3 Md)@sdZddlZddlmZddddgZeddd dZeddd dZed ed dddZed ed dddZ dddZ dS)zLaplacian matrix of graphs. N)not_implemented_forlaplacian_matrixnormalized_laplacian_matrixdirected_laplacian_matrix'directed_combinatorial_laplacian_matrixZdirectedweightc Csdddl}|dkrt|}tj|||dd}|j\}}|jdd}|jj|jdg||dd}||S)a\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 matrix The Laplacian matrix of G. Notes ----- For MultiGraph/MultiDiGraph, the edges weights are summed. See Also -------- to_numpy_array normalized_laplacian_matrix laplacian_spectrum rNcsr)nodelistrformat)axis)r ) scipy.sparselistnxto_scipy_sparse_matrixshapesumsparsespdiagsflatten) Gr rscipyAnmdiagsDrQ/var/www/html/virt/lib/python3.6/site-packages/networkx/linalg/laplacianmatrix.pyrs#  c Csddl}ddl}ddl}|dkr(t|}tj|||dd}|j\}}|jddj}|j j |dg||dd} | |} |j dd d |j |} WdQRXd| |j | <|j j | dg||dd} | j| j| S) a{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. 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 matrix The normalized Laplacian matrix of G. Notes ----- For MultiGraph/MultiDiGraph, the edges weights are summed. See 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 rr r )r )r ignore)divideg?)numpyrr rrrrrrrrZerrstatesqrtisinfdot) rr rnprrrrrrLZ diags_sqrtZDHrrrr<s4 Z undirectedZ multigraphffffff?cCsddl}ddlm}m}t|||||d}|j\} } |j|jdd\} } | jj } | | j }|j |}||dg| | ||d|dg| | }|j t |}|||jdS) a9Returns 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. 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 rN)rlinalg)r r walk_typealphar )kg?g@)r!r rr(_transition_matrixreigsTrrealrr"identitylen)rr rr)r*r%rr(PrrevalsevecsvpZsqrtpQIrrrrs<    (cCsddlm}m}t|||||d}|j\}} |j|jdd\} } | jj} | | j } || dg||}|j }||||j|dS)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 and `\Phi` a matrix with the Perron vector of `P` in the diagonal and zeros elsewhere. 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 r)rr()r rr)r*r )r+g@) r rr(r,rr-r.rr/rtodense)rr rr)r*rr(r2rrr3r4r5r6Phirrrrs;   cCsVddl}ddlm}m}|dkrDtj|r@tj|r:d}qDd}nd}tj|||td}|j \} } |dkr|d|j |j d d j dg| | } |dkr| |} n|| } | | |d } n|dkrHd|kod knstj d |j}|j|j d d dk}x|dD]}d| ||<q W||j d d }||d || } n tj 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) 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 matrix transition matrix of G. Raises ------ NetworkXError If walk_type not specified or alpha not in valid range rN)r0rrandomlazyZpagerank)r rZdtypeg?r )r g@zalpha must be between 0 and 1z+walk_type must be random, lazy, or pagerank)r;r<)r!r r0rrZis_strongly_connectedZ is_aperiodicrfloatrarrayrZflatZ NetworkXErrorr9where)rr rr)r*r%r0rMrrZDIr2r8Zdanglingdrrrr,#s6&   $    r,)Nr)Nr)NrNr')NrNr')NrNr') __doc__ZnetworkxrZnetworkx.utilsr__all__rrrrr,rrrrs"  - JLJ