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# Source code for networkx.linalg.laplacianmatrix

    """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