Gd4&dZddlZddlmZgdZedddZedddZdd Zed ed  dd Z ed ed  ddZ ddZ dS)zLaplacian matrix of graphs. N)not_implemented_for)laplacian_matrixnormalized_laplacian_matrixtotal_spanning_tree_weightdirected_laplacian_matrix'directed_combinatorial_laplacian_matrixdirectedweightc ddl}ddl}|t|}tj|||d}|j\}}|j|j| dd||d}||z S)aReturns 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. See Also -------- to_numpy_array normalized_laplacian_matrix 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]] rNcsrnodelistr formataxisr) scipy scipy.sparselistnxto_scipy_sparse_arrayshapesparse csr_arrayspdiagssum) Grr sprAnmDs ?lib/python3.11/site-packages/networkx/linalg/laplacianmatrix.pyrrsd77  XfUSSSA 7DAq BI--aeeemmQ1U-SSTTA q5Lc ^ddl}ddl}ddl}|t|}t j|||d}|j\}}|d} |j |j | d||d} | |z } | d 5d | | z } dddn #1swxYwYd| | | <|j |j | d||d} | | | zzS) aReturns 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]_. See Also -------- laplacian_matrix normalized_laplacian_spectrum 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. rNr r rrrignore)divide?)numpyrrrrrrrrrrerrstatesqrtisinf)rrr nprrr r!r"diagsr#L diags_sqrtDHs r$rrMsyh77  XfUSSSA 7DAq EEqEMME BI--eQ1U-KKLLA AA H % %**2775>>) ***************'(Jrxx ##$   RY..z1a5.QQ R RB R=s(C  CCcddl}tj||}t |j|ddddfS)as Returns the total weight of all spanning trees of `G`. Kirchoff's Tree Matrix Theorem 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. Parameters ---------- G : NetworkX Graph The graph to use Kirchhoff's theorem on. weight : string or None The key for the edge attribute holding the edge weight. If `None`, then each edge is assumed to have a weight of 1 and this function returns the total number of spanning trees in `G`. Returns ------- float The sum of the total multiplicative weights for all spanning trees in `G` rN)r r)r*rrtoarrayabslinalgdet)rr r. G_laplacians r$rrs^2%a777??AAK ry}}[QRR011 2 22r% undirected multigraphffffff?c ddl}ddl}ddl}ddl}t |||||}|j\} } |jj|j d\} } | j } | | z }| |}|j|j|d| | |z|j|jd|z d| | z}|t#|}|||j zdz z S)a>Returns the directed Laplacian matrix of G. The graph directed Laplacian is the matrix .. math:: L = I - (\Phi^{1/2} P \Phi^{-1/2} + \Phi^{-1/2} P^T \Phi^{1/2} ) / 2 where `I` is the identity matrix, `P` is the transition matrix of the graph, and `\Phi` a matrix with the Perron vector of `P` in the diagonal and zeros elsewhere [1]_. Depending on the value of walk_type, `P` can be the transition matrix induced by a random walk, a lazy random walk, or a random walk with teleportation (PageRank). Parameters ---------- G : DiGraph A NetworkX graph nodelist : list, optional The rows and columns are ordered according to the nodes in nodelist. If nodelist is None, then the ordering is produced by G.nodes(). weight : string or None, optional (default='weight') The edge data key used to compute each value in the matrix. If None, then each edge has weight 1. walk_type : string or None, optional (default=None) If None, `P` is selected depending on the properties of the graph. Otherwise is one of 'random', 'lazy', or 'pagerank' alpha : real (1 - alpha) is the teleportation probability used with pagerank Returns ------- L : NumPy matrix Normalized Laplacian of G. Notes ----- Only implemented for DiGraphs See Also -------- laplacian_matrix References ---------- .. [1] Fan Chung (2005). Laplacians and the Cheeger inequality for directed graphs. Annals of Combinatorics, 9(1), 2005 rNrr walk_typealpharkr)@)r*rrscipy.sparse.linalg_transition_matrixrrr6eigsTflattenrealrr,rridentitylen)rrr r>r?r.rrPr!r"evalsevecsvpsqrtpQIs r$rrsQx  HVy   A 7DAq9#(((22LE5 A AEEGG A GGAJJE BI--eQ1==>>   )  bi//e Q1EE F F G CFFA AC3 r%cddl}ddl}ddl}t|||||}|j\}} |jj|jd\} } | j } | | z } |j |j | d||}|||z|j|zzdz z S)aReturn the directed combinatorial Laplacian matrix of G. The graph directed combinatorial Laplacian is the matrix .. math:: L = \Phi - (\Phi P + P^T \Phi) / 2 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) If None, `P` is selected depending on the properties of the graph. Otherwise is one of 'random', 'lazy', or 'pagerank' alpha : real (1 - alpha) is the teleportation probability used with pagerank Returns ------- L : NumPy matrix Combinatorial Laplacian of G. Notes ----- Only implemented for DiGraphs See Also -------- laplacian_matrix References ---------- .. [1] Fan Chung (2005). Laplacians and the Cheeger inequality for directed graphs. Annals of Combinatorics, 9(1), 2005 rNr=rr@rB)rrrCrDrrr6rErFrGrHrrrr4)rrr r>r?rrrKr!r"rLrMrNrOPhis r$rrsv HVy   A 7DAq9#(((22LE5 A AEEGG A )  bi//1a;; < < D D F FC #'AC#I%, ,,r%cfddl}ddl}ddl}|0tj|rtj|rd}nd}nd}tj|||t}|j\} } |dvr|j |j d| d z d| | } |dkr| |z} n|j |j | } | | |zzd z } n|dkrd|cxkrd ksntjd |}d | z || d dkddf<|| d |jddfjz }||zd |z | z z} ntjd | S)aReturns the transition matrix of G. This is a row stochastic giving the transition probabilities while performing a random walk on the graph. Depending on the value of walk_type, P can be the transition matrix induced by a random walk, a lazy random walk, or a random walk with teleportation (PageRank). Parameters ---------- G : DiGraph A NetworkX graph nodelist : list, optional The rows and columns are ordered according to the nodes in nodelist. If nodelist is None, then the ordering is produced by G.nodes(). weight : string or None, optional (default='weight') The edge data key used to compute each value in the matrix. If None, then each edge has weight 1. walk_type : string or None, optional (default=None) If None, `P` is selected depending on the properties of the graph. Otherwise is one of 'random', 'lazy', or 'pagerank' alpha : real (1 - alpha) is the teleportation probability used with pagerank Returns ------- P : numpy.ndarray transition matrix of G. Raises ------ NetworkXError If walk_type not specified or alpha not in valid range rNrandomlazypagerank)rr dtype)rVrWr)rrrBzalpha must be between 0 and 1z+walk_type must be random, lazy, or pagerank)r*rrris_strongly_connected is_aperiodicrfloatrrrrrrI NetworkXErrorr4newaxisrF)rrr r>r?r.rrr r!r"DIrKrRs r$rDrDasL #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 )N)Nr Nr;) __doc__networkxrnetworkx.utilsr__all__rrrrrrDr%r$resi......   Z  :::! :zZ  DDD! DN3333H\""\""=ATTT#"#"Tn\""\""=AJ-J-J-#"#"J-ZL L L L L L r%