| """Bethe Hessian or deformed Laplacian matrix of graphs.""" |
|
|
| import networkx as nx |
| from networkx.utils import not_implemented_for |
|
|
| __all__ = ["bethe_hessian_matrix"] |
|
|
|
|
| @not_implemented_for("directed") |
| @not_implemented_for("multigraph") |
| @nx._dispatchable |
| def bethe_hessian_matrix(G, r=None, nodelist=None): |
| r"""Returns the Bethe Hessian matrix of G. |
| |
| The Bethe Hessian is a family of matrices parametrized by r, defined as |
| H(r) = (r^2 - 1) I - r A + D where A is the adjacency matrix, D is the |
| diagonal matrix of node degrees, and I is the identify matrix. It is equal |
| to the graph laplacian when the regularizer r = 1. |
| |
| The default choice of regularizer should be the ratio [2]_ |
| |
| .. math:: |
| r_m = \left(\sum k_i \right)^{-1}\left(\sum k_i^2 \right) - 1 |
| |
| Parameters |
| ---------- |
| G : Graph |
| A NetworkX graph |
| r : float |
| Regularizer parameter |
| nodelist : list, optional |
| The rows and columns are ordered according to the nodes in nodelist. |
| If nodelist is None, then the ordering is produced by ``G.nodes()``. |
| |
| Returns |
| ------- |
| H : scipy.sparse.csr_array |
| The Bethe Hessian matrix of `G`, with parameter `r`. |
| |
| Examples |
| -------- |
| >>> k = [3, 2, 2, 1, 0] |
| >>> G = nx.havel_hakimi_graph(k) |
| >>> H = nx.bethe_hessian_matrix(G) |
| >>> H.toarray() |
| array([[ 3.5625, -1.25 , -1.25 , -1.25 , 0. ], |
| [-1.25 , 2.5625, -1.25 , 0. , 0. ], |
| [-1.25 , -1.25 , 2.5625, 0. , 0. ], |
| [-1.25 , 0. , 0. , 1.5625, 0. ], |
| [ 0. , 0. , 0. , 0. , 0.5625]]) |
| |
| See Also |
| -------- |
| bethe_hessian_spectrum |
| adjacency_matrix |
| laplacian_matrix |
| |
| References |
| ---------- |
| .. [1] A. Saade, F. Krzakala and L. Zdeborová |
| "Spectral Clustering of Graphs with the Bethe Hessian", |
| Advances in Neural Information Processing Systems, 2014. |
| .. [2] C. M. Le, E. Levina |
| "Estimating the number of communities in networks by spectral methods" |
| arXiv:1507.00827, 2015. |
| """ |
| import scipy as sp |
|
|
| if nodelist is None: |
| nodelist = list(G) |
| if r is None: |
| r = sum(d**2 for v, d in nx.degree(G)) / sum(d for v, d in nx.degree(G)) - 1 |
| A = nx.to_scipy_sparse_array(G, nodelist=nodelist, format="csr") |
| n, m = A.shape |
| |
| D = sp.sparse.csr_array(sp.sparse.spdiags(A.sum(axis=1), 0, m, n, format="csr")) |
| |
| I = sp.sparse.csr_array(sp.sparse.eye(m, n, format="csr")) |
| return (r**2 - 1) * I - r * A + D |
|
|
