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wemm/lib/python3.10/site-packages/botocore/data/schemas/2019-12-02/endpoint-rule-set-1.json.gz

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+version https://git-lfs.github.com/spec/v1
+oid sha256:af42a70c3fed77a8cdfc96011b2830a6bacf903b8315b72adeda63d5a953ea72
+size 1146

wemm/lib/python3.10/site-packages/networkx/algorithms/traversal/tests/test_dfs.py

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+import networkx as nx
+
+
+class TestDFS:
+    @classmethod
+    def setup_class(cls):
+        # simple graph
+        G = nx.Graph()
+        G.add_edges_from([(0, 1), (1, 2), (1, 3), (2, 4), (3, 0), (0, 4)])
+        cls.G = G
+        # simple graph, disconnected
+        D = nx.Graph()
+        D.add_edges_from([(0, 1), (2, 3)])
+        cls.D = D
+
+    def test_preorder_nodes(self):
+        assert list(nx.dfs_preorder_nodes(self.G, source=0)) == [0, 1, 2, 4, 3]
+        assert list(nx.dfs_preorder_nodes(self.D)) == [0, 1, 2, 3]
+        assert list(nx.dfs_preorder_nodes(self.D, source=2)) == [2, 3]
+
+    def test_postorder_nodes(self):
+        assert list(nx.dfs_postorder_nodes(self.G, source=0)) == [4, 2, 3, 1, 0]
+        assert list(nx.dfs_postorder_nodes(self.D)) == [1, 0, 3, 2]
+        assert list(nx.dfs_postorder_nodes(self.D, source=0)) == [1, 0]
+
+    def test_successor(self):
+        assert nx.dfs_successors(self.G, source=0) == {0: [1], 1: [2, 3], 2: [4]}
+        assert nx.dfs_successors(self.G, source=1) == {0: [3, 4], 1: [0], 4: [2]}
+        assert nx.dfs_successors(self.D) == {0: [1], 2: [3]}
+        assert nx.dfs_successors(self.D, source=1) == {1: [0]}
+
+    def test_predecessor(self):
+        assert nx.dfs_predecessors(self.G, source=0) == {1: 0, 2: 1, 3: 1, 4: 2}
+        assert nx.dfs_predecessors(self.D) == {1: 0, 3: 2}
+
+    def test_dfs_tree(self):
+        exp_nodes = sorted(self.G.nodes())
+        exp_edges = [(0, 1), (1, 2), (1, 3), (2, 4)]
+        # Search from first node
+        T = nx.dfs_tree(self.G, source=0)
+        assert sorted(T.nodes()) == exp_nodes
+        assert sorted(T.edges()) == exp_edges
+        # Check source=None
+        T = nx.dfs_tree(self.G, source=None)
+        assert sorted(T.nodes()) == exp_nodes
+        assert sorted(T.edges()) == exp_edges
+        # Check source=None is the default
+        T = nx.dfs_tree(self.G)
+        assert sorted(T.nodes()) == exp_nodes
+        assert sorted(T.edges()) == exp_edges
+
+    def test_dfs_edges(self):
+        edges = nx.dfs_edges(self.G, source=0)
+        assert list(edges) == [(0, 1), (1, 2), (2, 4), (1, 3)]
+        edges = nx.dfs_edges(self.D)
+        assert list(edges) == [(0, 1), (2, 3)]
+
+    def test_dfs_edges_sorting(self):
+        G = nx.Graph([(0, 1), (1, 2), (1, 3), (2, 4), (3, 0), (0, 4)])
+        edges_asc = nx.dfs_edges(G, source=0, sort_neighbors=sorted)
+        sorted_desc = lambda x: sorted(x, reverse=True)
+        edges_desc = nx.dfs_edges(G, source=0, sort_neighbors=sorted_desc)
+        assert list(edges_asc) == [(0, 1), (1, 2), (2, 4), (1, 3)]
+        assert list(edges_desc) == [(0, 4), (4, 2), (2, 1), (1, 3)]
+
+    def test_dfs_labeled_edges(self):
+        edges = list(nx.dfs_labeled_edges(self.G, source=0))
+        forward = [(u, v) for (u, v, d) in edges if d == "forward"]
+        assert forward == [(0, 0), (0, 1), (1, 2), (2, 4), (1, 3)]
+        assert edges == [
+            (0, 0, "forward"),
+            (0, 1, "forward"),
+            (1, 0, "nontree"),
+            (1, 2, "forward"),
+            (2, 1, "nontree"),
+            (2, 4, "forward"),
+            (4, 2, "nontree"),
+            (4, 0, "nontree"),
+            (2, 4, "reverse"),
+            (1, 2, "reverse"),
+            (1, 3, "forward"),
+            (3, 1, "nontree"),
+            (3, 0, "nontree"),
+            (1, 3, "reverse"),
+            (0, 1, "reverse"),
+            (0, 3, "nontree"),
+            (0, 4, "nontree"),
+            (0, 0, "reverse"),
+        ]
+
+    def test_dfs_labeled_edges_sorting(self):
+        G = nx.Graph([(0, 1), (1, 2), (1, 3), (2, 4), (3, 0), (0, 4)])
+        edges_asc = nx.dfs_labeled_edges(G, source=0, sort_neighbors=sorted)
+        sorted_desc = lambda x: sorted(x, reverse=True)
+        edges_desc = nx.dfs_labeled_edges(G, source=0, sort_neighbors=sorted_desc)
+        assert list(edges_asc) == [
+            (0, 0, "forward"),
+            (0, 1, "forward"),
+            (1, 0, "nontree"),
+            (1, 2, "forward"),
+            (2, 1, "nontree"),
+            (2, 4, "forward"),
+            (4, 0, "nontree"),
+            (4, 2, "nontree"),
+            (2, 4, "reverse"),
+            (1, 2, "reverse"),
+            (1, 3, "forward"),
+            (3, 0, "nontree"),
+            (3, 1, "nontree"),
+            (1, 3, "reverse"),
+            (0, 1, "reverse"),
+            (0, 3, "nontree"),
+            (0, 4, "nontree"),
+            (0, 0, "reverse"),
+        ]
+        assert list(edges_desc) == [
+            (0, 0, "forward"),
+            (0, 4, "forward"),
+            (4, 2, "forward"),
+            (2, 4, "nontree"),
+            (2, 1, "forward"),
+            (1, 3, "forward"),
+            (3, 1, "nontree"),
+            (3, 0, "nontree"),
+            (1, 3, "reverse"),
+            (1, 2, "nontree"),
+            (1, 0, "nontree"),
+            (2, 1, "reverse"),
+            (4, 2, "reverse"),
+            (4, 0, "nontree"),
+            (0, 4, "reverse"),
+            (0, 3, "nontree"),
+            (0, 1, "nontree"),
+            (0, 0, "reverse"),
+        ]
+
+    def test_dfs_labeled_disconnected_edges(self):
+        edges = list(nx.dfs_labeled_edges(self.D))
+        forward = [(u, v) for (u, v, d) in edges if d == "forward"]
+        assert forward == [(0, 0), (0, 1), (2, 2), (2, 3)]
+        assert edges == [
+            (0, 0, "forward"),
+            (0, 1, "forward"),
+            (1, 0, "nontree"),
+            (0, 1, "reverse"),
+            (0, 0, "reverse"),
+            (2, 2, "forward"),
+            (2, 3, "forward"),
+            (3, 2, "nontree"),
+            (2, 3, "reverse"),
+            (2, 2, "reverse"),
+        ]
+
+    def test_dfs_tree_isolates(self):
+        G = nx.Graph()
+        G.add_node(1)
+        G.add_node(2)
+        T = nx.dfs_tree(G, source=1)
+        assert sorted(T.nodes()) == [1]
+        assert sorted(T.edges()) == []
+        T = nx.dfs_tree(G, source=None)
+        assert sorted(T.nodes()) == [1, 2]
+        assert sorted(T.edges()) == []
+
+
+class TestDepthLimitedSearch:
+    @classmethod
+    def setup_class(cls):
+        # a tree
+        G = nx.Graph()
+        nx.add_path(G, [0, 1, 2, 3, 4, 5, 6])
+        nx.add_path(G, [2, 7, 8, 9, 10])
+        cls.G = G
+        # a disconnected graph
+        D = nx.Graph()
+        D.add_edges_from([(0, 1), (2, 3)])
+        nx.add_path(D, [2, 7, 8, 9, 10])
+        cls.D = D
+
+    def test_dls_preorder_nodes(self):
+        assert list(nx.dfs_preorder_nodes(self.G, source=0, depth_limit=2)) == [0, 1, 2]
+        assert list(nx.dfs_preorder_nodes(self.D, source=1, depth_limit=2)) == ([1, 0])
+
+    def test_dls_postorder_nodes(self):
+        assert list(nx.dfs_postorder_nodes(self.G, source=3, depth_limit=3)) == [
+            1,
+            7,
+            2,
+            5,
+            4,
+            3,
+        ]
+        assert list(nx.dfs_postorder_nodes(self.D, source=2, depth_limit=2)) == (
+            [3, 7, 2]
+        )
+
+    def test_dls_successor(self):
+        result = nx.dfs_successors(self.G, source=4, depth_limit=3)
+        assert {n: set(v) for n, v in result.items()} == {
+            2: {1, 7},
+            3: {2},
+            4: {3, 5},
+            5: {6},
+        }
+        result = nx.dfs_successors(self.D, source=7, depth_limit=2)
+        assert {n: set(v) for n, v in result.items()} == {8: {9}, 2: {3}, 7: {8, 2}}
+
+    def test_dls_predecessor(self):
+        assert nx.dfs_predecessors(self.G, source=0, depth_limit=3) == {
+            1: 0,
+            2: 1,
+            3: 2,
+            7: 2,
+        }
+        assert nx.dfs_predecessors(self.D, source=2, depth_limit=3) == {
+            8: 7,
+            9: 8,
+            3: 2,
+            7: 2,
+        }
+
+    def test_dls_tree(self):
+        T = nx.dfs_tree(self.G, source=3, depth_limit=1)
+        assert sorted(T.edges()) == [(3, 2), (3, 4)]
+
+    def test_dls_edges(self):
+        edges = nx.dfs_edges(self.G, source=9, depth_limit=4)
+        assert list(edges) == [(9, 8), (8, 7), (7, 2), (2, 1), (2, 3), (9, 10)]
+
+    def test_dls_labeled_edges_depth_1(self):
+        edges = list(nx.dfs_labeled_edges(self.G, source=5, depth_limit=1))
+        forward = [(u, v) for (u, v, d) in edges if d == "forward"]
+        assert forward == [(5, 5), (5, 4), (5, 6)]
+        # Note: reverse-depth_limit edge types were not reported before gh-6240
+        assert edges == [
+            (5, 5, "forward"),
+            (5, 4, "forward"),
+            (5, 4, "reverse-depth_limit"),
+            (5, 6, "forward"),
+            (5, 6, "reverse-depth_limit"),
+            (5, 5, "reverse"),
+        ]
+
+    def test_dls_labeled_edges_depth_2(self):
+        edges = list(nx.dfs_labeled_edges(self.G, source=6, depth_limit=2))
+        forward = [(u, v) for (u, v, d) in edges if d == "forward"]
+        assert forward == [(6, 6), (6, 5), (5, 4)]
+        assert edges == [
+            (6, 6, "forward"),
+            (6, 5, "forward"),
+            (5, 4, "forward"),
+            (5, 4, "reverse-depth_limit"),
+            (5, 6, "nontree"),
+            (6, 5, "reverse"),
+            (6, 6, "reverse"),
+        ]
+
+    def test_dls_labeled_disconnected_edges(self):
+        edges = list(nx.dfs_labeled_edges(self.D, depth_limit=1))
+        assert edges == [
+            (0, 0, "forward"),
+            (0, 1, "forward"),
+            (0, 1, "reverse-depth_limit"),
+            (0, 0, "reverse"),
+            (2, 2, "forward"),
+            (2, 3, "forward"),
+            (2, 3, "reverse-depth_limit"),
+            (2, 7, "forward"),
+            (2, 7, "reverse-depth_limit"),
+            (2, 2, "reverse"),
+            (8, 8, "forward"),
+            (8, 7, "nontree"),
+            (8, 9, "forward"),
+            (8, 9, "reverse-depth_limit"),
+            (8, 8, "reverse"),
+            (10, 10, "forward"),
+            (10, 9, "nontree"),
+            (10, 10, "reverse"),
+        ]
+        # large depth_limit has no impact
+        edges = list(nx.dfs_labeled_edges(self.D, depth_limit=19))
+        assert edges == [
+            (0, 0, "forward"),
+            (0, 1, "forward"),
+            (1, 0, "nontree"),
+            (0, 1, "reverse"),
+            (0, 0, "reverse"),
+            (2, 2, "forward"),
+            (2, 3, "forward"),
+            (3, 2, "nontree"),
+            (2, 3, "reverse"),
+            (2, 7, "forward"),
+            (7, 2, "nontree"),
+            (7, 8, "forward"),
+            (8, 7, "nontree"),
+            (8, 9, "forward"),
+            (9, 8, "nontree"),
+            (9, 10, "forward"),
+            (10, 9, "nontree"),
+            (9, 10, "reverse"),
+            (8, 9, "reverse"),
+            (7, 8, "reverse"),
+            (2, 7, "reverse"),
+            (2, 2, "reverse"),
+        ]

wemm/lib/python3.10/site-packages/networkx/generators/tests/test_triads.py

@@ -0,0 +1,15 @@
+"""Unit tests for the :mod:`networkx.generators.triads` module."""
+
+import pytest
+
+from networkx import triad_graph
+
+
+def test_triad_graph():
+    G = triad_graph("030T")
+    assert [tuple(e) for e in ("ab", "ac", "cb")] == sorted(G.edges())
+
+
+def test_invalid_name():
+    with pytest.raises(ValueError):
+        triad_graph("bogus")

wemm/lib/python3.10/site-packages/networkx/linalg/__init__.py

@@ -0,0 +1,13 @@
+from networkx.linalg.attrmatrix import *
+from networkx.linalg import attrmatrix
+from networkx.linalg.spectrum import *
+from networkx.linalg import spectrum
+from networkx.linalg.graphmatrix import *
+from networkx.linalg import graphmatrix
+from networkx.linalg.laplacianmatrix import *
+from networkx.linalg import laplacianmatrix
+from networkx.linalg.algebraicconnectivity import *
+from networkx.linalg.modularitymatrix import *
+from networkx.linalg import modularitymatrix
+from networkx.linalg.bethehessianmatrix import *
+from networkx.linalg import bethehessianmatrix

wemm/lib/python3.10/site-packages/networkx/linalg/algebraicconnectivity.py

@@ -0,0 +1,657 @@
+"""
+Algebraic connectivity and Fiedler vectors of undirected graphs.
+"""
+
+from functools import partial
+
+import networkx as nx
+from networkx.utils import (
+    not_implemented_for,
+    np_random_state,
+    reverse_cuthill_mckee_ordering,
+)
+
+__all__ = [
+    "algebraic_connectivity",
+    "fiedler_vector",
+    "spectral_ordering",
+    "spectral_bisection",
+]
+
+
+class _PCGSolver:
+    """Preconditioned conjugate gradient method.
+
+    To solve Ax = b:
+        M = A.diagonal() # or some other preconditioner
+        solver = _PCGSolver(lambda x: A * x, lambda x: M * x)
+        x = solver.solve(b)
+
+    The inputs A and M are functions which compute
+    matrix multiplication on the argument.
+    A - multiply by the matrix A in Ax=b
+    M - multiply by M, the preconditioner surrogate for A
+
+    Warning: There is no limit on number of iterations.
+    """
+
+    def __init__(self, A, M):
+        self._A = A
+        self._M = M
+
+    def solve(self, B, tol):
+        import numpy as np
+
+        # Densifying step - can this be kept sparse?
+        B = np.asarray(B)
+        X = np.ndarray(B.shape, order="F")
+        for j in range(B.shape[1]):
+            X[:, j] = self._solve(B[:, j], tol)
+        return X
+
+    def _solve(self, b, tol):
+        import numpy as np
+        import scipy as sp
+
+        A = self._A
+        M = self._M
+        tol *= sp.linalg.blas.dasum(b)
+        # Initialize.
+        x = np.zeros(b.shape)
+        r = b.copy()
+        z = M(r)
+        rz = sp.linalg.blas.ddot(r, z)
+        p = z.copy()
+        # Iterate.
+        while True:
+            Ap = A(p)
+            alpha = rz / sp.linalg.blas.ddot(p, Ap)
+            x = sp.linalg.blas.daxpy(p, x, a=alpha)
+            r = sp.linalg.blas.daxpy(Ap, r, a=-alpha)
+            if sp.linalg.blas.dasum(r) < tol:
+                return x
+            z = M(r)
+            beta = sp.linalg.blas.ddot(r, z)
+            beta, rz = beta / rz, beta
+            p = sp.linalg.blas.daxpy(p, z, a=beta)
+
+
+class _LUSolver:
+    """LU factorization.
+
+    To solve Ax = b:
+        solver = _LUSolver(A)
+        x = solver.solve(b)
+
+    optional argument `tol` on solve method is ignored but included
+    to match _PCGsolver API.
+    """
+
+    def __init__(self, A):
+        import scipy as sp
+
+        self._LU = sp.sparse.linalg.splu(
+            A,
+            permc_spec="MMD_AT_PLUS_A",
+            diag_pivot_thresh=0.0,
+            options={"Equil": True, "SymmetricMode": True},
+        )
+
+    def solve(self, B, tol=None):
+        import numpy as np
+
+        B = np.asarray(B)
+        X = np.ndarray(B.shape, order="F")
+        for j in range(B.shape[1]):
+            X[:, j] = self._LU.solve(B[:, j])
+        return X
+
+
+def _preprocess_graph(G, weight):
+    """Compute edge weights and eliminate zero-weight edges."""
+    if G.is_directed():
+        H = nx.MultiGraph()
+        H.add_nodes_from(G)
+        H.add_weighted_edges_from(
+            ((u, v, e.get(weight, 1.0)) for u, v, e in G.edges(data=True) if u != v),
+            weight=weight,
+        )
+        G = H
+    if not G.is_multigraph():
+        edges = (
+            (u, v, abs(e.get(weight, 1.0))) for u, v, e in G.edges(data=True) if u != v
+        )
+    else:
+        edges = (
+            (u, v, sum(abs(e.get(weight, 1.0)) for e in G[u][v].values()))
+            for u, v in G.edges()
+            if u != v
+        )
+    H = nx.Graph()
+    H.add_nodes_from(G)
+    H.add_weighted_edges_from((u, v, e) for u, v, e in edges if e != 0)
+    return H
+
+
+def _rcm_estimate(G, nodelist):
+    """Estimate the Fiedler vector using the reverse Cuthill-McKee ordering."""
+    import numpy as np
+
+    G = G.subgraph(nodelist)
+    order = reverse_cuthill_mckee_ordering(G)
+    n = len(nodelist)
+    index = dict(zip(nodelist, range(n)))
+    x = np.ndarray(n, dtype=float)
+    for i, u in enumerate(order):
+        x[index[u]] = i
+    x -= (n - 1) / 2.0
+    return x
+
+
+def _tracemin_fiedler(L, X, normalized, tol, method):
+    """Compute the Fiedler vector of L using the TraceMIN-Fiedler algorithm.
+
+    The Fiedler vector of a connected undirected graph is the eigenvector
+    corresponding to the second smallest eigenvalue of the Laplacian matrix
+    of the graph. This function starts with the Laplacian L, not the Graph.
+
+    Parameters
+    ----------
+    L : Laplacian of a possibly weighted or normalized, but undirected graph
+
+    X : Initial guess for a solution. Usually a matrix of random numbers.
+        This function allows more than one column in X to identify more than
+        one eigenvector if desired.
+
+    normalized : bool
+        Whether the normalized Laplacian matrix is used.
+
+    tol : float
+        Tolerance of relative residual in eigenvalue computation.
+        Warning: There is no limit on number of iterations.
+
+    method : string
+        Should be 'tracemin_pcg' or 'tracemin_lu'.
+        Otherwise exception is raised.
+
+    Returns
+    -------
+    sigma, X : Two NumPy arrays of floats.
+        The lowest eigenvalues and corresponding eigenvectors of L.
+        The size of input X determines the size of these outputs.
+        As this is for Fiedler vectors, the zero eigenvalue (and
+        constant eigenvector) are avoided.
+    """
+    import numpy as np
+    import scipy as sp
+
+    n = X.shape[0]
+
+    if normalized:
+        # Form the normalized Laplacian matrix and determine the eigenvector of
+        # its nullspace.
+        e = np.sqrt(L.diagonal())
+        # TODO: rm csr_array wrapper when spdiags array creation becomes available
+        D = sp.sparse.csr_array(sp.sparse.spdiags(1 / e, 0, n, n, format="csr"))
+        L = D @ L @ D
+        e *= 1.0 / np.linalg.norm(e, 2)
+
+    if normalized:
+
+        def project(X):
+            """Make X orthogonal to the nullspace of L."""
+            X = np.asarray(X)
+            for j in range(X.shape[1]):
+                X[:, j] -= (X[:, j] @ e) * e
+
+    else:
+
+        def project(X):
+            """Make X orthogonal to the nullspace of L."""
+            X = np.asarray(X)
+            for j in range(X.shape[1]):
+                X[:, j] -= X[:, j].sum() / n
+
+    if method == "tracemin_pcg":
+        D = L.diagonal().astype(float)
+        solver = _PCGSolver(lambda x: L @ x, lambda x: D * x)
+    elif method == "tracemin_lu":
+        # Convert A to CSC to suppress SparseEfficiencyWarning.
+        A = sp.sparse.csc_array(L, dtype=float, copy=True)
+        # Force A to be nonsingular. Since A is the Laplacian matrix of a
+        # connected graph, its rank deficiency is one, and thus one diagonal
+        # element needs to modified. Changing to infinity forces a zero in the
+        # corresponding element in the solution.
+        i = (A.indptr[1:] - A.indptr[:-1]).argmax()
+        A[i, i] = np.inf
+        solver = _LUSolver(A)
+    else:
+        raise nx.NetworkXError(f"Unknown linear system solver: {method}")
+
+    # Initialize.
+    Lnorm = abs(L).sum(axis=1).flatten().max()
+    project(X)
+    W = np.ndarray(X.shape, order="F")
+
+    while True:
+        # Orthonormalize X.
+        X = np.linalg.qr(X)[0]
+        # Compute iteration matrix H.
+        W[:, :] = L @ X
+        H = X.T @ W
+        sigma, Y = sp.linalg.eigh(H, overwrite_a=True)
+        # Compute the Ritz vectors.
+        X = X @ Y
+        # Test for convergence exploiting the fact that L * X == W * Y.
+        res = sp.linalg.blas.dasum(W @ Y[:, 0] - sigma[0] * X[:, 0]) / Lnorm
+        if res < tol:
+            break
+        # Compute X = L \ X / (X' * (L \ X)).
+        # L \ X can have an arbitrary projection on the nullspace of L,
+        # which will be eliminated.
+        W[:, :] = solver.solve(X, tol)
+        X = (sp.linalg.inv(W.T @ X) @ W.T).T  # Preserves Fortran storage order.
+        project(X)
+
+    return sigma, np.asarray(X)
+
+
+def _get_fiedler_func(method):
+    """Returns a function that solves the Fiedler eigenvalue problem."""
+    import numpy as np
+
+    if method == "tracemin":  # old style keyword <v2.1
+        method = "tracemin_pcg"
+    if method in ("tracemin_pcg", "tracemin_lu"):
+
+        def find_fiedler(L, x, normalized, tol, seed):
+            q = 1 if method == "tracemin_pcg" else min(4, L.shape[0] - 1)
+            X = np.asarray(seed.normal(size=(q, L.shape[0]))).T
+            sigma, X = _tracemin_fiedler(L, X, normalized, tol, method)
+            return sigma[0], X[:, 0]
+
+    elif method == "lanczos" or method == "lobpcg":
+
+        def find_fiedler(L, x, normalized, tol, seed):
+            import scipy as sp
+
+            L = sp.sparse.csc_array(L, dtype=float)
+            n = L.shape[0]
+            if normalized:
+                # TODO: rm csc_array wrapping when spdiags array becomes available
+                D = sp.sparse.csc_array(
+                    sp.sparse.spdiags(
+                        1.0 / np.sqrt(L.diagonal()), [0], n, n, format="csc"
+                    )
+                )
+                L = D @ L @ D
+            if method == "lanczos" or n < 10:
+                # Avoid LOBPCG when n < 10 due to
+                # https://github.com/scipy/scipy/issues/3592
+                # https://github.com/scipy/scipy/pull/3594
+                sigma, X = sp.sparse.linalg.eigsh(
+                    L, 2, which="SM", tol=tol, return_eigenvectors=True
+                )
+                return sigma[1], X[:, 1]
+            else:
+                X = np.asarray(np.atleast_2d(x).T)
+                # TODO: rm csr_array wrapping when spdiags array becomes available
+                M = sp.sparse.csr_array(sp.sparse.spdiags(1.0 / L.diagonal(), 0, n, n))
+                Y = np.ones(n)
+                if normalized:
+                    Y /= D.diagonal()
+                sigma, X = sp.sparse.linalg.lobpcg(
+                    L, X, M=M, Y=np.atleast_2d(Y).T, tol=tol, maxiter=n, largest=False
+                )
+                return sigma[0], X[:, 0]
+
+    else:
+        raise nx.NetworkXError(f"unknown method {method!r}.")
+
+    return find_fiedler
+
+
+@not_implemented_for("directed")
+@np_random_state(5)
+@nx._dispatchable(edge_attrs="weight")
+def algebraic_connectivity(
+    G, weight="weight", normalized=False, tol=1e-8, method="tracemin_pcg", seed=None
+):
+    r"""Returns the algebraic connectivity of an undirected graph.
+
+    The algebraic connectivity of a connected undirected graph is the second
+    smallest eigenvalue of its Laplacian matrix.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+        An undirected graph.
+
+    weight : object, optional (default: None)
+        The data key used to determine the weight of each edge. If None, then
+        each edge has unit weight.
+
+    normalized : bool, optional (default: False)
+        Whether the normalized Laplacian matrix is used.
+
+    tol : float, optional (default: 1e-8)
+        Tolerance of relative residual in eigenvalue computation.
+
+    method : string, optional (default: 'tracemin_pcg')
+        Method of eigenvalue computation. It must be one of the tracemin
+        options shown below (TraceMIN), 'lanczos' (Lanczos iteration)
+        or 'lobpcg' (LOBPCG).
+
+        The TraceMIN algorithm uses a linear system solver. The following
+        values allow specifying the solver to be used.
+
+        =============== ========================================
+        Value           Solver
+        =============== ========================================
+        'tracemin_pcg'  Preconditioned conjugate gradient method
+        'tracemin_lu'   LU factorization
+        =============== ========================================
+
+    seed : integer, random_state, or None (default)
+        Indicator of random number generation state.
+        See :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    algebraic_connectivity : float
+        Algebraic connectivity.
+
+    Raises
+    ------
+    NetworkXNotImplemented
+        If G is directed.
+
+    NetworkXError
+        If G has less than two nodes.
+
+    Notes
+    -----
+    Edge weights are interpreted by their absolute values. For MultiGraph's,
+    weights of parallel edges are summed. Zero-weighted edges are ignored.
+
+    See Also
+    --------
+    laplacian_matrix
+
+    Examples
+    --------
+    For undirected graphs algebraic connectivity can tell us if a graph is connected or not
+    `G` is connected iff  ``algebraic_connectivity(G) > 0``:
+
+    >>> G = nx.complete_graph(5)
+    >>> nx.algebraic_connectivity(G) > 0
+    True
+    >>> G.add_node(10)  # G is no longer connected
+    >>> nx.algebraic_connectivity(G) > 0
+    False
+
+    """
+    if len(G) < 2:
+        raise nx.NetworkXError("graph has less than two nodes.")
+    G = _preprocess_graph(G, weight)
+    if not nx.is_connected(G):
+        return 0.0
+
+    L = nx.laplacian_matrix(G)
+    if L.shape[0] == 2:
+        return 2.0 * float(L[0, 0]) if not normalized else 2.0
+
+    find_fiedler = _get_fiedler_func(method)
+    x = None if method != "lobpcg" else _rcm_estimate(G, G)
+    sigma, fiedler = find_fiedler(L, x, normalized, tol, seed)
+    return float(sigma)
+
+
+@not_implemented_for("directed")
+@np_random_state(5)
+@nx._dispatchable(edge_attrs="weight")
+def fiedler_vector(
+    G, weight="weight", normalized=False, tol=1e-8, method="tracemin_pcg", seed=None
+):
+    """Returns the Fiedler vector of a connected undirected graph.
+
+    The Fiedler vector of a connected undirected graph is the eigenvector
+    corresponding to the second smallest eigenvalue of the Laplacian matrix
+    of the graph.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+        An undirected graph.
+
+    weight : object, optional (default: None)
+        The data key used to determine the weight of each edge. If None, then
+        each edge has unit weight.
+
+    normalized : bool, optional (default: False)
+        Whether the normalized Laplacian matrix is used.
+
+    tol : float, optional (default: 1e-8)
+        Tolerance of relative residual in eigenvalue computation.
+
+    method : string, optional (default: 'tracemin_pcg')
+        Method of eigenvalue computation. It must be one of the tracemin
+        options shown below (TraceMIN), 'lanczos' (Lanczos iteration)
+        or 'lobpcg' (LOBPCG).
+
+        The TraceMIN algorithm uses a linear system solver. The following
+        values allow specifying the solver to be used.
+
+        =============== ========================================
+        Value           Solver
+        =============== ========================================
+        'tracemin_pcg'  Preconditioned conjugate gradient method
+        'tracemin_lu'   LU factorization
+        =============== ========================================
+
+    seed : integer, random_state, or None (default)
+        Indicator of random number generation state.
+        See :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    fiedler_vector : NumPy array of floats.
+        Fiedler vector.
+
+    Raises
+    ------
+    NetworkXNotImplemented
+        If G is directed.
+
+    NetworkXError
+        If G has less than two nodes or is not connected.
+
+    Notes
+    -----
+    Edge weights are interpreted by their absolute values. For MultiGraph's,
+    weights of parallel edges are summed. Zero-weighted edges are ignored.
+
+    See Also
+    --------
+    laplacian_matrix
+
+    Examples
+    --------
+    Given a connected graph the signs of the values in the Fiedler vector can be
+    used to partition the graph into two components.
+
+    >>> G = nx.barbell_graph(5, 0)
+    >>> nx.fiedler_vector(G, normalized=True, seed=1)
+    array([-0.32864129, -0.32864129, -0.32864129, -0.32864129, -0.26072899,
+            0.26072899,  0.32864129,  0.32864129,  0.32864129,  0.32864129])
+
+    The connected components are the two 5-node cliques of the barbell graph.
+    """
+    import numpy as np
+
+    if len(G) < 2:
+        raise nx.NetworkXError("graph has less than two nodes.")
+    G = _preprocess_graph(G, weight)
+    if not nx.is_connected(G):
+        raise nx.NetworkXError("graph is not connected.")
+
+    if len(G) == 2:
+        return np.array([1.0, -1.0])
+
+    find_fiedler = _get_fiedler_func(method)
+    L = nx.laplacian_matrix(G)
+    x = None if method != "lobpcg" else _rcm_estimate(G, G)
+    sigma, fiedler = find_fiedler(L, x, normalized, tol, seed)
+    return fiedler
+
+
+@np_random_state(5)
+@nx._dispatchable(edge_attrs="weight")
+def spectral_ordering(
+    G, weight="weight", normalized=False, tol=1e-8, method="tracemin_pcg", seed=None
+):
+    """Compute the spectral_ordering of a graph.
+
+    The spectral ordering of a graph is an ordering of its nodes where nodes
+    in the same weakly connected components appear contiguous and ordered by
+    their corresponding elements in the Fiedler vector of the component.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+        A graph.
+
+    weight : object, optional (default: None)
+        The data key used to determine the weight of each edge. If None, then
+        each edge has unit weight.
+
+    normalized : bool, optional (default: False)
+        Whether the normalized Laplacian matrix is used.
+
+    tol : float, optional (default: 1e-8)
+        Tolerance of relative residual in eigenvalue computation.
+
+    method : string, optional (default: 'tracemin_pcg')
+        Method of eigenvalue computation. It must be one of the tracemin
+        options shown below (TraceMIN), 'lanczos' (Lanczos iteration)
+        or 'lobpcg' (LOBPCG).
+
+        The TraceMIN algorithm uses a linear system solver. The following
+        values allow specifying the solver to be used.
+
+        =============== ========================================
+        Value           Solver
+        =============== ========================================
+        'tracemin_pcg'  Preconditioned conjugate gradient method
+        'tracemin_lu'   LU factorization
+        =============== ========================================
+
+    seed : integer, random_state, or None (default)
+        Indicator of random number generation state.
+        See :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    spectral_ordering : NumPy array of floats.
+        Spectral ordering of nodes.
+
+    Raises
+    ------
+    NetworkXError
+        If G is empty.
+
+    Notes
+    -----
+    Edge weights are interpreted by their absolute values. For MultiGraph's,
+    weights of parallel edges are summed. Zero-weighted edges are ignored.
+
+    See Also
+    --------
+    laplacian_matrix
+    """
+    if len(G) == 0:
+        raise nx.NetworkXError("graph is empty.")
+    G = _preprocess_graph(G, weight)
+
+    find_fiedler = _get_fiedler_func(method)
+    order = []
+    for component in nx.connected_components(G):
+        size = len(component)
+        if size > 2:
+            L = nx.laplacian_matrix(G, component)
+            x = None if method != "lobpcg" else _rcm_estimate(G, component)
+            sigma, fiedler = find_fiedler(L, x, normalized, tol, seed)
+            sort_info = zip(fiedler, range(size), component)
+            order.extend(u for x, c, u in sorted(sort_info))
+        else:
+            order.extend(component)
+
+    return order
+
+
+@nx._dispatchable(edge_attrs="weight")
+def spectral_bisection(
+    G, weight="weight", normalized=False, tol=1e-8, method="tracemin_pcg", seed=None
+):
+    """Bisect the graph using the Fiedler vector.
+
+    This method uses the Fiedler vector to bisect a graph.
+    The partition is defined by the nodes which are associated with
+    either positive or negative values in the vector.
+
+    Parameters
+    ----------
+    G : NetworkX Graph
+
+    weight : str, optional (default: weight)
+        The data key used to determine the weight of each edge. If None, then
+        each edge has unit weight.
+
+    normalized : bool, optional (default: False)
+        Whether the normalized Laplacian matrix is used.
+
+    tol : float, optional (default: 1e-8)
+        Tolerance of relative residual in eigenvalue computation.
+
+    method : string, optional (default: 'tracemin_pcg')
+        Method of eigenvalue computation. It must be one of the tracemin
+        options shown below (TraceMIN), 'lanczos' (Lanczos iteration)
+        or 'lobpcg' (LOBPCG).
+
+        The TraceMIN algorithm uses a linear system solver. The following
+        values allow specifying the solver to be used.
+
+        =============== ========================================
+        Value           Solver
+        =============== ========================================
+        'tracemin_pcg'  Preconditioned conjugate gradient method
+        'tracemin_lu'   LU factorization
+        =============== ========================================
+
+    seed : integer, random_state, or None (default)
+        Indicator of random number generation state.
+        See :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    bisection : tuple of sets
+        Sets with the bisection of nodes
+
+    Examples
+    --------
+    >>> G = nx.barbell_graph(3, 0)
+    >>> nx.spectral_bisection(G)
+    ({0, 1, 2}, {3, 4, 5})
+
+    References
+    ----------
+    .. [1] M. E. J Newman 'Networks: An Introduction', pages 364-370
+       Oxford University Press 2011.
+    """
+    import numpy as np
+
+    v = nx.fiedler_vector(G, weight, normalized, tol, method, seed)
+    nodes = np.array(list(G))
+    pos_vals = v >= 0
+
+    return set(nodes[~pos_vals].tolist()), set(nodes[pos_vals].tolist())

wemm/lib/python3.10/site-packages/networkx/linalg/attrmatrix.py

@@ -0,0 +1,465 @@
+"""
+Functions for constructing matrix-like objects from graph attributes.
+"""
+
+import networkx as nx
+
+__all__ = ["attr_matrix", "attr_sparse_matrix"]
+
+
+def _node_value(G, node_attr):
+    """Returns a function that returns a value from G.nodes[u].
+
+    We return a function expecting a node as its sole argument. Then, in the
+    simplest scenario, the returned function will return G.nodes[u][node_attr].
+    However, we also handle the case when `node_attr` is None or when it is a
+    function itself.
+
+    Parameters
+    ----------
+    G : graph
+        A NetworkX graph
+
+    node_attr : {None, str, callable}
+        Specification of how the value of the node attribute should be obtained
+        from the node attribute dictionary.
+
+    Returns
+    -------
+    value : function
+        A function expecting a node as its sole argument. The function will
+        returns a value from G.nodes[u] that depends on `edge_attr`.
+
+    """
+    if node_attr is None:
+
+        def value(u):
+            return u
+
+    elif not callable(node_attr):
+        # assume it is a key for the node attribute dictionary
+        def value(u):
+            return G.nodes[u][node_attr]
+
+    else:
+        # Advanced:  Allow users to specify something else.
+        #
+        # For example,
+        #     node_attr = lambda u: G.nodes[u].get('size', .5) * 3
+        #
+        value = node_attr
+
+    return value
+
+
+def _edge_value(G, edge_attr):
+    """Returns a function that returns a value from G[u][v].
+
+    Suppose there exists an edge between u and v.  Then we return a function
+    expecting u and v as arguments.  For Graph and DiGraph, G[u][v] is
+    the edge attribute dictionary, and the function (essentially) returns
+    G[u][v][edge_attr].  However, we also handle cases when `edge_attr` is None
+    and when it is a function itself. For MultiGraph and MultiDiGraph, G[u][v]
+    is a dictionary of all edges between u and v.  In this case, the returned
+    function sums the value of `edge_attr` for every edge between u and v.
+
+    Parameters
+    ----------
+    G : graph
+       A NetworkX graph
+
+    edge_attr : {None, str, callable}
+        Specification of how the value of the edge attribute should be obtained
+        from the edge attribute dictionary, G[u][v].  For multigraphs, G[u][v]
+        is a dictionary of all the edges between u and v.  This allows for
+        special treatment of multiedges.
+
+    Returns
+    -------
+    value : function
+        A function expecting two nodes as parameters. The nodes should
+        represent the from- and to- node of an edge. The function will
+        return a value from G[u][v] that depends on `edge_attr`.
+
+    """
+
+    if edge_attr is None:
+        # topological count of edges
+
+        if G.is_multigraph():
+
+            def value(u, v):
+                return len(G[u][v])
+
+        else:
+
+            def value(u, v):
+                return 1
+
+    elif not callable(edge_attr):
+        # assume it is a key for the edge attribute dictionary
+
+        if edge_attr == "weight":
+            # provide a default value
+            if G.is_multigraph():
+
+                def value(u, v):
+                    return sum(d.get(edge_attr, 1) for d in G[u][v].values())
+
+            else:
+
+                def value(u, v):
+                    return G[u][v].get(edge_attr, 1)
+
+        else:
+            # otherwise, the edge attribute MUST exist for each edge
+            if G.is_multigraph():
+
+                def value(u, v):
+                    return sum(d[edge_attr] for d in G[u][v].values())
+
+            else:
+
+                def value(u, v):
+                    return G[u][v][edge_attr]
+
+    else:
+        # Advanced:  Allow users to specify something else.
+        #
+        # Alternative default value:
+        #     edge_attr = lambda u,v: G[u][v].get('thickness', .5)
+        #
+        # Function on an attribute:
+        #     edge_attr = lambda u,v: abs(G[u][v]['weight'])
+        #
+        # Handle Multi(Di)Graphs differently:
+        #     edge_attr = lambda u,v: numpy.prod([d['size'] for d in G[u][v].values()])
+        #
+        # Ignore multiple edges
+        #     edge_attr = lambda u,v: 1 if len(G[u][v]) else 0
+        #
+        value = edge_attr
+
+    return value
+
+
+@nx._dispatchable(edge_attrs={"edge_attr": None}, node_attrs="node_attr")
+def attr_matrix(
+    G,
+    edge_attr=None,
+    node_attr=None,
+    normalized=False,
+    rc_order=None,
+    dtype=None,
+    order=None,
+):
+    """Returns the attribute matrix using attributes from `G` as a numpy array.
+
+    If only `G` is passed in, then the adjacency matrix is constructed.
+
+    Let A be a discrete set of values for the node attribute `node_attr`. Then
+    the elements of A represent the rows and columns of the constructed matrix.
+    Now, iterate through every edge e=(u,v) in `G` and consider the value
+    of the edge attribute `edge_attr`.  If ua and va are the values of the
+    node attribute `node_attr` for u and v, respectively, then the value of
+    the edge attribute is added to the matrix element at (ua, va).
+
+    Parameters
+    ----------
+    G : graph
+        The NetworkX graph used to construct the attribute matrix.
+
+    edge_attr : str, optional
+        Each element of the matrix represents a running total of the
+        specified edge attribute for edges whose node attributes correspond
+        to the rows/cols of the matrix. The attribute must be present for
+        all edges in the graph. If no attribute is specified, then we
+        just count the number of edges whose node attributes correspond
+        to the matrix element.
+
+    node_attr : str, optional
+        Each row and column in the matrix represents a particular value
+        of the node attribute.  The attribute must be present for all nodes
+        in the graph. Note, the values of this attribute should be reliably
+        hashable. So, float values are not recommended. If no attribute is
+        specified, then the rows and columns will be the nodes of the graph.
+
+    normalized : bool, optional
+        If True, then each row is normalized by the summation of its values.
+
+    rc_order : list, optional
+        A list of the node attribute values. This list specifies the ordering
+        of rows and columns of the array. If no ordering is provided, then
+        the ordering will be random (and also, a return value).
+
+    Other Parameters
+    ----------------
+    dtype : NumPy data-type, optional
+        A valid NumPy dtype used to initialize the array. Keep in mind certain
+        dtypes can yield unexpected results if the array is to be normalized.
+        The parameter is passed to numpy.zeros(). If unspecified, the NumPy
+        default is used.
+
+    order : {'C', 'F'}, optional
+        Whether to store multidimensional data in C- or Fortran-contiguous
+        (row- or column-wise) order in memory. This parameter is passed to
+        numpy.zeros(). If unspecified, the NumPy default is used.
+
+    Returns
+    -------
+    M : 2D NumPy ndarray
+        The attribute matrix.
+
+    ordering : list
+        If `rc_order` was specified, then only the attribute matrix is returned.
+        However, if `rc_order` was None, then the ordering used to construct
+        the matrix is returned as well.
+
+    Examples
+    --------
+    Construct an adjacency matrix:
+
+    >>> G = nx.Graph()
+    >>> G.add_edge(0, 1, thickness=1, weight=3)
+    >>> G.add_edge(0, 2, thickness=2)
+    >>> G.add_edge(1, 2, thickness=3)
+    >>> nx.attr_matrix(G, rc_order=[0, 1, 2])
+    array([[0., 1., 1.],
+           [1., 0., 1.],
+           [1., 1., 0.]])
+
+    Alternatively, we can obtain the matrix describing edge thickness.
+
+    >>> nx.attr_matrix(G, edge_attr="thickness", rc_order=[0, 1, 2])
+    array([[0., 1., 2.],
+           [1., 0., 3.],
+           [2., 3., 0.]])
+
+    We can also color the nodes and ask for the probability distribution over
+    all edges (u,v) describing:
+
+        Pr(v has color Y | u has color X)
+
+    >>> G.nodes[0]["color"] = "red"
+    >>> G.nodes[1]["color"] = "red"
+    >>> G.nodes[2]["color"] = "blue"
+    >>> rc = ["red", "blue"]
+    >>> nx.attr_matrix(G, node_attr="color", normalized=True, rc_order=rc)
+    array([[0.33333333, 0.66666667],
+           [1.        , 0.        ]])
+
+    For example, the above tells us that for all edges (u,v):
+
+        Pr( v is red  | u is red)  = 1/3
+        Pr( v is blue | u is red)  = 2/3
+
+        Pr( v is red  | u is blue) = 1
+        Pr( v is blue | u is blue) = 0
+
+    Finally, we can obtain the total weights listed by the node colors.
+
+    >>> nx.attr_matrix(G, edge_attr="weight", node_attr="color", rc_order=rc)
+    array([[3., 2.],
+           [2., 0.]])
+
+    Thus, the total weight over all edges (u,v) with u and v having colors:
+
+        (red, red)   is 3   # the sole contribution is from edge (0,1)
+        (red, blue)  is 2   # contributions from edges (0,2) and (1,2)
+        (blue, red)  is 2   # same as (red, blue) since graph is undirected
+        (blue, blue) is 0   # there are no edges with blue endpoints
+
+    """
+    import numpy as np
+
+    edge_value = _edge_value(G, edge_attr)
+    node_value = _node_value(G, node_attr)
+
+    if rc_order is None:
+        ordering = list({node_value(n) for n in G})
+    else:
+        ordering = rc_order
+
+    N = len(ordering)
+    undirected = not G.is_directed()
+    index = dict(zip(ordering, range(N)))
+    M = np.zeros((N, N), dtype=dtype, order=order)
+
+    seen = set()
+    for u, nbrdict in G.adjacency():
+        for v in nbrdict:
+            # Obtain the node attribute values.
+            i, j = index[node_value(u)], index[node_value(v)]
+            if v not in seen:
+                M[i, j] += edge_value(u, v)
+                if undirected:
+                    M[j, i] = M[i, j]
+
+        if undirected:
+            seen.add(u)
+
+    if normalized:
+        M /= M.sum(axis=1).reshape((N, 1))
+
+    if rc_order is None:
+        return M, ordering
+    else:
+        return M
+
+
+@nx._dispatchable(edge_attrs={"edge_attr": None}, node_attrs="node_attr")
+def attr_sparse_matrix(
+    G, edge_attr=None, node_attr=None, normalized=False, rc_order=None, dtype=None
+):
+    """Returns a SciPy sparse array using attributes from G.
+
+    If only `G` is passed in, then the adjacency matrix is constructed.
+
+    Let A be a discrete set of values for the node attribute `node_attr`. Then
+    the elements of A represent the rows and columns of the constructed matrix.
+    Now, iterate through every edge e=(u,v) in `G` and consider the value
+    of the edge attribute `edge_attr`.  If ua and va are the values of the
+    node attribute `node_attr` for u and v, respectively, then the value of
+    the edge attribute is added to the matrix element at (ua, va).
+
+    Parameters
+    ----------
+    G : graph
+        The NetworkX graph used to construct the NumPy matrix.
+
+    edge_attr : str, optional
+        Each element of the matrix represents a running total of the
+        specified edge attribute for edges whose node attributes correspond
+        to the rows/cols of the matrix. The attribute must be present for
+        all edges in the graph. If no attribute is specified, then we
+        just count the number of edges whose node attributes correspond
+        to the matrix element.
+
+    node_attr : str, optional
+        Each row and column in the matrix represents a particular value
+        of the node attribute.  The attribute must be present for all nodes
+        in the graph. Note, the values of this attribute should be reliably
+        hashable. So, float values are not recommended. If no attribute is
+        specified, then the rows and columns will be the nodes of the graph.
+
+    normalized : bool, optional
+        If True, then each row is normalized by the summation of its values.
+
+    rc_order : list, optional
+        A list of the node attribute values. This list specifies the ordering
+        of rows and columns of the array. If no ordering is provided, then
+        the ordering will be random (and also, a return value).
+
+    Other Parameters
+    ----------------
+    dtype : NumPy data-type, optional
+        A valid NumPy dtype used to initialize the array. Keep in mind certain
+        dtypes can yield unexpected results if the array is to be normalized.
+        The parameter is passed to numpy.zeros(). If unspecified, the NumPy
+        default is used.
+
+    Returns
+    -------
+    M : SciPy sparse array
+        The attribute matrix.
+
+    ordering : list
+        If `rc_order` was specified, then only the matrix is returned.
+        However, if `rc_order` was None, then the ordering used to construct
+        the matrix is returned as well.
+
+    Examples
+    --------
+    Construct an adjacency matrix:
+
+    >>> G = nx.Graph()
+    >>> G.add_edge(0, 1, thickness=1, weight=3)
+    >>> G.add_edge(0, 2, thickness=2)
+    >>> G.add_edge(1, 2, thickness=3)
+    >>> M = nx.attr_sparse_matrix(G, rc_order=[0, 1, 2])
+    >>> M.toarray()
+    array([[0., 1., 1.],
+           [1., 0., 1.],
+           [1., 1., 0.]])
+
+    Alternatively, we can obtain the matrix describing edge thickness.
+
+    >>> M = nx.attr_sparse_matrix(G, edge_attr="thickness", rc_order=[0, 1, 2])
+    >>> M.toarray()
+    array([[0., 1., 2.],
+           [1., 0., 3.],
+           [2., 3., 0.]])
+
+    We can also color the nodes and ask for the probability distribution over
+    all edges (u,v) describing:
+
+        Pr(v has color Y | u has color X)
+
+    >>> G.nodes[0]["color"] = "red"
+    >>> G.nodes[1]["color"] = "red"
+    >>> G.nodes[2]["color"] = "blue"
+    >>> rc = ["red", "blue"]
+    >>> M = nx.attr_sparse_matrix(G, node_attr="color", normalized=True, rc_order=rc)
+    >>> M.toarray()
+    array([[0.33333333, 0.66666667],
+           [1.        , 0.        ]])
+
+    For example, the above tells us that for all edges (u,v):
+
+        Pr( v is red  | u is red)  = 1/3
+        Pr( v is blue | u is red)  = 2/3
+
+        Pr( v is red  | u is blue) = 1
+        Pr( v is blue | u is blue) = 0
+
+    Finally, we can obtain the total weights listed by the node colors.
+
+    >>> M = nx.attr_sparse_matrix(G, edge_attr="weight", node_attr="color", rc_order=rc)
+    >>> M.toarray()
+    array([[3., 2.],
+           [2., 0.]])
+
+    Thus, the total weight over all edges (u,v) with u and v having colors:
+
+        (red, red)   is 3   # the sole contribution is from edge (0,1)
+        (red, blue)  is 2   # contributions from edges (0,2) and (1,2)
+        (blue, red)  is 2   # same as (red, blue) since graph is undirected
+        (blue, blue) is 0   # there are no edges with blue endpoints
+
+    """
+    import numpy as np
+    import scipy as sp
+
+    edge_value = _edge_value(G, edge_attr)
+    node_value = _node_value(G, node_attr)
+
+    if rc_order is None:
+        ordering = list({node_value(n) for n in G})
+    else:
+        ordering = rc_order
+
+    N = len(ordering)
+    undirected = not G.is_directed()
+    index = dict(zip(ordering, range(N)))
+    M = sp.sparse.lil_array((N, N), dtype=dtype)
+
+    seen = set()
+    for u, nbrdict in G.adjacency():
+        for v in nbrdict:
+            # Obtain the node attribute values.
+            i, j = index[node_value(u)], index[node_value(v)]
+            if v not in seen:
+                M[i, j] += edge_value(u, v)
+                if undirected:
+                    M[j, i] = M[i, j]
+
+        if undirected:
+            seen.add(u)
+
+    if normalized:
+        M *= 1 / M.sum(axis=1)[:, np.newaxis]  # in-place mult preserves sparse
+
+    if rc_order is None:
+        return M, ordering
+    else:
+        return M

wemm/lib/python3.10/site-packages/networkx/linalg/bethehessianmatrix.py

@@ -0,0 +1,79 @@
+"""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
+    # 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

wemm/lib/python3.10/site-packages/networkx/linalg/graphmatrix.py

@@ -0,0 +1,168 @@
+"""
+Adjacency matrix and incidence matrix of graphs.
+"""
+
+import networkx as nx
+
+__all__ = ["incidence_matrix", "adjacency_matrix"]
+
+
+@nx._dispatchable(edge_attrs="weight")
+def incidence_matrix(
+    G, nodelist=None, edgelist=None, oriented=False, weight=None, *, dtype=None
+):
+    """Returns incidence matrix of G.
+
+    The incidence matrix assigns each row to a node and each column to an edge.
+    For a standard incidence matrix a 1 appears wherever a row's node is
+    incident on the column's edge.  For an oriented incidence matrix each
+    edge is assigned an orientation (arbitrarily for undirected and aligning to
+    direction for directed).  A -1 appears for the source (tail) of an edge and
+    1 for the destination (head) of the edge.  The elements are zero otherwise.
+
+    Parameters
+    ----------
+    G : graph
+       A NetworkX graph
+
+    nodelist : list, optional   (default= all nodes in G)
+       The rows are ordered according to the nodes in nodelist.
+       If nodelist is None, then the ordering is produced by G.nodes().
+
+    edgelist : list, optional (default= all edges in G)
+       The columns are ordered according to the edges in edgelist.
+       If edgelist is None, then the ordering is produced by G.edges().
+
+    oriented: bool, optional (default=False)
+       If True, matrix elements are +1 or -1 for the head or tail node
+       respectively of each edge.  If False, +1 occurs at both nodes.
+
+    weight : string or None, optional (default=None)
+       The edge data key used to provide each value in the matrix.
+       If None, then each edge has weight 1.  Edge weights, if used,
+       should be positive so that the orientation can provide the sign.
+
+    dtype : a NumPy dtype or None (default=None)
+        The dtype of the output sparse array. This type should be a compatible
+        type of the weight argument, eg. if weight would return a float this
+        argument should also be a float.
+        If None, then the default for SciPy is used.
+
+    Returns
+    -------
+    A : SciPy sparse array
+      The incidence matrix of G.
+
+    Notes
+    -----
+    For MultiGraph/MultiDiGraph, the edges in edgelist should be
+    (u,v,key) 3-tuples.
+
+    "Networks are the best discrete model for so many problems in
+    applied mathematics" [1]_.
+
+    References
+    ----------
+    .. [1] Gil Strang, Network applications: A = incidence matrix,
+       http://videolectures.net/mit18085f07_strang_lec03/
+    """
+    import scipy as sp
+
+    if nodelist is None:
+        nodelist = list(G)
+    if edgelist is None:
+        if G.is_multigraph():
+            edgelist = list(G.edges(keys=True))
+        else:
+            edgelist = list(G.edges())
+    A = sp.sparse.lil_array((len(nodelist), len(edgelist)), dtype=dtype)
+    node_index = {node: i for i, node in enumerate(nodelist)}
+    for ei, e in enumerate(edgelist):
+        (u, v) = e[:2]
+        if u == v:
+            continue  # self loops give zero column
+        try:
+            ui = node_index[u]
+            vi = node_index[v]
+        except KeyError as err:
+            raise nx.NetworkXError(
+                f"node {u} or {v} in edgelist but not in nodelist"
+            ) from err
+        if weight is None:
+            wt = 1
+        else:
+            if G.is_multigraph():
+                ekey = e[2]
+                wt = G[u][v][ekey].get(weight, 1)
+            else:
+                wt = G[u][v].get(weight, 1)
+        if oriented:
+            A[ui, ei] = -wt
+            A[vi, ei] = wt
+        else:
+            A[ui, ei] = wt
+            A[vi, ei] = wt
+    return A.asformat("csc")
+
+
+@nx._dispatchable(edge_attrs="weight")
+def adjacency_matrix(G, nodelist=None, dtype=None, weight="weight"):
+    """Returns adjacency matrix of `G`.
+
+    Parameters
+    ----------
+    G : graph
+       A NetworkX graph
+
+    nodelist : list, optional
+       The rows and columns are ordered according to the nodes in `nodelist`.
+       If ``nodelist=None`` (the default), then the ordering is produced by
+       ``G.nodes()``.
+
+    dtype : NumPy data-type, optional
+        The desired data-type for the array.
+        If `None`, then the NumPy default is used.
+
+    weight : string or None, optional (default='weight')
+       The edge data key used to provide each value in the matrix.
+       If None, then each edge has weight 1.
+
+    Returns
+    -------
+    A : SciPy sparse array
+      Adjacency matrix representation of G.
+
+    Notes
+    -----
+    For directed graphs, entry ``i, j`` corresponds to an edge from ``i`` to ``j``.
+
+    If you want a pure Python adjacency matrix representation try
+    :func:`~networkx.convert.to_dict_of_dicts` which will return a
+    dictionary-of-dictionaries format that can be addressed as a
+    sparse matrix.
+
+    For multigraphs with parallel edges the weights are summed.
+    See :func:`networkx.convert_matrix.to_numpy_array` for other options.
+
+    The convention used for self-loop edges in graphs is to assign the
+    diagonal matrix entry value to the edge weight attribute
+    (or the number 1 if the edge has no weight attribute).  If the
+    alternate convention of doubling the edge weight is desired the
+    resulting SciPy sparse array can be modified as follows::
+
+        >>> G = nx.Graph([(1, 1)])
+        >>> A = nx.adjacency_matrix(G)
+        >>> A.toarray()
+        array([[1]])
+        >>> A.setdiag(A.diagonal() * 2)
+        >>> A.toarray()
+        array([[2]])
+
+    See Also
+    --------
+    to_numpy_array
+    to_scipy_sparse_array
+    to_dict_of_dicts
+    adjacency_spectrum
+    """
+    return nx.to_scipy_sparse_array(G, nodelist=nodelist, dtype=dtype, weight=weight)

wemm/lib/python3.10/site-packages/networkx/linalg/laplacianmatrix.py

@@ -0,0 +1,617 @@
+"""Laplacian matrix of graphs.
+
+All calculations here are done using the out-degree. For Laplacians using
+in-degree, use `G.reverse(copy=False)` instead of `G` and take the transpose.
+
+The `laplacian_matrix` function provides an unnormalized matrix,
+while `normalized_laplacian_matrix`, `directed_laplacian_matrix`,
+and `directed_combinatorial_laplacian_matrix` are all normalized.
+"""
+
+import networkx as nx
+from networkx.utils import not_implemented_for
+
+__all__ = [
+    "laplacian_matrix",
+    "normalized_laplacian_matrix",
+    "total_spanning_tree_weight",
+    "directed_laplacian_matrix",
+    "directed_combinatorial_laplacian_matrix",
+]
+
+
+@nx._dispatchable(edge_attrs="weight")
+def laplacian_matrix(G, nodelist=None, weight="weight"):
+    """Returns the Laplacian matrix of G.
+
+    The graph Laplacian is the matrix L = D - A, where
+    A is the adjacency matrix and D is the diagonal matrix of node degrees.
+
+    Parameters
+    ----------
+    G : graph
+       A NetworkX graph
+
+    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().
+
+    weight : string or None, optional (default='weight')
+       The edge data key used to compute each value in the matrix.
+       If None, then each edge has weight 1.
+
+    Returns
+    -------
+    L : SciPy sparse array
+      The Laplacian matrix of G.
+
+    Notes
+    -----
+    For MultiGraph, the edges weights are summed.
+
+    This returns an unnormalized matrix. For a normalized output,
+    use `normalized_laplacian_matrix`, `directed_laplacian_matrix`,
+    or `directed_combinatorial_laplacian_matrix`.
+
+    This calculation uses the out-degree of the graph `G`. To use the
+    in-degree for calculations instead, use `G.reverse(copy=False)` and
+    take the transpose.
+
+    See Also
+    --------
+    :func:`~networkx.convert_matrix.to_numpy_array`
+    normalized_laplacian_matrix
+    directed_laplacian_matrix
+    directed_combinatorial_laplacian_matrix
+    :func:`~networkx.linalg.spectrum.laplacian_spectrum`
+
+    Examples
+    --------
+    For graphs with multiple connected components, L is permutation-similar
+    to a block diagonal matrix where each block is the respective Laplacian
+    matrix for each component.
+
+    >>> G = nx.Graph([(1, 2), (2, 3), (4, 5)])
+    >>> print(nx.laplacian_matrix(G).toarray())
+    [[ 1 -1  0  0  0]
+     [-1  2 -1  0  0]
+     [ 0 -1  1  0  0]
+     [ 0  0  0  1 -1]
+     [ 0  0  0 -1  1]]
+
+    >>> edges = [
+    ...     (1, 2),
+    ...     (2, 1),
+    ...     (2, 4),
+    ...     (4, 3),
+    ...     (3, 4),
+    ... ]
+    >>> DiG = nx.DiGraph(edges)
+    >>> print(nx.laplacian_matrix(DiG).toarray())
+    [[ 1 -1  0  0]
+     [-1  2 -1  0]
+     [ 0  0  1 -1]
+     [ 0  0 -1  1]]
+
+    Notice that node 4 is represented by the third column and row. This is because
+    by default the row/column order is the order of `G.nodes` (i.e. the node added
+    order -- in the edgelist, 4 first appears in (2, 4), before node 3 in edge (4, 3).)
+    To control the node order of the matrix, use the `nodelist` argument.
+
+    >>> print(nx.laplacian_matrix(DiG, nodelist=[1, 2, 3, 4]).toarray())
+    [[ 1 -1  0  0]
+     [-1  2  0 -1]
+     [ 0  0  1 -1]
+     [ 0  0 -1  1]]
+
+    This calculation uses the out-degree of the graph `G`. To use the
+    in-degree for calculations instead, use `G.reverse(copy=False)` and
+    take the transpose.
+
+    >>> print(nx.laplacian_matrix(DiG.reverse(copy=False)).toarray().T)
+    [[ 1 -1  0  0]
+     [-1  1 -1  0]
+     [ 0  0  2 -1]
+     [ 0  0 -1  1]]
+
+    References
+    ----------
+    .. [1] Langville, Amy N., and Carl D. Meyer. Google’s PageRank and Beyond:
+       The Science of Search Engine Rankings. Princeton University Press, 2006.
+
+    """
+    import scipy as sp
+
+    if nodelist is None:
+        nodelist = list(G)
+    A = nx.to_scipy_sparse_array(G, nodelist=nodelist, weight=weight, format="csr")
+    n, m = A.shape
+    # TODO: rm csr_array wrapper when spdiags can produce arrays
+    D = sp.sparse.csr_array(sp.sparse.spdiags(A.sum(axis=1), 0, m, n, format="csr"))
+    return D - A
+
+
+@nx._dispatchable(edge_attrs="weight")
+def normalized_laplacian_matrix(G, nodelist=None, weight="weight"):
+    r"""Returns the normalized Laplacian matrix of G.
+
+    The normalized graph Laplacian is the matrix
+
+    .. math::
+
+        N = D^{-1/2} L D^{-1/2}
+
+    where `L` is the graph Laplacian and `D` is the diagonal matrix of
+    node degrees [1]_.
+
+    Parameters
+    ----------
+    G : graph
+       A NetworkX graph
+
+    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().
+
+    weight : string or None, optional (default='weight')
+       The edge data key used to compute each value in the matrix.
+       If None, then each edge has weight 1.
+
+    Returns
+    -------
+    N : SciPy sparse array
+      The normalized Laplacian matrix of G.
+
+    Notes
+    -----
+    For MultiGraph, the edges weights are summed.
+    See :func:`to_numpy_array` for other options.
+
+    If the Graph contains selfloops, D is defined as ``diag(sum(A, 1))``, where A is
+    the adjacency matrix [2]_.
+
+    This calculation uses the out-degree of the graph `G`. To use the
+    in-degree for calculations instead, use `G.reverse(copy=False)` and
+    take the transpose.
+
+    For an unnormalized output, use `laplacian_matrix`.
+
+    Examples
+    --------
+
+    >>> import numpy as np
+    >>> edges = [
+    ...     (1, 2),
+    ...     (2, 1),
+    ...     (2, 4),
+    ...     (4, 3),
+    ...     (3, 4),
+    ... ]
+    >>> DiG = nx.DiGraph(edges)
+    >>> print(nx.normalized_laplacian_matrix(DiG).toarray())
+    [[ 1.         -0.70710678  0.          0.        ]
+     [-0.70710678  1.         -0.70710678  0.        ]
+     [ 0.          0.          1.         -1.        ]
+     [ 0.          0.         -1.          1.        ]]
+
+    Notice that node 4 is represented by the third column and row. This is because
+    by default the row/column order is the order of `G.nodes` (i.e. the node added
+    order -- in the edgelist, 4 first appears in (2, 4), before node 3 in edge (4, 3).)
+    To control the node order of the matrix, use the `nodelist` argument.
+
+    >>> print(nx.normalized_laplacian_matrix(DiG, nodelist=[1, 2, 3, 4]).toarray())
+    [[ 1.         -0.70710678  0.          0.        ]
+     [-0.70710678  1.          0.         -0.70710678]
+     [ 0.          0.          1.         -1.        ]
+     [ 0.          0.         -1.          1.        ]]
+    >>> G = nx.Graph(edges)
+    >>> print(nx.normalized_laplacian_matrix(G).toarray())
+    [[ 1.         -0.70710678  0.          0.        ]
+     [-0.70710678  1.         -0.5         0.        ]
+     [ 0.         -0.5         1.         -0.70710678]
+     [ 0.          0.         -0.70710678  1.        ]]
+
+    See Also
+    --------
+    laplacian_matrix
+    normalized_laplacian_spectrum
+    directed_laplacian_matrix
+    directed_combinatorial_laplacian_matrix
+
+    References
+    ----------
+    .. [1] Fan Chung-Graham, Spectral Graph Theory,
+       CBMS Regional Conference Series in Mathematics, Number 92, 1997.
+    .. [2] Steve Butler, Interlacing For Weighted Graphs Using The Normalized
+       Laplacian, Electronic Journal of Linear Algebra, Volume 16, pp. 90-98,
+       March 2007.
+    .. [3] Langville, Amy N., and Carl D. Meyer. Google’s PageRank and Beyond:
+       The Science of Search Engine Rankings. Princeton University Press, 2006.
+    """
+    import numpy as np
+    import scipy as sp
+
+    if nodelist is None:
+        nodelist = list(G)
+    A = nx.to_scipy_sparse_array(G, nodelist=nodelist, weight=weight, format="csr")
+    n, _ = A.shape
+    diags = A.sum(axis=1)
+    # TODO: rm csr_array wrapper when spdiags can produce arrays
+    D = sp.sparse.csr_array(sp.sparse.spdiags(diags, 0, n, n, format="csr"))
+    L = D - A
+    with np.errstate(divide="ignore"):
+        diags_sqrt = 1.0 / np.sqrt(diags)
+    diags_sqrt[np.isinf(diags_sqrt)] = 0
+    # TODO: rm csr_array wrapper when spdiags can produce arrays
+    DH = sp.sparse.csr_array(sp.sparse.spdiags(diags_sqrt, 0, n, n, format="csr"))
+    return DH @ (L @ DH)
+
+
+@nx._dispatchable(edge_attrs="weight")
+def total_spanning_tree_weight(G, weight=None, root=None):
+    """
+    Returns the total weight of all spanning trees of `G`.
+
+    Kirchoff's Tree Matrix Theorem [1]_, [2]_ states that the determinant of any
+    cofactor of the Laplacian matrix of a graph is the number of spanning trees
+    in the graph. For a weighted Laplacian matrix, it is the sum across all
+    spanning trees of the multiplicative weight of each tree. That is, the
+    weight of each tree is the product of its edge weights.
+
+    For unweighted graphs, the total weight equals the number of spanning trees in `G`.
+
+    For directed graphs, the total weight follows by summing over all directed
+    spanning trees in `G` that start in the `root` node [3]_.
+
+    .. deprecated:: 3.3
+
+       ``total_spanning_tree_weight`` is deprecated and will be removed in v3.5.
+       Use ``nx.number_of_spanning_trees(G)`` instead.
+
+    Parameters
+    ----------
+    G : NetworkX Graph
+
+    weight : string or None, optional (default=None)
+        The key for the edge attribute holding the edge weight.
+        If None, then each edge has weight 1.
+
+    root : node (only required for directed graphs)
+       A node in the directed graph `G`.
+
+    Returns
+    -------
+    total_weight : float
+        Undirected graphs:
+            The sum of the total multiplicative weights for all spanning trees in `G`.
+        Directed graphs:
+            The sum of the total multiplicative weights for all spanning trees of `G`,
+            rooted at node `root`.
+
+    Raises
+    ------
+    NetworkXPointlessConcept
+        If `G` does not contain any nodes.
+
+    NetworkXError
+        If the graph `G` is not (weakly) connected,
+        or if `G` is directed and the root node is not specified or not in G.
+
+    Examples
+    --------
+    >>> G = nx.complete_graph(5)
+    >>> round(nx.total_spanning_tree_weight(G))
+    125
+
+    >>> G = nx.Graph()
+    >>> G.add_edge(1, 2, weight=2)
+    >>> G.add_edge(1, 3, weight=1)
+    >>> G.add_edge(2, 3, weight=1)
+    >>> round(nx.total_spanning_tree_weight(G, "weight"))
+    5
+
+    Notes
+    -----
+    Self-loops are excluded. Multi-edges are contracted in one edge
+    equal to the sum of the weights.
+
+    References
+    ----------
+    .. [1] Wikipedia
+       "Kirchhoff's theorem."
+       https://en.wikipedia.org/wiki/Kirchhoff%27s_theorem
+    .. [2] Kirchhoff, G. R.
+        Über die Auflösung der Gleichungen, auf welche man
+        bei der Untersuchung der linearen Vertheilung
+        Galvanischer Ströme geführt wird
+        Annalen der Physik und Chemie, vol. 72, pp. 497-508, 1847.
+    .. [3] Margoliash, J.
+        "Matrix-Tree Theorem for Directed Graphs"
+        https://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Margoliash.pdf
+    """
+    import warnings
+
+    warnings.warn(
+        (
+            "\n\ntotal_spanning_tree_weight is deprecated and will be removed in v3.5.\n"
+            "Use `nx.number_of_spanning_trees(G)` instead."
+        ),
+        category=DeprecationWarning,
+        stacklevel=3,
+    )
+
+    return nx.number_of_spanning_trees(G, weight=weight, root=root)
+
+
+###############################################################################
+# Code based on work from https://github.com/bjedwards
+
+
+@not_implemented_for("undirected")
+@not_implemented_for("multigraph")
+@nx._dispatchable(edge_attrs="weight")
+def directed_laplacian_matrix(
+    G, nodelist=None, weight="weight", walk_type=None, alpha=0.95
+):
+    r"""Returns the directed Laplacian matrix of G.
+
+    The graph directed Laplacian is the matrix
+
+    .. math::
+
+        L = I - \frac{1}{2} \left (\Phi^{1/2} P \Phi^{-1/2} + \Phi^{-1/2} P^T \Phi^{1/2} \right )
+
+    where `I` is the identity matrix, `P` is the transition matrix of the
+    graph, and `\Phi` a matrix with the Perron vector of `P` in the diagonal and
+    zeros elsewhere [1]_.
+
+    Depending on the value of walk_type, `P` can be the transition matrix
+    induced by a random walk, a lazy random walk, or a random walk with
+    teleportation (PageRank).
+
+    Parameters
+    ----------
+    G : DiGraph
+       A NetworkX graph
+
+    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().
+
+    weight : string or None, optional (default='weight')
+       The edge data key used to compute each value in the matrix.
+       If None, then each edge has weight 1.
+
+    walk_type : string or None, optional (default=None)
+       One of ``"random"``, ``"lazy"``, or ``"pagerank"``. If ``walk_type=None``
+       (the default), then a value is selected according to the properties of `G`:
+       - ``walk_type="random"`` if `G` is strongly connected and aperiodic
+       - ``walk_type="lazy"`` if `G` is strongly connected but not aperiodic
+       - ``walk_type="pagerank"`` for all other cases.
+
+    alpha : real
+       (1 - alpha) is the teleportation probability used with pagerank
+
+    Returns
+    -------
+    L : NumPy matrix
+      Normalized Laplacian of G.
+
+    Notes
+    -----
+    Only implemented for DiGraphs
+
+    The result is always a symmetric matrix.
+
+    This calculation uses the out-degree of the graph `G`. To use the
+    in-degree for calculations instead, use `G.reverse(copy=False)` and
+    take the transpose.
+
+    See Also
+    --------
+    laplacian_matrix
+    normalized_laplacian_matrix
+    directed_combinatorial_laplacian_matrix
+
+    References
+    ----------
+    .. [1] Fan Chung (2005).
+       Laplacians and the Cheeger inequality for directed graphs.
+       Annals of Combinatorics, 9(1), 2005
+    """
+    import numpy as np
+    import scipy as sp
+
+    # NOTE: P has type ndarray if walk_type=="pagerank", else csr_array
+    P = _transition_matrix(
+        G, nodelist=nodelist, weight=weight, walk_type=walk_type, alpha=alpha
+    )
+
+    n, m = P.shape
+
+    evals, evecs = sp.sparse.linalg.eigs(P.T, k=1)
+    v = evecs.flatten().real
+    p = v / v.sum()
+    # p>=0 by Perron-Frobenius Thm. Use abs() to fix roundoff across zero gh-6865
+    sqrtp = np.sqrt(np.abs(p))
+    Q = (
+        # TODO: rm csr_array wrapper when spdiags creates arrays
+        sp.sparse.csr_array(sp.sparse.spdiags(sqrtp, 0, n, n))
+        @ P
+        # TODO: rm csr_array wrapper when spdiags creates arrays
+        @ sp.sparse.csr_array(sp.sparse.spdiags(1.0 / sqrtp, 0, n, n))
+    )
+    # NOTE: This could be sparsified for the non-pagerank cases
+    I = np.identity(len(G))
+
+    return I - (Q + Q.T) / 2.0
+
+
+@not_implemented_for("undirected")
+@not_implemented_for("multigraph")
+@nx._dispatchable(edge_attrs="weight")
+def directed_combinatorial_laplacian_matrix(
+    G, nodelist=None, weight="weight", walk_type=None, alpha=0.95
+):
+    r"""Return the directed combinatorial Laplacian matrix of G.
+
+    The graph directed combinatorial Laplacian is the matrix
+
+    .. math::
+
+        L = \Phi - \frac{1}{2} \left (\Phi P + P^T \Phi \right)
+
+    where `P` is the transition matrix of the graph and `\Phi` a matrix
+    with the Perron vector of `P` in the diagonal and zeros elsewhere [1]_.
+
+    Depending on the value of walk_type, `P` can be the transition matrix
+    induced by a random walk, a lazy random walk, or a random walk with
+    teleportation (PageRank).
+
+    Parameters
+    ----------
+    G : DiGraph
+       A NetworkX graph
+
+    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().
+
+    weight : string or None, optional (default='weight')
+       The edge data key used to compute each value in the matrix.
+       If None, then each edge has weight 1.
+
+    walk_type : string or None, optional (default=None)
+        One of ``"random"``, ``"lazy"``, or ``"pagerank"``. If ``walk_type=None``
+        (the default), then a value is selected according to the properties of `G`:
+        - ``walk_type="random"`` if `G` is strongly connected and aperiodic
+        - ``walk_type="lazy"`` if `G` is strongly connected but not aperiodic
+        - ``walk_type="pagerank"`` for all other cases.
+
+    alpha : real
+       (1 - alpha) is the teleportation probability used with pagerank
+
+    Returns
+    -------
+    L : NumPy matrix
+      Combinatorial Laplacian of G.
+
+    Notes
+    -----
+    Only implemented for DiGraphs
+
+    The result is always a symmetric matrix.
+
+    This calculation uses the out-degree of the graph `G`. To use the
+    in-degree for calculations instead, use `G.reverse(copy=False)` and
+    take the transpose.
+
+    See Also
+    --------
+    laplacian_matrix
+    normalized_laplacian_matrix
+    directed_laplacian_matrix
+
+    References
+    ----------
+    .. [1] Fan Chung (2005).
+       Laplacians and the Cheeger inequality for directed graphs.
+       Annals of Combinatorics, 9(1), 2005
+    """
+    import scipy as sp
+
+    P = _transition_matrix(
+        G, nodelist=nodelist, weight=weight, walk_type=walk_type, alpha=alpha
+    )
+
+    n, m = P.shape
+
+    evals, evecs = sp.sparse.linalg.eigs(P.T, k=1)
+    v = evecs.flatten().real
+    p = v / v.sum()
+    # NOTE: could be improved by not densifying
+    # TODO: Rm csr_array wrapper when spdiags array creation becomes available
+    Phi = sp.sparse.csr_array(sp.sparse.spdiags(p, 0, n, n)).toarray()
+
+    return Phi - (Phi @ P + P.T @ Phi) / 2.0
+
+
+def _transition_matrix(G, nodelist=None, weight="weight", walk_type=None, alpha=0.95):
+    """Returns the transition matrix of G.
+
+    This is a row stochastic giving the transition probabilities while
+    performing a random walk on the graph. Depending on the value of walk_type,
+    P can be the transition matrix induced by a random walk, a lazy random walk,
+    or a random walk with teleportation (PageRank).
+
+    Parameters
+    ----------
+    G : DiGraph
+       A NetworkX graph
+
+    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().
+
+    weight : string or None, optional (default='weight')
+       The edge data key used to compute each value in the matrix.
+       If None, then each edge has weight 1.
+
+    walk_type : string or None, optional (default=None)
+       One of ``"random"``, ``"lazy"``, or ``"pagerank"``. If ``walk_type=None``
+       (the default), then a value is selected according to the properties of `G`:
+        - ``walk_type="random"`` if `G` is strongly connected and aperiodic
+        - ``walk_type="lazy"`` if `G` is strongly connected but not aperiodic
+        - ``walk_type="pagerank"`` for all other cases.
+
+    alpha : real
+       (1 - alpha) is the teleportation probability used with pagerank
+
+    Returns
+    -------
+    P : numpy.ndarray
+      transition matrix of G.
+
+    Raises
+    ------
+    NetworkXError
+        If walk_type not specified or alpha not in valid range
+    """
+    import numpy as np
+    import scipy as sp
+
+    if walk_type is None:
+        if nx.is_strongly_connected(G):
+            if nx.is_aperiodic(G):
+                walk_type = "random"
+            else:
+                walk_type = "lazy"
+        else:
+            walk_type = "pagerank"
+
+    A = nx.to_scipy_sparse_array(G, nodelist=nodelist, weight=weight, dtype=float)
+    n, m = A.shape
+    if walk_type in ["random", "lazy"]:
+        # TODO: Rm csr_array wrapper when spdiags array creation becomes available
+        DI = sp.sparse.csr_array(sp.sparse.spdiags(1.0 / A.sum(axis=1), 0, n, n))
+        if walk_type == "random":
+            P = DI @ A
+        else:
+            # TODO: Rm csr_array wrapper when identity array creation becomes available
+            I = sp.sparse.csr_array(sp.sparse.identity(n))
+            P = (I + DI @ A) / 2.0
+
+    elif walk_type == "pagerank":
+        if not (0 < alpha < 1):
+            raise nx.NetworkXError("alpha must be between 0 and 1")
+        # this is using a dense representation. NOTE: This should be sparsified!
+        A = A.toarray()
+        # add constant to dangling nodes' row
+        A[A.sum(axis=1) == 0, :] = 1 / n
+        # normalize
+        A = A / A.sum(axis=1)[np.newaxis, :].T
+        P = alpha * A + (1 - alpha) / n
+    else:
+        raise nx.NetworkXError("walk_type must be random, lazy, or pagerank")
+
+    return P

wemm/lib/python3.10/site-packages/networkx/linalg/spectrum.py

@@ -0,0 +1,186 @@
+"""
+Eigenvalue spectrum of graphs.
+"""
+
+import networkx as nx
+
+__all__ = [
+    "laplacian_spectrum",
+    "adjacency_spectrum",
+    "modularity_spectrum",
+    "normalized_laplacian_spectrum",
+    "bethe_hessian_spectrum",
+]
+
+
+@nx._dispatchable(edge_attrs="weight")
+def laplacian_spectrum(G, weight="weight"):
+    """Returns eigenvalues of the Laplacian of G
+
+    Parameters
+    ----------
+    G : graph
+       A NetworkX graph
+
+    weight : string or None, optional (default='weight')
+       The edge data key used to compute each value in the matrix.
+       If None, then each edge has weight 1.
+
+    Returns
+    -------
+    evals : NumPy array
+      Eigenvalues
+
+    Notes
+    -----
+    For MultiGraph/MultiDiGraph, the edges weights are summed.
+    See :func:`~networkx.convert_matrix.to_numpy_array` for other options.
+
+    See Also
+    --------
+    laplacian_matrix
+
+    Examples
+    --------
+    The multiplicity of 0 as an eigenvalue of the laplacian matrix is equal
+    to the number of connected components of G.
+
+    >>> G = nx.Graph()  # Create a graph with 5 nodes and 3 connected components
+    >>> G.add_nodes_from(range(5))
+    >>> G.add_edges_from([(0, 2), (3, 4)])
+    >>> nx.laplacian_spectrum(G)
+    array([0., 0., 0., 2., 2.])
+
+    """
+    import scipy as sp
+
+    return sp.linalg.eigvalsh(nx.laplacian_matrix(G, weight=weight).todense())
+
+
+@nx._dispatchable(edge_attrs="weight")
+def normalized_laplacian_spectrum(G, weight="weight"):
+    """Return eigenvalues of the normalized Laplacian of G
+
+    Parameters
+    ----------
+    G : graph
+       A NetworkX graph
+
+    weight : string or None, optional (default='weight')
+       The edge data key used to compute each value in the matrix.
+       If None, then each edge has weight 1.
+
+    Returns
+    -------
+    evals : NumPy array
+      Eigenvalues
+
+    Notes
+    -----
+    For MultiGraph/MultiDiGraph, the edges weights are summed.
+    See to_numpy_array for other options.
+
+    See Also
+    --------
+    normalized_laplacian_matrix
+    """
+    import scipy as sp
+
+    return sp.linalg.eigvalsh(
+        nx.normalized_laplacian_matrix(G, weight=weight).todense()
+    )
+
+
+@nx._dispatchable(edge_attrs="weight")
+def adjacency_spectrum(G, weight="weight"):
+    """Returns eigenvalues of the adjacency matrix of G.
+
+    Parameters
+    ----------
+    G : graph
+       A NetworkX graph
+
+    weight : string or None, optional (default='weight')
+       The edge data key used to compute each value in the matrix.
+       If None, then each edge has weight 1.
+
+    Returns
+    -------
+    evals : NumPy array
+      Eigenvalues
+
+    Notes
+    -----
+    For MultiGraph/MultiDiGraph, the edges weights are summed.
+    See to_numpy_array for other options.
+
+    See Also
+    --------
+    adjacency_matrix
+    """
+    import scipy as sp
+
+    return sp.linalg.eigvals(nx.adjacency_matrix(G, weight=weight).todense())
+
+
+@nx._dispatchable
+def modularity_spectrum(G):
+    """Returns eigenvalues of the modularity matrix of G.
+
+    Parameters
+    ----------
+    G : Graph
+       A NetworkX Graph or DiGraph
+
+    Returns
+    -------
+    evals : NumPy array
+      Eigenvalues
+
+    See Also
+    --------
+    modularity_matrix
+
+    References
+    ----------
+    .. [1] M. E. J. Newman, "Modularity and community structure in networks",
+       Proc. Natl. Acad. Sci. USA, vol. 103, pp. 8577-8582, 2006.
+    """
+    import scipy as sp
+
+    if G.is_directed():
+        return sp.linalg.eigvals(nx.directed_modularity_matrix(G))
+    else:
+        return sp.linalg.eigvals(nx.modularity_matrix(G))
+
+
+@nx._dispatchable
+def bethe_hessian_spectrum(G, r=None):
+    """Returns eigenvalues of the Bethe Hessian matrix of G.
+
+    Parameters
+    ----------
+    G : Graph
+       A NetworkX Graph or DiGraph
+
+    r : float
+       Regularizer parameter
+
+    Returns
+    -------
+    evals : NumPy array
+      Eigenvalues
+
+    See Also
+    --------
+    bethe_hessian_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.
+    """
+    import scipy as sp
+
+    return sp.linalg.eigvalsh(nx.bethe_hessian_matrix(G, r).todense())