hySrSSKJr SSKrSSKJr SS/r\RS5r Sr S'S jr S r S r S rS(S jr\R"SSS9S)Sj5r\RS5r\R"SS9S)Sj5r\RS5rSrSrSrSrSrSrSrSrS*SjrSrSrS r\"S!5S)S"j5r S#r!S$r"\"S%5S+S&j5r#g),z? Threshold Graphs - Creation, manipulation and identification. )sqrtN)py_random_stateis_threshold_graphfind_threshold_graphcj[UR5VVs/sHupUPM snn5$s snnf)a  Returns `True` if `G` is a threshold graph. Parameters ---------- G : NetworkX graph instance An instance of `Graph`, `DiGraph`, `MultiGraph` or `MultiDiGraph` Returns ------- bool `True` if `G` is a threshold graph, `False` otherwise. Examples -------- >>> from networkx.algorithms.threshold import is_threshold_graph >>> G = nx.path_graph(3) >>> is_threshold_graph(G) True >>> G = nx.barbell_graph(3, 3) >>> is_threshold_graph(G) False References ---------- .. [1] Threshold graphs: https://en.wikipedia.org/wiki/Threshold_graph )is_threshold_sequencedegree)Gnds O/var/www/html/env/lib/python3.13/site-packages/networkx/algorithms/threshold.pyrr s): ! !; ! !; <??/4((5D5J5J5L M5L/5vo5L M!*?!; >> from networkx.algorithms.threshold import make_compact >>> make_compact(["d", "i", "i", "d", "d", "i", "i", "i"]) [1, 2, 2, 3] >>> make_compact(["d", "d", "d", "i", "d", "d"]) [3, 1, 2] Notice that the first number is the first vertex to be used for construction and so is always 'd'. Labeled creation sequences lose their labels in the compact representation. >>> make_compact([3, 1, 2]) [3, 1, 2] rNr"Not a valid creation sequence type)rstrtupleint TypeErrorrangerappend)r%firstr#sccscountrs r rrs, a E% q ! E5 ! !- .-qd- . E3    <== C E 1c"g  5B1uI  QJE JJu E  JJu J/sCczUSn[U[5(aU$[U[5(aU$[U[5(aUSSnO [ S5e/nU(aXUR UR S5S/-5 U(a$UR UR S5S/-5 U(aMXU$)z Converts a compact creation sequence for a threshold graph to a standard creation sequence (unlabeled). If the creation_sequence is already standard, return it. See creation_sequence. rNr'r r)rr(r)r*r+extendr)r%r.ccscopyr#s r uncompactr5s a E%  E5 ! !  E3  #A&<== B  '++a.C5()  IIgkk!nu, - ' IcUSn[U[5(a'[U[5(aUSSnOb[U5nOV[U[5(aUVs/sHo3SPM nnO,[U[5(a [ U5nO [ S5eUR5 SnSn[U5H"upgUS:XaXBU'UnMUS:XdMUnUS- nM$ UR5 [U5H"upgUS:XaXBU'UnMUS:XdMUnUS- nM$ US:XaUS- nSU- nUV s/sHoU-PM sn $s snfs sn f)z Returns a list of node weights which create the threshold graph designated by the creation sequence. The weights are scaled so that the threshold is 1.0. The order of the nodes is the same as that in the creation sequence. rNrr'rr ) rr(listr)r*r5r+reverser) r%r.wseqr$wprevjr/wscalewws r creation_sequence_to_weightsr@sS a E% ' . .$Q'D)*D E5 ! !/0/!/0 E3  *+<==LLN A D$ 8GD S[D FA   LLN$ 8GD S[D FA   s{ Q UF"& '$BK$ ''918 (s E2EcU(aU(a [S5e[U[5(a&UR5VVs/sHupEXT/PM nnnO [ U5VVs/sHupuXW/PM nnnUR 5 /nXSS- n U(aUSSU :a'UR S5upTURUS45 O/UR 5upTURUS45 XSS- n [U5S:Xa%UR 5upTURUS45 U(aMUR5 U(aU$U(a [U5$UV s/sHoSPM sn $s snnfs snnfs sn f)a Returns a creation sequence for a threshold graph determined by the weights and threshold given as input. If the sum of two node weights is greater than the threshold value, an edge is created between these nodes. The creation sequence is a list of single characters 'd' or 'i': 'd' for dominating or 'i' for isolated vertices. Dominating vertices are connected to all vertices present when it is added. The first node added is by convention 'd'. If with_labels==True: Returns a list of 2-tuples containing the vertex number and a character 'd' or 'i' which describes the type of vertex. If compact==True: Returns the creation sequence in a compact form that is the number of 'i's and 'd's alternating. Examples: [1,2,2,3] represents d,i,i,d,d,i,i,i [3,1,2] represents d,d,d,i,d,d Notice that the first number is the first vertex to be used for construction and so is always 'd'. with_labels and compact cannot both be True. rrrrr r) rrrrrrrr-rr9r) weights thresholdr r!r"r;r:rr#cutoffr$s r weights_to_creation_sequencerEsT<w>??'4  -4]]_=_z _=#,W#56#541#56IIK B b! $F  71: !JQ IIucl #JQ IIucl #b!,F t9>JQ IIucl # $JJL B "QaD" 1>6, sE7#E=&FT)graphs returns_graphchUSn[U[5(a[[U55nO\[U[5(aUSSnOA[U[ 5(a [ U5n[[U55nO [S5 g[R"SU5nUR5(a[R"S5eSUl U(aVURS5upgUS:Xa#[U5HnURXh5 M URU5 U(aMVU$)a Create a threshold graph from the creation sequence or compact creation_sequence. The input sequence can be a creation sequence (e.g. ['d','i','d','d','d','i']) labeled creation sequence (e.g. [(0,'d'),(2,'d'),(1,'i')]) compact creation sequence (e.g. [2,1,1,2,0]) Use cs=creation_sequence(degree_sequence,labeled=True) to convert a degree sequence to a creation sequence. Returns None if the sequence is not valid rNz"not a valid creation sequence typezDirected Graph not supportedzThreshold Graphr )rr(r8rr)r*r5printnx empty_graph is_directed NetworkXErrornameradd_edgeadd_node) r% create_usingr.cir#r r$ node_typeus r threshold_graphrU1s$ a E% )-. / E5 ! ! q ! E3   ( ) )B-  23 q,'A}}=>> AF    !W 1  1  " Hr6c UR5HzupUR5HanURX5(aMX:wdM"URU5H*nURX$5(aMX$:wdM"XX4/s s s $ Mc M| g)z Returns False if there aren't any alternating 4 cycles. Otherwise returns the cycle as [a,b,c,d] where (a,b) and (c,d) are edges and (a,c) and (b,d) are not. F)edgesnodeshas_edge neighbors)r rTr$r;xs r find_alternating_4_cycler\esm A::a##QA::a++ !a|+( r6)rGc,[[U5U5$)a Returns a threshold subgraph that is close to largest in `G`. The threshold graph will contain the largest degree node in G. Parameters ---------- G : NetworkX graph instance An instance of `Graph`, or `MultiDiGraph` create_using : NetworkX graph class or `None` (default), optional Type of graph to use when constructing the threshold graph. If `None`, infer the appropriate graph type from the input. Returns ------- graph : A graph instance representing the threshold graph Examples -------- >>> from networkx.algorithms.threshold import find_threshold_graph >>> G = nx.barbell_graph(3, 3) >>> T = find_threshold_graph(G) >>> T.nodes # may vary NodeView((7, 8, 5, 6)) References ---------- .. [1] Threshold graphs: https://en.wikipedia.org/wiki/Threshold_graph )rUfind_creation_sequence)r rQs r rrus@ 1!4l CCr6c /nUnUR5S:Ga&[UR55nUR5VVs/sHupEXT4PM nnnUR 5 USSS:Xa0UR [ US/[U5S- -S/-55 OUSSS:Xa4URS5upWURUS45 USSS:XaM4UR5upXURUS45 URURU55nUR5S:aGM&UR5 U$s snnf)zy Find a threshold subgraph that is close to largest in G. Returns the labeled creation sequence of that threshold graph. rrrrr ) orderrr rrr3ziprrr-subgraphrZr9) r r#Hdsdictr$r risobigvs r r^r^s& B A '')a-ahhj!!' 0qf 0  b6!9> IIc&3%3r7Q;"73%"?@ A eAh!mvvayHQ IIsCj !eAh!mFFH  4+ JJq{{4( )% '')a-&JJL I#1sEcUnURS5nX"S- -US- -S- n[U5HupEUS:XaX2US- -S- - nMUS-nM! U$)zV Compute number of triangles in the threshold graph with the given creation sequence. r rr)r1r)r%r#drntrirtyps r trianglesrmsm B #B a=BF #a 'DB- #: "q&MA% %D !GB  Kr6cUn/nURS5nUS- US- -S-nSnSn[U5HUupxUS:XaUS- nXCS- U--n O'W S:XaXCS- U-- nSnX6-nSnUS- nX3S- -S-n URU 5 Un MW U$)zL Return triangle sequence for the given threshold graph creation sequence. r rrhrr1rr-) r%r#seqrjdcurirundrunrsymtriprevsyms r triangle_sequencerws B C #B FrAv ! #D D DB- #: AIDq&D(C#~a4'  AIDQ-1$C 3  Jr6c[U5n[U5n/n[U5H?upEXnUS::aURS5 M"XUS- -S-nURXg- 5 MA U$)zJ Return cluster sequence for the given threshold graph creation sequence. rrrh)rwrrr-)r%triseqdegseqcseqrdegrumax_sizes r cluster_sequencer~sq0 1F . /F DF#i !8 KKN 7O) CN# $ Kr6cUn/nURS5n[U5H6upEUS:XaUS-nURX4-5 M%URU5 M8 U$)zQ Return degree sequence for the threshold graph with the given creation sequence r rro)r%r#rprdrrts r rrs[ B C #BB- #: !GB JJrv  JJrN  Jr6cZ[U5n[[U55nXS- -nX#- nU$)zu Return the density of the graph with this creation_sequence. The density is the fraction of possible edges present. r)rsumr)r%Ntwo_size two_possibledens r densityr s7 A?#456HA;L  !C Jr6cUnSnSnSnSnURS5n[U5VVs/sHupxUS:XdM UPM n nn[U5n [U5HgupxUS:Xa+XyS:wa[SXy5 [eU R S5 Xn U H'n Xn X-U -- nX;S-U S--- nXKU -- nUS- nM) Mi SU-U-XD-- nSU-U-XD-- nUS:XaUS:Xag[ SU35eX- $s snnf)z6 Return the degree-degree correlation over all edges. rr z!Logic error in degree_correlationrhrz"Zero Denominator but Numerator is )r1rrrIrr)r%r#s1s2s3mrrrtrdirdegidjdegjdenomnumers r degree_correlationrs9 B B B B A #B$R= 7=C3J1=C 7  BB- #:F{91B  GGAJuB6D + B 'D!G# #B + B FA   EBJ E EBJ E z A:=eWEFF =+ 8s C=C=cUSn[U[5(a([[U55Vs/sHoDX4PM nnOn[U[5(aUSSnOS[U[ 5(a3[ U5n[[U55Vs/sHoDXd4PM nnO [S5eUVs/sHowSPM nnX(;a[SUS35eX;a[SUS35eX:XaU/$URU5n URU5n [X5n X[SS:XaX/$X[SnU(a)UR5n U SS:XaXSU/$U(aM)gs snfs snfs snf) a Find the shortest path between u and v in a threshold graph G with the given creation_sequence. For an unlabeled creation_sequence, the vertices u and v must be integers in (0,len(sequence)) referring to the position of the desired vertices in the sequence. For a labeled creation_sequence, u and v are labels of vertices. Use cs=creation_sequence(degree_sequence,with_labels=True) to convert a degree sequence to a creation sequence. Returns a list of vertices from u to v. Example: if they are neighbors, it returns [u,v] rNr'zVertex z$ not in graph from creation_sequencerr r) rr(r,rr)r*r5r+rindexmaxr) r%rTr$r.rr#rRr/vertsuindexvindexbigindverts r shortest_pathr9s}$ a E%16s;L7M1N O1NA#&'1N O E5 ! ! q ! E3   ( )"'B. 1.Q"%j. 1<== 2aqT2E ~71#%IJKK~71#%IJKKvs [[^F [[^F  F z!}v GB vvx 7c>Aw? " " =P 2 sE%E*5E/cUSn[U[5(a'[U[5(aUSSnO[U5nOz[U[5(a9UVs/sHoDSPM nnUVs/sHoDSPM snR U5nO,[U[ 5(a [ U5nO [S5e[U5nS/U-nSXa'[US-U5HnX7S:XdM SXg'M X1S:Xa[U5HnSXg'M [US- SS5HnX7S:Xa U$SXg'M U$s snfs snf)al Return the shortest path length from indicated node to every other node for the threshold graph with the given creation sequence. Node is indicated by index i in creation_sequence unless creation_sequence is labeled in which case, i is taken to be the label of the node. Paths lengths in threshold graphs are at most 2. Length to unreachable nodes is set to -1. rNrr'rhr r) rr(r8r)rr*r5r+rr,)r%rr.r#r$rsplr=s r shortest_path_lengthrnsM a E% ' . ."1%B'(B E5 ! !- .-qd- ., -,aqT, - 3 3A 6 E3   ( )<== BA #'C CF 1q5!_ 5C<CF u|qACF1q5!R  5C<  J! J// -s D;/EcUn/nSn[URS55nSnSnSn[U5HbupU S:Xa$XS- U-U- -SU-X- U- -U- -n US- nOUS:Xa W nXW-nSnSnSn US- nUR[U 55 U nMd U(a-[ U5n SU S- U S- -- n UVs/sHoU -PM nnU$s snf)z Return betweenness for the threshold graph with the given creation sequence. The result is unscaled. To scale the values to the interval [0,1] divide by (n-1)*(n-2). r rgrrh?)floatr1rr-r)r% normalizedr#rplastcharrjrrrsdlastrcbr`scaler/s r betweenness_sequencers  B CH rxx} B D D E"  8T)B..TQX_1MPR1RRA AID3 A AID 58!&B eai01"%&#Q5y#& J's<Cc[U5n[U5nS/U-nUSSn[USSS25nUSnS[U5- /U-US'SUS'UnXV-nSnSn Sn X:aBS[X-U -5- n X*/-X-/-S/X)- S- --X9'XtU 'U S- n U S- n X:aMB[U5S:XaXC4$USSHnS[Xi-X--5- n X*U -/-XiU -/--S/X)- U- --X9'U(+nU(a X-nXV-nOUnXtU 'U n U S- n Sn X:dMeS[X- X--5- n S/U -X- U */--X-/-S/X)- S- --X9'XtU 'U S- n U S- n X:aMNM XC4$)a Return a 2-tuple of Laplacian eigenvalues and eigenvectors for the threshold network with creation_sequence. The first value is a list of eigenvalues. The second value is a list of eigenvectors. The lists are in the same order so corresponding eigenvectors and eigenvalues are in the same position in the two lists. Notice that the order of the eigenvalues returned by eigenvalues(cs) may not correspond to the order of these eigenvectors. rNrhrTr)rrrr) r%r0rvecvalrjnnetype_drddrsts r eigenvectorsrs  ( )C CA #'C a&C S1XB QBDGm_q CF CF AHB F A B 'd27Q;''fX ,saeai/@@A Q a '  3x1}z!"gd26QV,--cEk]"Ru9+%55qurz8JJ A HBAA  Q g$qv/00ES2XUF8 33rzlBaSAETUIEVVCFF FA !GB g( :r6c~/nUSnUH/n[S[X@555nURU5 M1 U$)a| Returns the coefficients of each eigenvector in a projection of the vector u onto the normalized eigenvectors which are contained in eigenpairs. eigenpairs should be a list of two objects. The first is a list of eigenvalues and the second a list of eigenvectors. The eigenvectors should be lists. There's not a lot of error checking on lengths of arrays, etc. so be careful. rc3.# UH upX-v M g7fN).0evvuvs r &spectral_projection..s5*Yc*s)rrar-)rT eigenpairscoeffevectevrs r spectral_projectionrsB E qME 5#b*5 5 Q Lr6c[U5nUR5 /nSn[U5nUR5nU(aDXT:aUR U5 US-nOUS- nU(aUR5nOSnU(aMDU$)a Return sequence of eigenvalues of the Laplacian of the threshold graph for the given creation_sequence. Based on the Ferrer's diagram method. The spectrum is integral and is the conjugate of the degree sequence. See:: @Article{degree-merris-1994, author = {Russel Merris}, title = {Degree maximal graphs are Laplacian integral}, journal = {Linear Algebra Appl.}, year = {1994}, volume = {199}, pages = {381--389}, } rr)rrrrr-)r%rzeiglisteigrowbigdegs r eigenvaluesrs}(. /F KKMG C f+C ZZ\F < NN3  1HC 1HC # Nr6rhcSUs=::aS::d O [S5eS/n[SU5H;nUR5U:aURS5 M*URS5 M= U$)a Create a random threshold sequence of size n. A creation sequence is built by randomly choosing d's with probability p and i's with probability 1-p. s=nx.random_threshold_sequence(10,0.5) returns a threshold sequence of length 10 with equal probably of an i or a d at each position. A "random" threshold graph can be built with G=nx.threshold_graph(s) seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness`. rrzp must be in [0,1]r r)rr,randomr-)r pseedr#rs r random_threshold_sequencer=s]( KaK-.. B 1a[ ;;=1  IIcN IIcN  Ir6cS/S/US- --nX:aSX!'U$XUS- -S- :a [S5eUS- nUS- nXA:aSX#'US-nXC- nXA:aMXU- - nSX#'U$z Create a skewed threshold graph with a given number of vertices (n) and a given number of edges (m). The routine returns an unlabeled creation sequence for the threshold graph. FIXME: describe algorithm r rrrhz#Too many edges for this many nodes.r)r rr#indrs r right_d_threshold_sequencer`s #!a% B u  A;?>?? a%C a%C ' q  ' Sy/CBG Ir6cS/S/US- --nX:aSX!'U$XUS- -S- :a [S5eSX S- 'US- nSnX1:aSX$'X4- nUS- nX1:aMX1:aSX#U- 'U$rr)r rr#rrs r left_d_threshold_sequencers #!a% B u  A;?>??B1uI a%C C '  q ' w7 Ir6c&[USS5VVs/sHupEUS:XdM UPM nnnUR5U:aaURU5nUR[U55nXx- n X:wa-XS:Xa%X S:XaSX'SX'SX 'UR U5 UR5U:aWU(aPURU5n URU5n X-n U [ U5:d X S:XdX:XaU$SX 'SX 'SX 'U$s snnf)aN Perform a "swap" operation on a threshold sequence. The swap preserves the number of nodes and edges in the graph for the given sequence. The resulting sequence is still a threshold sequence. Perform one split and one combine operation on the 'd's of a creation sequence for a threshold graph. This operation maintains the number of nodes and edges in the graph, but shifts the edges from node to node maintaining the threshold quality of the graph. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness`. rrr r)rrchoicer,remover) r#p_split p_combinerrrSdlistrsplit_to flip_side first_choice second_choicetargets r swap_drs(&/r!Bx%8 M%8>AIrs[* !7 8==>.8v)X0+(`6;;~T20 30 f  %D&DD@(8"" D2j+\%P7t*$TDD F..r6