 1. ]])
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which now instead of the diagonal returns the scaling coefficients
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>>> d
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array([1.41421356, 1.41421356, 1.41421356])
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Zero scaling coefficients are substituted with 1s, where scaling
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has thus no effect, e.g.,
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>>> G = np.array([[0, 0, 0], [0, 0, 1], [0, 1, 0]])
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>>> G
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array([[0, 0, 0],
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[0, 0, 1],
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[0, 1, 0]])
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>>> L, d = csgraph.laplacian(G, return_diag=True, normed=True)
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>>> L
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array([[ 0., -0., -0.],
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[-0., 1., -1.],
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[-0., -1., 1.]])
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>>> d
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array([1., 1., 1.])
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Only the symmetric normalization is implemented, resulting
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in a symmetric Laplacian matrix if and only if its graph is symmetric
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and has all non-negative degrees, like in the examples above.
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The output Laplacian matrix is by default a dense array or a sparse matrix
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inferring its shape, format, and dtype from the input graph matrix:
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>>> G = np.array([[0, 1, 1], [1, 0, 1], [1, 1, 0]]).astype(np.float32)
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>>> G
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array([[0., 1., 1.],
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[1., 0., 1.],
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[1., 1., 0.]], dtype=float32)
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>>> csgraph.laplacian(G)
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array([[ 2., -1., -1.],
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[-1., 2., -1.],
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[-1., -1., 2.]], dtype=float32)
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but can alternatively be generated matrix-free as a LinearOperator:
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>>> L = csgraph.laplacian(G, form="lo")
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>>> L
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<3x3 _CustomLinearOperator with dtype=float32>
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>>> L(np.eye(3))
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array([[ 2., -1., -1.],
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[-1., 2., -1.],
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[-1., -1., 2.]])
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or as a lambda-function:
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>>> L = csgraph.laplacian(G, form="function")
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>>> L
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<function _laplace.<locals>.<lambda> at 0x0000012AE6F5A598>
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>>> L(np.eye(3))
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array([[ 2., -1., -1.],
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[-1., 2., -1.],
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[-1., -1., 2.]])
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The Laplacian matrix is used for
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spectral data clustering and embedding
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as well as for spectral graph partitioning.
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Our final example illustrates the latter
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for a noisy directed linear graph.
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>>> from scipy.sparse import diags, random
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>>> from scipy.sparse.linalg import lobpcg
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Create a directed linear graph with ``N=35`` vertices
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using a sparse adjacency matrix ``G``:
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>>> N = 35
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>>> G = diags(np.ones(N-1), 1, format="csr")
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Fix a random seed ``rng`` and add a random sparse noise to the graph ``G``:
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>>> rng = np.random.default_rng()
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>>> G += 1e-2 * random(N, N, density=0.1, random_state=rng)
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Set initial approximations for eigenvectors:
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>>> X = rng.random((N, 2))
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The constant vector of ones is always a trivial eigenvector
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of the non-normalized Laplacian to be filtered out:
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>>> Y = np.ones((N, 1))
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Alternating (1) the sign of the graph weights allows determining
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labels for spectral max- and min- cuts in a single loop.
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Since the graph is undirected, the option ``symmetrized=True``
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must be used in the construction of the Laplacian.
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The option ``normed=True`` cannot be used in (2) for the negative weights
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here as the symmetric normalization evaluates square roots.
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The option ``form="lo"`` in (2) is matrix-free, i.e., guarantees
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a fixed memory footprint and read-only access to the graph.
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Calling the eigenvalue solver ``lobpcg`` (3) computes the Fiedler vector
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that determines the labels as the signs of its components in (5).
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Since the sign in an eigenvector is not deterministic and can flip,
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we fix the sign of the first component to be always +1 in (4).
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>>> for cut in ["max", "min"]:
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... G = -G # 1.
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... L = csgraph.laplacian(G, symmetrized=True, form="lo") # 2.
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... _, eves = lobpcg(L, X, Y=Y, largest=False, tol=1e-3) # 3.
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... eves *= np.sign(eves[0, 0]) # 4.
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... print(cut + "-cut labels:\\n", 1 * (eves[:, 0]>0)) # 5.
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max-cut labels:
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[1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1]
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min-cut labels:
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[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
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As anticipated for a (slightly noisy) linear graph,
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the max-cut strips all the edges of the graph coloring all
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odd vertices into one color and all even vertices into another one,
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while the balanced min-cut partitions the graph
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in the middle by deleting a single edge.
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Both determined partitions are optimal.
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"""
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is_pydata_sparse = is_pydata_spmatrix(csgraph)
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if is_pydata_sparse:
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pydata_sparse_cls = csgraph.__class__
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csgraph = convert_pydata_sparse_to_scipy(csgraph)
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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 (
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np.issubdtype(csgraph.dtype, np.signedinteger)
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or np.issubdtype(csgraph.dtype, np.uint)
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):
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csgraph = csgraph.astype(np.float64)
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if form == "array":
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create_lap = (
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_laplacian_sparse if issparse(csgraph) else _laplacian_dense
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)
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else:
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create_lap = (
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_laplacian_sparse_flo
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if issparse(csgraph)
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else _laplacian_dense_flo
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)
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degree_axis = 1 if use_out_degree else 0
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lap, d = create_lap(
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|
csgraph,
|
||
|
|
normed=normed,
|
||
|
|
axis=degree_axis,
|
||
|
|
copy=copy,
|
||
|
|
form=form,
|
||
|
|
dtype=dtype,
|
||
|
|
symmetrized=symmetrized,
|
||
|
|
)
|
||
|
|
if is_pydata_sparse:
|
||
|
|
lap = pydata_sparse_cls.from_scipy_sparse(lap)
|
||
|
|
if return_diag:
|
||
|
|
return lap, d
|
||
|
|
return lap
|
||
|
|
|
||
|
|
|
||
|
|
def _setdiag_dense(m, d):
|
||
|
|
step = len(d) + 1
|
||
|
|
m.flat[::step] = d
|
||
|
|
|
||
|
|
|
||
|
|
def _laplace(m, d):
|
||
|
|
return lambda v: v * d[:, np.newaxis] - m @ v
|
||
|
|
|
||
|
|
|
||
|
|
def _laplace_normed(m, d, nd):
|
||
|
|
laplace = _laplace(m, d)
|
||
|
|
return lambda v: nd[:, np.newaxis] * laplace(v * nd[:, np.newaxis])
|
||
|
|
|
||
|
|
|
||
|
|
def _laplace_sym(m, d):
|
||
|
|
return (
|
||
|
|
lambda v: v * d[:, np.newaxis]
|
||
|
|
- m @ v
|
||
|
|
- np.transpose(np.conjugate(np.transpose(np.conjugate(v)) @ m))
|
||
|
|
)
|
||
|
|
|
||
|
|
|
||
|
|
def _laplace_normed_sym(m, d, nd):
|
||
|
|
laplace_sym = _laplace_sym(m, d)
|
||
|
|
return lambda v: nd[:, np.newaxis] * laplace_sym(v * nd[:, np.newaxis])
|
||
|
|
|
||
|
|
|
||
|
|
def _linearoperator(mv, shape, dtype):
|
||
|
|
return LinearOperator(matvec=mv, matmat=mv, shape=shape, dtype=dtype)
|
||
|
|
|
||
|
|
|
||
|
|
def _laplacian_sparse_flo(graph, normed, axis, copy, form, dtype, symmetrized):
|
||
|
|
# The keyword argument `copy` is unused and has no effect here.
|
||
|
|
del copy
|
||
|
|
|
||
|
|
if dtype is None:
|
||
|
|
dtype = graph.dtype
|
||
|
|
|
||
|
|
graph_sum = np.asarray(graph.sum(axis=axis)).ravel()
|
||
|
|
graph_diagonal = graph.diagonal()
|
||
|
|
diag = graph_sum - graph_diagonal
|
||
|
|
if symmetrized:
|
||
|
|
graph_sum += np.asarray(graph.sum(axis=1 - axis)).ravel()
|
||
|
|
diag = graph_sum - graph_diagonal - graph_diagonal
|
||
|
|
|
||
|
|
if normed:
|
||
|
|
isolated_node_mask = diag == 0
|
||
|
|
w = np.where(isolated_node_mask, 1, np.sqrt(diag))
|
||
|
|
if symmetrized:
|
||
|
|
md = _laplace_normed_sym(graph, graph_sum, 1.0 / w)
|
||
|
|
else:
|
||
|
|
md = _laplace_normed(graph, graph_sum, 1.0 / w)
|
||
|
|
if form == "function":
|
||
|
|
return md, w.astype(dtype, copy=False)
|
||
|
|
elif form == "lo":
|
||
|
|
m = _linearoperator(md, shape=graph.shape, dtype=dtype)
|
||
|
|
return m, w.astype(dtype, copy=False)
|
||
|
|
else:
|
||
|
|
raise ValueError(f"Invalid form: {form!r}")
|
||
|
|
else:
|
||
|
|
if symmetrized:
|
||
|
|
md = _laplace_sym(graph, graph_sum)
|
||
|
|
else:
|
||
|
|
md = _laplace(graph, graph_sum)
|
||
|
|
if form == "function":
|
||
|
|
return md, diag.astype(dtype, copy=False)
|
||
|
|
elif form == "lo":
|
||
|
|
m = _linearoperator(md, shape=graph.shape, dtype=dtype)
|
||
|
|
return m, diag.astype(dtype, copy=False)
|
||
|
|
else:
|
||
|
|
raise ValueError(f"Invalid form: {form!r}")
|
||
|
|
|
||
|
|
|
||
|
|
def _laplacian_sparse(graph, normed, axis, copy, form, dtype, symmetrized):
|
||
|
|
# The keyword argument `form` is unused and has no effect here.
|
||
|
|
del form
|
||
|
|
|
||
|
|
if dtype is None:
|
||
|
|
dtype = graph.dtype
|
||
|
|
|
||
|
|
needs_copy = False
|
||
|
|
if graph.format in ('lil', 'dok'):
|
||
|
|
m = graph.tocoo()
|
||
|
|
else:
|
||
|
|
m = graph
|
||
|
|
if copy:
|
||
|
|
needs_copy = True
|
||
|
|
|
||
|
|
if symmetrized:
|
||
|
|
m += m.T.conj()
|
||
|
|
|
||
|
|
w = np.asarray(m.sum(axis=axis)).ravel() - m.diagonal()
|
||
|
|
if normed:
|
||
|
|
m = m.tocoo(copy=needs_copy)
|
||
|
|
isolated_node_mask = (w == 0)
|
||
|
|
w = np.where(isolated_node_mask, 1, np.sqrt(w))
|
||
|
|
m.data /= w[m.row]
|
||
|
|
m.data /= w[m.col]
|
||
|
|
m.data *= -1
|
||
|
|
m.setdiag(1 - isolated_node_mask)
|
||
|
|
else:
|
||
|
|
if m.format == 'dia':
|
||
|
|
m = m.copy()
|
||
|
|
else:
|
||
|
|
m = m.tocoo(copy=needs_copy)
|
||
|
|
m.data *= -1
|
||
|
|
m.setdiag(w)
|
||
|
|
|
||
|
|
return m.astype(dtype, copy=False), w.astype(dtype)
|
||
|
|
|
||
|
|
|
||
|
|
def _laplacian_dense_flo(graph, normed, axis, copy, form, dtype, symmetrized):
|
||
|
|
|
||
|
|
if copy:
|
||
|
|
m = np.array(graph)
|
||
|
|
else:
|
||
|
|
m = np.asarray(graph)
|
||
|
|
|
||
|
|
if dtype is None:
|
||
|
|
dtype = m.dtype
|
||
|
|
|
||
|
|
graph_sum = m.sum(axis=axis)
|
||
|
|
graph_diagonal = m.diagonal()
|
||
|
|
diag = graph_sum - graph_diagonal
|
||
|
|
if symmetrized:
|
||
|
|
graph_sum += m.sum(axis=1 - axis)
|
||
|
|
diag = graph_sum - graph_diagonal - graph_diagonal
|
||
|
|
|
||
|
|
if normed:
|
||
|
|
isolated_node_mask = diag == 0
|
||
|
|
w = np.where(isolated_node_mask, 1, np.sqrt(diag))
|
||
|
|
if symmetrized:
|
||
|
|
md = _laplace_normed_sym(m, graph_sum, 1.0 / w)
|
||
|
|
else:
|
||
|
|
md = _laplace_normed(m, graph_sum, 1.0 / w)
|
||
|
|
if form == "function":
|
||
|
|
return md, w.astype(dtype, copy=False)
|
||
|
|
elif form == "lo":
|
||
|
|
m = _linearoperator(md, shape=graph.shape, dtype=dtype)
|
||
|
|
return m, w.astype(dtype, copy=False)
|
||
|
|
else:
|
||
|
|
raise ValueError(f"Invalid form: {form!r}")
|
||
|
|
else:
|
||
|
|
if symmetrized:
|
||
|
|
md = _laplace_sym(m, graph_sum)
|
||
|
|
else:
|
||
|
|
md = _laplace(m, graph_sum)
|
||
|
|
if form == "function":
|
||
|
|
return md, diag.astype(dtype, copy=False)
|
||
|
|
elif form == "lo":
|
||
|
|
m = _linearoperator(md, shape=graph.shape, dtype=dtype)
|
||
|
|
return m, diag.astype(dtype, copy=False)
|
||
|
|
else:
|
||
|
|
raise ValueError(f"Invalid form: {form!r}")
|
||
|
|
|
||
|
|
|
||
|
|
def _laplacian_dense(graph, normed, axis, copy, form, dtype, symmetrized):
|
||
|
|
|
||
|
|
if form != "array":
|
||
|
|
raise ValueError(f'{form!r} must be "array"')
|
||
|
|
|
||
|
|
if dtype is None:
|
||
|
|
dtype = graph.dtype
|
||
|
|
|
||
|
|
if copy:
|
||
|
|
m = np.array(graph)
|
||
|
|
else:
|
||
|
|
m = np.asarray(graph)
|
||
|
|
|
||
|
|
if dtype is None:
|
||
|
|
dtype = m.dtype
|
||
|
|
|
||
|
|
if symmetrized:
|
||
|
|
m += m.T.conj()
|
||
|
|
np.fill_diagonal(m, 0)
|
||
|
|
w = m.sum(axis=axis)
|
||
|
|
if normed:
|
||
|
|
isolated_node_mask = (w == 0)
|
||
|
|
w = np.where(isolated_node_mask, 1, np.sqrt(w))
|
||
|
|
m /= w
|
||
|
|
m /= w[:, np.newaxis]
|
||
|
|
m *= -1
|
||
|
|
_setdiag_dense(m, 1 - isolated_node_mask)
|
||
|
|
else:
|
||
|
|
m *= -1
|
||
|
|
_setdiag_dense(m, w)
|
||
|
|
|
||
|
|
return m.astype(dtype, copy=False), w.astype(dtype, copy=False)
|