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79 lines
2.2 KiB
79 lines
2.2 KiB
4 years ago
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"""Bethe Hessian or deformed Laplacian matrix of graphs."""
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import networkx as nx
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from networkx.utils import not_implemented_for
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__all__ = ["bethe_hessian_matrix"]
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@not_implemented_for("directed")
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@not_implemented_for("multigraph")
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def bethe_hessian_matrix(G, r=None, nodelist=None):
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r"""Returns the Bethe Hessian matrix of G.
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The Bethe Hessian is a family of matrices parametrized by r, defined as
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H(r) = (r^2 - 1) I - r A + D where A is the adjacency matrix, D is the
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diagonal matrix of node degrees, and I is the identify matrix. It is equal
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to the graph laplacian when the regularizer r = 1.
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The default choice of regularizer should be the ratio [2]
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.. math::
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r_m = \left(\sum k_i \right)^{-1}\left(\sum k_i^2 \right) - 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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r : float
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Regularizer parameter
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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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Returns
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-------
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H : Numpy matrix
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The Bethe Hessian matrix of G, with paramter r.
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Examples
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--------
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>>> k = [3, 2, 2, 1, 0]
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>>> G = nx.havel_hakimi_graph(k)
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>>> H = nx.modularity_matrix(G)
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See Also
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--------
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bethe_hessian_spectrum
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to_numpy_array
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adjacency_matrix
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laplacian_matrix
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References
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----------
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.. [1] A. Saade, F. Krzakala and L. Zdeborová
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"Spectral clustering of graphs with the bethe hessian",
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Advances in Neural Information Processing Systems. 2014.
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.. [2] C. M. Lee, E. Levina
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"Estimating the number of communities in networks by spectral methods"
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arXiv:1507.00827, 2015.
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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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if r is None:
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r = (
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sum([d ** 2 for v, d in nx.degree(G)]) / sum([d for v, d in nx.degree(G)])
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- 1
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)
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A = nx.to_scipy_sparse_matrix(G, nodelist=nodelist, 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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I = scipy.sparse.eye(m, n, format="csr")
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return (r ** 2 - 1) * I - r * A + D
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