hRSrSSKJr SSKrSSKJrJrJr /SQr "SS5r "SS 5r S r S r S rS r\"S5\"S5\R "SS9SSj555r\"S5\"S5\R "SS9SSj555r\"S5\R "SS9SSj55r\R "SS9SSj5rg)zB Algebraic connectivity and Fiedler vectors of undirected graphs. )partialN)not_implemented_fornp_random_statereverse_cuthill_mckee_ordering)algebraic_connectivityfiedler_vectorspectral_orderingspectral_bisectionc*\rSrSrSrSrSrSrSrg) _PCGSolveraPreconditioned conjugate gradient method. To solve Ax = b: M = A.diagonal() # or some other preconditioner solver = _PCGSolver(lambda x: A * x, lambda x: M * x) x = solver.solve(b) The inputs A and M are functions which compute matrix multiplication on the argument. A - multiply by the matrix A in Ax=b M - multiply by M, the preconditioner surrogate for A Warning: There is no limit on number of iterations. cXlX lgN_A_M)selfAMs W/var/www/html/env/lib/python3.13/site-packages/networkx/linalg/algebraicconnectivity.py__init___PCGSolver.__init__&s cSSKnURU5nURURSS9n[ URS5H#nUR USS2U4U5USS2U4'M% U$NrForder)numpyasarrayndarrayshaperange_solverBtolnpXjs rsolve_PCGSolver.solve*sf JJqM JJqwwcJ *qwwqz"Akk!AqD'3/AadG#rcXSSKnSSKnURnURnX$RR R U5-nURUR5nUR5nU"U5n URR RX5n U R5n U"U 5n XRR RX5- n URR RXU S9nURR RXU *S9nURR R U5U:aU$U"U5n URR RX5nX- UpURR RXUS9n M)Nr)a) r scipyrrlinalgblasdasumzerosr#copyddotdaxpy)rbr(r)sprrxrzrzpApalphabetas rr%_PCGSolver._solve4sM GG GG yy~~##A&& HHQWW  FFH aD YY^^  & FFH1B,,Q33E $$QU$3A $$Ruf$5Ayy~~##A&,!A99>>&&q,Dy$" $$QT$2ArrN) __name__ __module__ __qualname____firstlineno____doc__rr,r%__static_attributes__rrr r s 3rr c(\rSrSrSrSrSSjrSrg) _LUSolverOzLU factorization. To solve Ax = b: solver = _LUSolver(A) x = solver.solve(b) optional argument `tol` on solve method is ignored but included to match _PCGsolver API. chSSKnURRRUSSSSS.S9Ulg)Nr MMD_AT_PLUS_AT)Equil SymmetricMode) permc_specdiag_pivot_threshoptions)r0sparser1splu_LU)rrr9s rr_LUSolver.__init__Zs799##(( &!"T: ) rNcSSKnURU5nURURSS9n[ URS5H,nUR R USS2U45USS2U4'M. U$r)r r!r"r#r$rWr,r&s rr,_LUSolver.solvedsf JJqM JJqwwcJ *qwwqz"AhhnnQq!tW-AadG#r)rWr)rCrDrErFrGrr,rHrIrrrKrKOs rrKc^^TR5(aO[R"5nURT5 UR U4SjTR SS95TS9 UmTR 5(dU4SjTR SS95nOUU4SjTR 55n[R"5nURT5 UR SU55 U$)z5Compute edge weights and eliminate zero-weight edges.c3`># UH#upo1U:wdM XURTS54v M% g7f?N)get.0uveweights r $_preprocess_graph..ts/ T7IGA!RSV 'aAEE&#& '7Is ..T)data)rec 3r># UH,upo1U:wdM X[URTS554v M. g7fr]absr_r`s rrfrgys7 ;MaVWQW +Q3quuVS)* +;Ms 7$7c3># UH9upX:wdM X[U4SjTUUR5554v M; g7f)c3Z># UH n[URTS55v M" g7fr]rj)rardres rrf._preprocess_graph...~s&I8H1s155-..8Hs(+N)sumvalues)rarbrcGres rrfrg}sA !v KQ3I!Q8HII J!s A2Ac3># UHupo3S:wdM XU4v M g7f)rNrI)rarbrcrds rrfrgsGuGA!QiqQius  ) is_directednx MultiGraphadd_nodes_fromadd_weighted_edges_fromedges is_multigraphGraph)rqreHrxs`` r_preprocess_graphr|ns}} MMO  !! TqwwDw7I T "   ??   ;<777;M      AQGuGG HrcSSKnURU5n[U5n[U5n[ [ U[ U555nURU[S9n[U5H upxXvXX'M XdS- S- -nU$)zEEstimate the Fiedler vector using the reverse Cuthill-McKee ordering.rNdtyper@) r subgraphrlendictzipr$r"float enumerate) rqnodelistr)rnindexr:irbs r _rcm_estimaters| 8A *1 -E H A XuQx( )E 1E "A% %( !a%3A Hrc ^^^^^SSKmSSKnURSmU(aTRTR 55mUR R UR RST- STTSS95mTT-T-mTSTRRTS5- -mU(aUU4SjnOUU4S jnUS :Xa8TR 5R[5m[U4S jU4S j5nOUS :XahUR RT[SS9nURSSURSS- R5n TR XU 4'[#U5nO[$R&"SU35e[)T5R+SS9R-5R/5n U"U5 TR1URSS9n TRR3U5SnTU-U SS2SS24'UR4U -n URR7U SS9upX-nURR8R;XSS2S4-U SUSS2S4-- 5U - nX:aOdUR=X5U SS2SS24'URR?U R4U-5U R4-R4nU"U5 GMU TRAU54$)auCompute the Fiedler vector of L using the TraceMIN-Fiedler algorithm. The Fiedler vector of a connected undirected graph is the eigenvector corresponding to the second smallest eigenvalue of the Laplacian matrix of the graph. This function starts with the Laplacian L, not the Graph. Parameters ---------- L : Laplacian of a possibly weighted or normalized, but undirected graph X : Initial guess for a solution. Usually a matrix of random numbers. This function allows more than one column in X to identify more than one eigenvector if desired. normalized : bool Whether the normalized Laplacian matrix is used. tol : float Tolerance of relative residual in eigenvalue computation. Warning: There is no limit on number of iterations. method : string Should be 'tracemin_pcg' or 'tracemin_lu'. Otherwise exception is raised. Returns ------- sigma, X : Two NumPy arrays of floats. The lowest eigenvalues and corresponding eigenvectors of L. The size of input X determines the size of these outputs. As this is for Fiedler vectors, the zero eigenvalue (and constant eigenvector) are avoided. rNrcsrformatr^c>TRU5n[URS5H!nUSS2U4==USS2U4T-T--ss'M# gz(Make X orthogonal to the nullspace of L.rN)r!r$r#)r*r+rdr)s rproject"_tracemin_fiedler..projectsJ 1 A1771:&!Q$AadGaK1,,'rc>TRU5n[URS5H,nUSS2U4==USS2U4R5T- -ss'M. gr)r!r$r#ro)r*r+rr)s rrrsL 1 A1771:&!Q$1QT7;;=1,,'r tracemin_pcgc>TU-$rrI)r:Ls r#_tracemin_fiedler..s a!erc>TU-$rrI)r:Ds rrrs q1ur tracemin_luT)rr5zUnknown linear system solver: )axisrr) overwrite_a)!r r0r#sqrtdiagonalrU csr_arrayspdiagsr1normastyperr csc_arrayindptrargmaxinfrKrt NetworkXErrorrkroflattenmaxr"qrTeighr2r3r,invr!)rr* normalizedr(methodr9rsolverrrLnormWr{sigmaYresrrdrr)s` @@@@r_tracemin_fiedlerrsD  A GGAJJL ! II   1 1!a%Aq 1 O P EAI S299>>!Q' '' - -  JJL   &O_= = II  T  : XXab\AHHSbM ) 1 1 3&&Q$1!?xHII FJJAJ  & & ( , , .E AJ 177# &A  IILLOA a%!Q$ CC!G99>>!>6 Eiinn""1Aw;qAadG1C#CDuL 9 ,,q&!Q$ YY]]1337 #acc ) , , % ( "**Q- rc^^SSKmTS:XaSmTS;a UU4SjnU$TS:XdTS:Xa UU4S jnU$[R"S T<S 35e) z>Returns a function that solves the Fiedler eigenvalue problem.rNtraceminr)rrc>TS:XaSO[SURSS- 5nT RURXPRS4S95Rn[ XX#T5upvUSUSS2S44$)Nrrr)size)minr#r!normalrr) rr:rr(seedqr*rrr)s r find_fiedler'_get_fiedler_func..find_fiedler st~-3q!''!*q.3IA 4;;Q O;<=??A(zGHE8Qq!tW$ $rlanczoslobpcgc >SSKnURRU[S9nURSnU(a]URRURR ST R UR55- S/XfSS95nXp-U-nT S:XdUS:a7URRRUSS US S 9upUS U SS2S 44$T RT RU5R5n URRURR SUR5- SXf55n T RU5n U(aU WR5-n URRRX U T RU 5RX6S S9upUSU SS2S44$)Nrr~r^cscrr rSMT)whichr(return_eigenvectorsrF)rrr(maxiterlargest)r0rUrrr#rrrr1eigshr! atleast_2drronesr)rr:rr(rr9rrrr*rrrr)s rrrs  ##AU#3A AII''II%%bggajjl33aS!u& EAI"a"f99++11q#42Qx1a4((JJr}}Q/112II'' (9(9# :LaQR(VWGGAJ%A99++22Aq!1!3!3QV3Qx1a4((rzunknown method .)r rtr)rrr)s` @r_get_fiedler_funcrsf  00 %X M 9 ( 2 )H  !<==rdirectedre) edge_attrsc[U5S:a[R"S5e[X5n[R"U5(dg[R "U5nUR SS:XaU(dS[US5-$S$[U5nUS:waSO [X5nU"XhX#U5up[U 5$) a Returns the algebraic connectivity of an undirected graph. The algebraic connectivity of a connected undirected graph is the second smallest eigenvalue of its Laplacian matrix. Parameters ---------- G : NetworkX graph An undirected graph. weight : object, optional (default: None) The data key used to determine the weight of each edge. If None, then each edge has unit weight. normalized : bool, optional (default: False) Whether the normalized Laplacian matrix is used. tol : float, optional (default: 1e-8) Tolerance of relative residual in eigenvalue computation. method : string, optional (default: 'tracemin_pcg') Method of eigenvalue computation. It must be one of the tracemin options shown below (TraceMIN), 'lanczos' (Lanczos iteration) or 'lobpcg' (LOBPCG). The TraceMIN algorithm uses a linear system solver. The following values allow specifying the solver to be used. =============== ======================================== Value Solver =============== ======================================== 'tracemin_pcg' Preconditioned conjugate gradient method 'tracemin_lu' LU factorization =============== ======================================== seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness`. Returns ------- algebraic_connectivity : float Algebraic connectivity. Raises ------ NetworkXNotImplemented If G is directed. NetworkXError If G has less than two nodes. Notes ----- Edge weights are interpreted by their absolute values. For MultiGraph's, weights of parallel edges are summed. Zero-weighted edges are ignored. See Also -------- laplacian_matrix Examples -------- For undirected graphs algebraic connectivity can tell us if a graph is connected or not `G` is connected iff ``algebraic_connectivity(G) > 0``: >>> G = nx.complete_graph(5) >>> nx.algebraic_connectivity(G) > 0 True >>> G.add_node(10) # G is no longer connected >>> nx.algebraic_connectivity(G) > 0 False rgraph has less than two nodes.rOrr)rrrN) rrtrr| is_connectedlaplacian_matrixr#rrr) rqrerr(rrrrr:rfiedlers rrr:s` 1vz?@@!$A ??1   AAwwqzQ+5sU1T7^#>3>$V,L(" a(;A!! >NE <rcSSKn[U5S:a[R"S5e[ X5n[R "U5(d[R"S5e[U5S:XaUR SS/5$[U5n[R"U5nUS:waSO [X5n U"XX#U5upU $) aReturns the Fiedler vector of a connected undirected graph. The Fiedler vector of a connected undirected graph is the eigenvector corresponding to the second smallest eigenvalue of the Laplacian matrix of the graph. Parameters ---------- G : NetworkX graph An undirected graph. weight : object, optional (default: None) The data key used to determine the weight of each edge. If None, then each edge has unit weight. normalized : bool, optional (default: False) Whether the normalized Laplacian matrix is used. tol : float, optional (default: 1e-8) Tolerance of relative residual in eigenvalue computation. method : string, optional (default: 'tracemin_pcg') Method of eigenvalue computation. It must be one of the tracemin options shown below (TraceMIN), 'lanczos' (Lanczos iteration) or 'lobpcg' (LOBPCG). The TraceMIN algorithm uses a linear system solver. The following values allow specifying the solver to be used. =============== ======================================== Value Solver =============== ======================================== 'tracemin_pcg' Preconditioned conjugate gradient method 'tracemin_lu' LU factorization =============== ======================================== seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness`. Returns ------- fiedler_vector : NumPy array of floats. Fiedler vector. Raises ------ NetworkXNotImplemented If G is directed. NetworkXError If G has less than two nodes or is not connected. Notes ----- Edge weights are interpreted by their absolute values. For MultiGraph's, weights of parallel edges are summed. Zero-weighted edges are ignored. See Also -------- laplacian_matrix Examples -------- Given a connected graph the signs of the values in the Fiedler vector can be used to partition the graph into two components. >>> G = nx.barbell_graph(5, 0) >>> nx.fiedler_vector(G, normalized=True, seed=1) array([-0.32864129, -0.32864129, -0.32864129, -0.32864129, -0.26072899, 0.26072899, 0.32864129, 0.32864129, 0.32864129, 0.32864129]) The connected components are the two 5-node cliques of the barbell graph. rNrrzgraph is not connected.r^gr) r rrtrr|rarrayrrr) rqrerr(rrr)rrr:rrs rrrs` 1vz?@@!$A ??1  899 1v{xxd $$$V,L AA(" a(;A!! >NE Nrc[U5S:Xa[R"S5e[X5n[ U5n/n[R "U5Hn[U5n U S:an[R "X5n US:waSO [X5n U"XX#U5up[U [U 5U5nURS[U555 MURU5 M U$)aCompute the spectral_ordering of a graph. The spectral ordering of a graph is an ordering of its nodes where nodes in the same weakly connected components appear contiguous and ordered by their corresponding elements in the Fiedler vector of the component. Parameters ---------- G : NetworkX graph A graph. weight : object, optional (default: None) The data key used to determine the weight of each edge. If None, then each edge has unit weight. normalized : bool, optional (default: False) Whether the normalized Laplacian matrix is used. tol : float, optional (default: 1e-8) Tolerance of relative residual in eigenvalue computation. method : string, optional (default: 'tracemin_pcg') Method of eigenvalue computation. It must be one of the tracemin options shown below (TraceMIN), 'lanczos' (Lanczos iteration) or 'lobpcg' (LOBPCG). The TraceMIN algorithm uses a linear system solver. The following values allow specifying the solver to be used. =============== ======================================== Value Solver =============== ======================================== 'tracemin_pcg' Preconditioned conjugate gradient method 'tracemin_lu' LU factorization =============== ======================================== seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness`. Returns ------- spectral_ordering : NumPy array of floats. Spectral ordering of nodes. Raises ------ NetworkXError If G is empty. Notes ----- Edge weights are interpreted by their absolute values. For MultiGraph's, weights of parallel edges are summed. Zero-weighted edges are ignored. See Also -------- laplacian_matrix rzgraph is empty.rrNc3*# UH upo3v M g7frrI)rar:crbs rrf$spectral_ordering..Is<*;wqQ*;s) rrtrr|rconnected_componentsrrrr$extendsorted)rqrerr(rrrr componentrrr:rr sort_infos rr r s@ 1v{011!$A$V,L E,,Q/ 9~ !8##A1A(* a0KA)! FNEGU4[)`. Returns ------- bisection : tuple of sets Sets with the bisection of nodes Examples -------- >>> G = nx.barbell_graph(3, 0) >>> nx.spectral_bisection(G) ({0, 1, 2}, {3, 4, 5}) References ---------- .. [1] M. E. J Newman 'Networks: An Introduction', pages 364-370 Oxford University Press 2011. rN)r rtrrlistsettolist) rqrerr(rrr)rcnodespos_valss rr r Psev !ZfCA HHT!W EAvH uY&&( )3u/E/E/G+H HHr)reFg:0yE>rN)rG functoolsrnetworkxrtnetworkx.utilsrrr__all__r rKr|rrr _dispatchablerrr r rIrrrs%  6363r> 4  i X4nZ X&PTZ'!ZzZ X&PT\'!\~X&PTO'OdX&PT@I'@Ir