VhCXdZddlZddlmZgdZej dddZej dddZed ed ej d dd Z ed ed ej d dd Z dd Z y)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 |j j|jdd||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 O/home/dcms/DCMS/lib/python3.12/site-packages/networkx/linalg/laplacianmatrix.pyrrs{H7   XfUSA 77DAq BII--aeeemQ1U-STA q5Lc <ddl}ddl}| t|}tj|||d}|j \}}|j d}|jj|jj|d||d} | |z } |jd 5d |j|z } dddd |j| <|jj|jj| d||d} | | | zzS#1swY^xYw) 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   XfUSA 77DAq EEqEME BII--eQ1U-KLA AA H %*2775>) *'(Jrxx #$   RYY..z1a5.Q RB R= **s DD undirected multigraphc ddl}ddl}t|||||}|j\}} |jj j |jd\} } | jj} | | jz } |j|j| }|jj|jj|d|||z|jj|jjd|z d||z}|jt!|}|||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 rr6r7r,rPrr evalsevecsvpsqrtpQIs r"rrs"P  HVy A 77DAq99##(((2LE5 A AEEG A GGBFF1I E BII--eQ1=>   ))  bii//e Q1E F G CFA ACC3 r#cddl}t|||||}|j\}}|jjj |j d\} } | jj} | | jz } |jj|jj| d||j} | | |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 rNr5rr8r:) rr;rrr<r=r>r?r@rrrtoarray)rr rr6r7rrDrr rErFrGrHPhis r"rrasN HVy A 77DAq99##(((2LE5 A AEEG A ))  bii//1a; < D D FC #'ACC#I%, ,,r#c^ddl}ddl}|2tj|rtj|rd}nd}nd}tj |||t }|j\}} |dvr|jj|jjd|jd z d||} |dk(r| |z} | S|jj|jj|} | | |zzd z } | S|dk(rd|cxkrd ksntjd |j}d |z ||jd dk(ddf<||jd |jddfj z }||zd |z |z z} | Stjd )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)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   XfERA 77DAq&& YY !2!23A3F1a!P Q  QA& H! ##BII$6$6q$9:AR!Vs"A H j E A ""#BC C IIK#$q5!%%Q%-1 a  1 bjj!m,.. . AIUa ' 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#