\MhCdZddlZddlmZgdZejdddZejdddZed ed ejd dd Z ed ed ejd dd Z ddZ dS)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 ddl}|t|}tj|||d}|j\}}|j|j|dd||d}||z S)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 _/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/networkx/linalg/laplacianmatrix.pyrrsH77  XfUSSSA 7DAq BI--aeeemmQ1U-SSTTA q5Lc Vddl}ddl}|t|}tj|||d}|j\}}|d}|j|j |d||d} | |z } | d 5d | |z } dddn #1swxYwYd| | | <|j|j | d||d} | | | zzS) 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"rrsmB77  XfUSSSA 7DAq EEqEMME BI--eQ1U-KKLLA AA H % %**2775>>) ***************'(Jrxx ##$   RY..z1a5.QQ R RB R=s$C  C C  undirected multigraphffffff?c ddl}ddl}t|||||}|j\}} |jj|jd\} } | j } | | z } | | | }|j |j|d|||z|j |jd|z d||z}|t!|}|||jzdz z S)abReturns 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 rr7r8r,rPrr evalsevecsvpsqrtpQIs r"rrsEP  HVy   A 7DAq9#(((22LE5 A AEEGG A GGBFF1II  E BI--eQ1==>>   )  bi//e Q1EE F F G CFFA AC3 r#cddl}t|||||}|j\}}|jj|jd\} } | j} | | z } |j |j | d|| } | | |z|j| zzdz z S)a4Return 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 rNr6rr9r;) rr<rrr=r>r?r@rArrrtoarray)rr rr7r8rrErr rFrGrHrIPhis r"rrasN HVy   A 7DAq9#(((22LE5 A AEEGG A )  bi//1a;; < < D D F FC #'AC#I%, ,,r#c^ddl}ddl}|0tj|rtj|rd}nd}nd}tj|||t }|j\}} |dvr|j |j d| d z d||} |dkr| |z} n|j |j |} | | |zzd z } n|dkrd|cxkrd ksntj d |}d |z || d dkddf<|| d |jddfjz }||zd |z |z z} ntj d | S)aReturns 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)rQrRr'rrr;zalpha must be between 0 and 1z+walk_type must be random, lazy, or pagerank)r(rris_strongly_connected is_aperiodicrfloatrrrrrrC NetworkXErrorrNnewaxisr?) rr rr7r8r,rrrr DIrErLs r"r<r<sR #A & & #q!! #$ " "I  XfERRRA 7DAq&&& Y !2!23A3F1a!P!P Q Q  QAA ##BI$6$6q$9$9::AR!Vs"AA j E A "#BCC C IIKK#$q5!%%Q%--1 aaa  1 bj!!!m,. . AIUa 'LMMM Hr#)Nr)NrNr4) __doc__networkxrnetworkx.utilsr__all__ _dispatchablerrrrr<r#r"ras......   X&&&kkk'&k\X&&&ppp'&pn\""\""X&&&=A^^^'&#"#"^B\""\""X&&&=AS-S-S-'&#"#"S-lN N N N N N r#