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annotate env/lib/python3.7/site-packages/networkx/linalg/bethehessianmatrix.py @ 0:26e78fe6e8c4 draft

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"planemo upload commit c699937486c35866861690329de38ec1a5d9f783"
author shellac
date Sat, 02 May 2020 07:14:21 -0400
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1 # -*- coding: utf-8 -*-
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2 # Copyright (C) 2004-2019 by
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3 # Aric Hagberg <hagberg@lanl.gov>
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4 # Dan Schult <dschult@colgate.edu>
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5 # Pieter Swart <swart@lanl.gov>
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6 # Jean-Gabriel Young <jeangabriel.young@gmail.com>
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7 # All rights reserved.
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8 # BSD license.
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9 #
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10 # Authors: Jean-Gabriel Young (jeangabriel.young@gmail.com)
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11 """Bethe Hessian or deformed Laplacian matrix of graphs."""
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12 import networkx as nx
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13 from networkx.utils import not_implemented_for
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14
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15 __all__ = ['bethe_hessian_matrix']
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16
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17
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18 @not_implemented_for('directed')
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19 @not_implemented_for('multigraph')
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20 def bethe_hessian_matrix(G, r=None, nodelist=None):
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21 r"""Returns the Bethe Hessian matrix of G.
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22
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23 The Bethe Hessian is a family of matrices parametrized by r, defined as
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24 H(r) = (r^2 - 1) I - r A + D where A is the adjacency matrix, D is the
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25 diagonal matrix of node degrees, and I is the identify matrix. It is equal
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26 to the graph laplacian when the regularizer r = 1.
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27
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28 The default choice of regularizer should be the ratio [2]
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29
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30 .. math::
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31 r_m = \left(\sum k_i \right)^{-1}\left(\sum k_i^2 \right) - 1
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32
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33 Parameters
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34 ----------
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35 G : Graph
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36 A NetworkX graph
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37
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38 r : float
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39 Regularizer parameter
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40
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41 nodelist : list, optional
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42 The rows and columns are ordered according to the nodes in nodelist.
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43 If nodelist is None, then the ordering is produced by G.nodes().
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44
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45
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46 Returns
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47 -------
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48 H : Numpy matrix
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49 The Bethe Hessian matrix of G, with paramter r.
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50
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51 Examples
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52 --------
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53 >>> import networkx as nx
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54 >>> k =[3, 2, 2, 1, 0]
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55 >>> G = nx.havel_hakimi_graph(k)
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56 >>> H = nx.modularity_matrix(G)
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57
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58
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59 See Also
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60 --------
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61 bethe_hessian_spectrum
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62 to_numpy_matrix
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63 adjacency_matrix
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64 laplacian_matrix
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65
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66 References
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67 ----------
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68 .. [1] A. Saade, F. Krzakala and L. Zdeborová
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69 "Spectral clustering of graphs with the bethe hessian",
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70 Advances in Neural Information Processing Systems. 2014.
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71 .. [2] C. M. Lee, E. Levina
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72 "Estimating the number of communities in networks by spectral methods"
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73 arXiv:1507.00827, 2015.
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74 """
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75 import scipy.sparse
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76 if nodelist is None:
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77 nodelist = list(G)
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78 if r is None:
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79 r = sum([d ** 2 for v, d in nx.degree(G)]) /\
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80 sum([d for v, d in nx.degree(G)]) - 1
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81 A = nx.to_scipy_sparse_matrix(G, nodelist=nodelist, format='csr')
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82 n, m = A.shape
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83 diags = A.sum(axis=1)
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84 D = scipy.sparse.spdiags(diags.flatten(), [0], m, n, format='csr')
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85 I = scipy.sparse.eye(m, n, format='csr')
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86 return (r ** 2 - 1) * I - r * A + D
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87
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88
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89 # fixture for pytest
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90 def setup_module(module):
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91 import pytest
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92 numpy = pytest.importorskip('numpy')