shCJSrSSKrSSKJr /SQr\R "SS9SSj5r\R "SS9SSj5r\"S 5\"S 5\R "SS9SS j555r \"S 5\"S 5\R "SS9SS j555r SS jr 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_matrixdirected_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. rNcsrnodelistrformataxisr) scipylistnxto_scipy_sparse_arrayshapesparse csr_arrayspdiagssum)Gr rspAnmDs s/Users/tiagomarins/Projetos/claudeai/copy_bank/venv/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) rr rnprrr_diagsr!L diags_sqrtDHs r"rrsB7   fUSA 77DA EEqEME BII--eQU-KLA A H %2775>)  &'(Jrxx #$   RYY..z1a5.Q RB R= & %s C?? D  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 rNr r walk_typealpharkr'@)r(r_transition_matrixrrlinalgeigsTflattenrealrr*absrridentitylen)rr rr6r7r,rPrr evalsevecsvpsqrtpQIs r"rrsP  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 rNr5rr8r:) rr;rrr<r=r>r?r@rrrtoarray)rr rr6r7rrDrr rErFrGrHPhis r"rrasN 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 $[R"S 5e)a*Returns the transition matrix of G. This is a row stochastic giving the transition probabilities while performing a random walk on the graph. Depending on the value of walk_type, P can be the transition matrix induced by a random walk, a lazy random walk, or a random walk with teleportation (PageRank). Parameters ---------- G : DiGraph A NetworkX graph nodelist : list, optional The rows and columns are ordered according to the nodes in nodelist. If nodelist is None, then the ordering is produced by G.nodes(). weight : string or None, optional (default='weight') The edge data key used to compute each value in the matrix. If None, then each edge has weight 1. walk_type : string or None, optional (default=None) One of ``"random"``, ``"lazy"``, or ``"pagerank"``. If ``walk_type=None`` (the default), then a value is selected according to the properties of `G`: - ``walk_type="random"`` if `G` is strongly connected and aperiodic - ``walk_type="lazy"`` if `G` is strongly connected but not aperiodic - ``walk_type="pagerank"`` for all other cases. alpha : real (1 - alpha) is the teleportation probability used with pagerank Returns ------- P : numpy.ndarray transition matrix of G. Raises ------ NetworkXError If walk_type not specified or alpha not in valid range rNrandomlazypagerank)r rdtype)rPrQr'rrr:zalpha must be between 0 and 1z+walk_type must be random, lazy, or pagerank)r(rris_strongly_connected is_aperiodicrfloatrrrrrrB NetworkXErrorrMnewaxisr>) rr rr6r7r,rrrr DIrDrKs r"r;r;sR # #A & &q!!$ " "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)NrNgffffff?) __doc__networkxrnetworkx.utilsr__all__ _dispatchablerrrrr;r#r"r`s. X&k'k\X&p'pn\"\"X&=A^'##^B\"\"X&=AS-'##S-lN r#