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.CondaPkg/env/lib/python3.11/site-packages/networkx/linalg/bethehessianmatrix.py

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+"""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")
+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
+    import scipy.sparse  # call as sp.sparse
+
+    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
+    # TODO: Rm csr_array wrapper when spdiags array creation becomes available
+    D = sp.sparse.csr_array(sp.sparse.spdiags(A.sum(axis=1), 0, m, n, format="csr"))
+    # TODO: Rm csr_array wrapper when eye array creation becomes available
+    I = sp.sparse.csr_array(sp.sparse.eye(m, n, format="csr"))
+    return (r**2 - 1) * I - r * A + D