Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/linalg/bethehessianmatrix.py @ 0:4f3585e2f14b draft default tip
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author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |
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1 """Bethe Hessian or deformed Laplacian matrix of graphs.""" |
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2 import networkx as nx |
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3 from networkx.utils import not_implemented_for |
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4 |
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5 __all__ = ["bethe_hessian_matrix"] |
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6 |
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7 |
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8 @not_implemented_for("directed") |
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9 @not_implemented_for("multigraph") |
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10 def bethe_hessian_matrix(G, r=None, nodelist=None): |
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11 r"""Returns the Bethe Hessian matrix of G. |
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12 |
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13 The Bethe Hessian is a family of matrices parametrized by r, defined as |
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14 H(r) = (r^2 - 1) I - r A + D where A is the adjacency matrix, D is the |
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15 diagonal matrix of node degrees, and I is the identify matrix. It is equal |
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16 to the graph laplacian when the regularizer r = 1. |
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17 |
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18 The default choice of regularizer should be the ratio [2] |
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19 |
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20 .. math:: |
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21 r_m = \left(\sum k_i \right)^{-1}\left(\sum k_i^2 \right) - 1 |
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22 |
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23 Parameters |
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24 ---------- |
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25 G : Graph |
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26 A NetworkX graph |
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27 |
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28 r : float |
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29 Regularizer parameter |
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30 |
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31 nodelist : list, optional |
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32 The rows and columns are ordered according to the nodes in nodelist. |
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33 If nodelist is None, then the ordering is produced by G.nodes(). |
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34 |
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35 |
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36 Returns |
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37 ------- |
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38 H : Numpy matrix |
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39 The Bethe Hessian matrix of G, with paramter r. |
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40 |
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41 Examples |
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42 -------- |
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43 >>> k = [3, 2, 2, 1, 0] |
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44 >>> G = nx.havel_hakimi_graph(k) |
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45 >>> H = nx.modularity_matrix(G) |
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46 |
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47 |
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48 See Also |
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49 -------- |
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50 bethe_hessian_spectrum |
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51 to_numpy_array |
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52 adjacency_matrix |
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53 laplacian_matrix |
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54 |
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55 References |
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56 ---------- |
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57 .. [1] A. Saade, F. Krzakala and L. Zdeborová |
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58 "Spectral clustering of graphs with the bethe hessian", |
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59 Advances in Neural Information Processing Systems. 2014. |
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60 .. [2] C. M. Lee, E. Levina |
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61 "Estimating the number of communities in networks by spectral methods" |
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62 arXiv:1507.00827, 2015. |
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63 """ |
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64 import scipy.sparse |
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65 |
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66 if nodelist is None: |
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67 nodelist = list(G) |
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68 if r is None: |
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69 r = ( |
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70 sum([d ** 2 for v, d in nx.degree(G)]) / sum([d for v, d in nx.degree(G)]) |
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71 - 1 |
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72 ) |
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73 A = nx.to_scipy_sparse_matrix(G, nodelist=nodelist, format="csr") |
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74 n, m = A.shape |
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75 diags = A.sum(axis=1) |
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76 D = scipy.sparse.spdiags(diags.flatten(), [0], m, n, format="csr") |
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77 I = scipy.sparse.eye(m, n, format="csr") |
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78 return (r ** 2 - 1) * I - r * A + D |