B ,[P@s~dZddlZddlmZdddddgZd d d gZed ddd Zed ddd Z ededddd Z ddZ dS)zLaplacian matrix of graphs. N)not_implemented_for z%Aric Hagberg zPieter Swart (swart@lanl.gov)z Dan Schult (dschult@colgate.edu)z3Alejandro Weinstein laplacian_matrixnormalized_laplacian_matrixdirected_laplacian_matrixZdirectedweightc Csdddl}|dkrt|}tj|||dd}|j\}}|jdd}|jj|dg||dd}||S)aEReturn 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_matrix normalized_laplacian_matrix rNcsr)nodelistrformat)axis)r ) scipy.sparselistnxto_scipy_sparse_matrixshapesumsparsespdiagsflatten) Gr rscipyAnmdiagsDr>lib/python3.7/site-packages/networkx/linalg/laplacianmatrix.pyrs"   c Csddl}ddl}|dkr t|}tj|||dd}|j\}}|jdd}|jj |dg||dd}||} |j dd d | |} WdQRXd| | | <|jj | dg||dd} | | | S) aRReturn 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 : NumPy matrix The normalized Laplacian matrix of G. Notes ----- For MultiGraph/MultiDiGraph, the edges weights are summed. See to_numpy_matrix 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 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)Zdivideg?)rr rrrrrrrrZerrstatesqrtZisinfdot) rr rrrrrrrLZ diags_sqrtZDHrrrrAs3  Z undirectedZ multigraphffffff?cCsddl}ddlm}m}m}|dkrHt|rDt|r>d}qHd}nd}tj|||t d} | j \} } |dkr|d | | j d d j dg| | } |dkr| | } n|| }|| | d } n|dkrJd|krd ksntd | } || j d d dk}x|dD]}d | | |<qW| | j d d } || d || } n td|j| jd d\}}|j}|| }||}||dg| | | |d |dg| | }|t|}|||jd S)aReturn 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 array Normalized Laplacian of G. Raises ------ NetworkXError If NumPy cannot be imported NetworkXNotImplemnted If G is not a DiGraph 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)identityrlinalgrandomlazyZpagerank)r rZdtype)r&r'g?r )r g@zalpha must be between 0 and 1z+walk_type must be random, lazy, or pagerank)k)rr r$rr%rZis_strongly_connectedZ is_aperiodicrfloatrZarrayrZflatZ NetworkXErrorZtodensewhereZeigsTrrealr len)rr rZ walk_typeZalphaZspr$rr%MrrZDIPIZdanglingdZevalsZevecsvpZsqrtpQrrrrsDC    $       (cCs2ddlm}y ddl}Wn|dYnXdS)Nr)SkipTestzNumPy not available)Znoser5numpy)moduler5r6rrr setup_modules   r8)Nr)Nr)NrNr#) __doc__ZnetworkxrZnetworkx.utilsrjoin __author____all__rrrr8rrrrs$  , Gq