h8PzSrSSKrSSKJr /SQr\R "SS9SSj5r\R "SS9SSj5r\R "SS9SS j5r \"S 5\"S 5\R "SS9SS j555r \"S 5\"S 5\R "SS9SS j555r SSjr g)a{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. N)not_implemented_for)laplacian_matrixnormalized_laplacian_matrixtotal_spanning_tree_weightdirected_laplacian_matrix'directed_combinatorial_laplacian_matrixweight) edge_attrsc SSKnUc [U5n[R"XUSS9nURupVUR R UR RURSS9SXeSS95nXt- $)u 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. rNcsrnodelistr formataxisr) scipylistnxto_scipy_sparse_arrayshapesparse csr_arrayspdiagssum)Grr spAnmDs Q/var/www/html/env/lib/python3.13/site-packages/networkx/linalg/laplacianmatrix.pyrrssH7   fUSA 77DA BII--aeeemQU-STA 5Lc  SSKnSSKnUc [U5n[R"XUSS9nUR upgUR SS9nURRURRUSXfSS95n X- n URSS 9 S URU5- n SSS5 SW URU 5'URRURRU SXfSS95n XU --$!,(df  N^=f) u 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. rNr r rrrignore)divide?) numpyrrrrrrrrrerrstatesqrtisinf) rrr nprrr _diagsr"L diags_sqrtDHs r#rrsB7   fUSA 77DA EEqEME BII--eQU-KLA A H %2775>)  &'(Jrxx #$   RYY..z1a5.Q RB R= & %s C?? D c^SSKnURS[SS9 [R"XUS9$)u~ 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 rNzu total_spanning_tree_weight is deprecated and will be removed in v3.5. Use `nx.number_of_spanning_trees(G)` instead.)category stacklevel)r root)warningswarnDeprecationWarningrnumber_of_spanning_trees)rr r7r8s r#rrs;d MM <$   & &qd CCr$ undirected multigraphc ~SSKnSSKn[XX#US9nURupURR R URSS9upU R5Rn XR5- n URURU 55nURRURRUSX55U-URRURRSU- SX55-nUR[!U55nUXR-S- - $)aReturns 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 rNrr walk_typealpharkr(@)r)r_transition_matrixrrlinalgeigsTflattenrealrr+absrridentitylen)rrr r@rAr-rPr r!evalsevecsvpsqrtpQIs r#rr^sP  V A 77DA99##(((2LE A EEG A GGBFF1I E BII--eQ=>   ))  bii//e QE F G CFA CC3 r$cSSKn[XX#US9nURupxURRR UR SS9upU R5Rn XR5- n URRURRU SXw55R5n XU-UR U --S- - $)atReturn the directed combinatorial Laplacian matrix of G. The graph directed combinatorial Laplacian is the matrix .. math:: L = \Phi - \frac{1}{2} \left (\Phi P + P^T \Phi \right) where `P` is the transition matrix of the graph and `\Phi` a matrix with the Perron vector of `P` in the diagonal and zeros elsewhere [1]_. Depending on the value of walk_type, `P` can be the transition matrix induced by a random walk, a lazy random walk, or a random walk with teleportation (PageRank). Parameters ---------- G : DiGraph A NetworkX graph nodelist : list, optional The rows and columns are ordered according to the nodes in nodelist. If nodelist is None, then the ordering is produced by G.nodes(). weight : string or None, optional (default='weight') The edge data key used to compute each value in the matrix. If None, then each edge has weight 1. walk_type : string or None, optional (default=None) One of ``"random"``, ``"lazy"``, or ``"pagerank"``. If ``walk_type=None`` (the default), then a value is selected according to the properties of `G`: - ``walk_type="random"`` if `G` is strongly connected and aperiodic - ``walk_type="lazy"`` if `G` is strongly connected but not aperiodic - ``walk_type="pagerank"`` for all other cases. alpha : real (1 - alpha) is the teleportation probability used with pagerank Returns ------- L : NumPy matrix Combinatorial Laplacian of G. Notes ----- Only implemented for DiGraphs The result is always a symmetric matrix. This calculation uses the out-degree of the graph `G`. To use the in-degree for calculations instead, use `G.reverse(copy=False)` and take the transpose. See Also -------- laplacian_matrix normalized_laplacian_matrix directed_laplacian_matrix References ---------- .. [1] Fan Chung (2005). Laplacians and the Cheeger inequality for directed graphs. Annals of Combinatorics, 9(1), 2005 rNr?rrBrD) rrErrrFrGrHrIrJrrrtoarray)rrr r@rArrNr r!rOrPrQrRPhis r#rrsN V A 77DA99##(((2LE A EEG A ))  bii//1a; < D D FC 'ACC#I%, ,,r$c dSSKnSSKnUc>[R"U5(a![R"U5(aSnOSnOSn[R "XU[ S9nURupUS;aURRURRSURS S 9- SX55n US:XaX-n U $URRURRU55n XU--S - n U $US:XaSUs=:aS :dO [R"S 5eUR5nS U- XwRS S 9S:HSS24'XwRS S 9URSS24R - nXG-S U- U- -n U $[R"S 5e)a*Returns the transition matrix of G. This is a row stochastic giving the transition probabilities while performing a random walk on the graph. Depending on the value of walk_type, P can be the transition matrix induced by a random walk, a lazy random walk, or a random walk with teleportation (PageRank). Parameters ---------- G : DiGraph A NetworkX graph nodelist : list, optional The rows and columns are ordered according to the nodes in nodelist. If nodelist is None, then the ordering is produced by G.nodes(). weight : string or None, optional (default='weight') The edge data key used to compute each value in the matrix. If None, then each edge has weight 1. walk_type : string or None, optional (default=None) One of ``"random"``, ``"lazy"``, or ``"pagerank"``. If ``walk_type=None`` (the default), then a value is selected according to the properties of `G`: - ``walk_type="random"`` if `G` is strongly connected and aperiodic - ``walk_type="lazy"`` if `G` is strongly connected but not aperiodic - ``walk_type="pagerank"`` for all other cases. alpha : real (1 - alpha) is the teleportation probability used with pagerank Returns ------- P : numpy.ndarray transition matrix of G. Raises ------ NetworkXError If walk_type not specified or alpha not in valid range rNrandomlazypagerank)rr dtype)rZr[r(rrrDzalpha must be between 0 and 1z+walk_type must be random, lazy, or pagerank)r)rris_strongly_connected is_aperiodicrfloatrrrrrrL NetworkXErrorrWnewaxisrH) rrr r@rAr-rrr r!DIrNrUs r#rErEsR # #A & &q!!$ " "I   fERA 77DA&& YY !2!23A3F1!P Q  A& H! ##BII$6$6q$9:A!Vs"A H j E A ""#BC C IIK#$q5%%Q%-1 a  1 bjj!m,.. . IUa ' HLMMr$)Nr )NN)Nr Ngffffff?) __doc__networkxrnetworkx.utilsr__all__ _dispatchablerrrrrrEr$r#rjs. X&k'k\X&p'pfX&\D'\DF\"\"X&=A^'##^B\"\"X&=AS-'##S-lN r$