79 lines
2.6 KiB
Python
79 lines
2.6 KiB
Python
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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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@nx._dispatchable
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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 : scipy.sparse.csr_array
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The Bethe Hessian matrix of `G`, with parameter `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.bethe_hessian_matrix(G)
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>>> H.toarray()
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array([[ 3.5625, -1.25 , -1.25 , -1.25 , 0. ],
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[-1.25 , 2.5625, -1.25 , 0. , 0. ],
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[-1.25 , -1.25 , 2.5625, 0. , 0. ],
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[-1.25 , 0. , 0. , 1.5625, 0. ],
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[ 0. , 0. , 0. , 0. , 0.5625]])
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See Also
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--------
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bethe_hessian_spectrum
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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. Le, 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 as sp
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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 = sum(d**2 for v, d in nx.degree(G)) / sum(d for v, d in nx.degree(G)) - 1
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A = nx.to_scipy_sparse_array(G, nodelist=nodelist, format="csr")
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n, m = A.shape
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# TODO: Rm csr_array wrapper when spdiags array creation becomes available
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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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# TODO: Rm csr_array wrapper when eye array creation becomes available
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I = sp.sparse.csr_array(sp.sparse.eye(m, n, format="csr"))
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return (r**2 - 1) * I - r * A + D
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