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Clique Mining Algorithms for Graph Analysis

This text explains various algorithms for mining cliques in graphs, including Breadth-First and Depth-First approaches, as well as specialized techniques for bipartite and hyperpartite graphs. It also explores tripartite and quadrapartite graphs and their potential applications in real-life scenarios.

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Clique Mining Algorithms for Graph Analysis

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  1. Edge Count Clique Alg (EC): A graph C is a clique iff |EC||PUC|=COMB(|VC|,2)|VC|!/((|VC|-2)!2!) SubGraph existence thm (SGE): (VC,EC) is a k-clique iff every induced k-1 subgraph, (VD,ED) is a (k-1)-clique. Apriori Clique Mining Alg (AP): finds all cliques in a graph. For Clique-Mining we can use an ARM-Apriori-like downward closure property: CSkkCliqueSet, CCSk+1Candidatek+1CliqueSet. By SGE, CCSk+1= all s of CSk pairs having k-1 common vertices. Let CCCSk+1 be a union of two k-cliques with k-1 common vertices. Let v and w be the kth vertices (different) of the two k-cliques, then CCSk+1 iff (PE)(v,w)=1. Breadth-1st Clique Alg: CLQK=all Kcliques. Find CLQ3 using CS0. Induction theorem: A Kclique and 3clique that share an edge form a (K+1)clique iff all K-2 edges from the non-shared Kclique vertices to the non-shared 3clique vertex exist. Next find CLQ4, then CLQ5, … Depth-1st Clique Alg: Find a Largest Maximal Clique v. Let (x,y)CLQ3pTree(v,w). If (x,y)E and Count(NewPtSet(v,w,x,y)CLQ3pTree(v,w)&CLQ3pTree(x,y)) is: 0, the 4 vertices form a maximal 4Clique (i.e., v,w,x,y). 1, the 5 vertices form a maximal 5Clique (i.e., v,w,x,y and the NewPt) 2, the 6 vertices form a maximal 6Clique if the NewPair is an edge, else they form 2 maximal 5Cliques. 3, the 7 vertices form a maximal 7Clique if each of the 3 NewPairs is an edge, elseif 1 or 2 of the NewPairs are edges then each of the 6VertexSets (vwxy + 2 EdgeEndpts) form Max6Clique, elseif 0 NewPairs is an edge, then each 5VertexSet (vwxy + 1 NewVertex) forms a maximal 5Clique…. Theorem:  hCliqueNewPtSet, those h vertices together with v,w,x,y form a maximal h+4Clique, where NPS(v,w,x,y)=CLQ3(v,w)&CLQ3(x,y). We can determine if each maximal kClique found is a “Largest” from counts (or find them all) but determining “Largest” early can save time (can move on to another v immediately). E.g., if there aren’t enough siblings left or a large enough 1-count among CLQ3pTrees… Bipartite Clique Mining Algorithm finds many of the Maximal Cliques (MCLQs) in a bipartite graph at a low cost (only selected pairwise ANDs). Each LETTERpTree is a MCLQ unless there are pairwise ANDs with the same count. If so, all LETTERs involved in those pairwise ANDs form a MCLQ with the set of NUMBERS making up that common count (The same is true for NUMBERpTree.).  pTree, A, the AND of all pairwise ANDs. A&B, B with Ct(A&B)=Ct(A) is a MCLQ (incl. A as one such B takes care of case when there are no other Bs s.t. Ct(A&B)=Ct(A)). There is potential for a k-plex [k-core] mining algorithm in this vein. Instead of Ct(A&B)=Ct(A), we would consider. E.g., Ct(A&B)=Ct(A)-1. Each such pTree, C, would be missing just 1vertex (Therefore 1 edge). Thus taking any MCLQ found as above, ANDing in CpTree would produce a 1-plex. ANDing in k such C’s would produce a k-plex. In fact, suppose we have produced a k-plex in such a manner, then ANDing in any C with Ct(C)=Ct(A)-h would produce a (K+h)-plex. &i=1..nAi is a [i=1..nCt(Ai)]-Core Tripartite Clique Mining Algorithm? A Tripartite Graph has 3 separate PARTs to its vertex set and edges must start and end in different parts. An example: PART1=tweeters; PART2=hashtags; PART3=tweets. Tweeters-to-hashtags is many-to-many? Tweeters-to-tweets is many-to-many (incl. retweets)?; hashtags-to-tweets is many-to-many? Multipartite Clique Mining Algorithm? Bipartite, Tripartite, Quadrapartite… graphs have 2,3,4… PARTs respectively and regular line edges with the rule that no edge can start and end in the same PART. Hyperpartite Clique Mining: A hyperpartite graph, e.g., a 3PARThyperpartite graph has 3 vertex PARTS and each edge is planar triangle (a triple of vertices), one from each PART. A stock recommender as a real-life example of a 3PARThyperpartite graph (Investors, Stocks, Days are the three parts and a triangular "edge" connects Investor #k, Stock X, and Day n means Investor k recommended (as determined by Azure Tweet Sentiment Analyzer or?) stock X on day n.  Then a 3PARThyperpartite clique would be a community in which all the investors in the clique recommend all the stocks in the clique on each of the days in the clique (A strong signal? - depending upon the quality of the investors, etc.) Conjecture: KPART multicliques and KPARThypercliques are in 1-1 correspondence (both are characterized by a set of K vertices)? So, we can concentrate on one of the mining processes only?

  2. A Tripartite Graph has 3 separate PARTs to its vertex set. Each edge consists of 2 vertices (from different PARTs) and a line between them. A Three PART Hypergraph has 3 PARTs to its vertex set. Each edge consists of 3 vertices (from different PARTs) and the planar triangle they form. Example1: PART1=tweeters; PART2=hashtags; PART3=tweets. Tweeters-to-hashtags is many-to-many? Tweeters-to-tweets is many-to-many (incl. retweets)?; hashtags-to-tweets is many-to-many? Example2: Users-Tags-Blogs? Example3: Users-Tags-Websites? Figure 5: Decomposition of a gene-disease-protein complex network. We integrated the gene-disease network from [3] with human protein complexes from the CORUM database [16]. This resulted in a layered tripartite graph, which is schematically drawn in (a). We performed a 10-fold approximation of this graph to estimate appropriate numbers of clusters. The boxplot curve (b) shows how the cost function f(H, C) from equation (1) depends on the number of gene clusters mg. The true minima of the cost function are decreasing with mg, and this is also visible in the approximated minima using our proposed algorithm. Therefore, we are able to identify structures on various resolution levels. The details represent the cost function course for large-scale clustering (i) and a decomposition on small scale (ii), respectively. For our detailed analyses, we used the decompositions showing steep drops in the cost function marked by the red and green boxes. A potential theorem? A clique in a Tripartite graph IS a clique in a 3PART Hypergraph and vice versa?

  3. Bipartite Graph cTrees B A 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 D C 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 A B 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 A C 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 A D 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 A E 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 A F 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 A G 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 A H 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 D E 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 D F 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 D G 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 3 D H 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 D I 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 G H 0 1 1 1 0 0 1 0 1 1 0 0 1 0 1 0 0 0 8 G I 0 0 1 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 5 B C 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 B D 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 C E 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 6 C F 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 5 C G 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 C H 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 C I 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 E F 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 6 E G 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 6 E H 1 1 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 7 E I 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 3 F E 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 6 F G 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 4 F H 1 1 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 7 F I 1 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 4 H G 0 1 1 1 0 0 1 0 1 1 0 0 1 0 1 0 0 0 8 I H 1 0 1 0 0 0 0 1 1 1 1 1 1 0 0 1 0 0 9 K F 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 K G 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 2 K H 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 K I 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 3 K J 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 2 K L 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 2 L F 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 L G 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 4 L H 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 5 L I 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 5 M G 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 2 M H 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 M I 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 M J 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 M K 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 M L 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 M N 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 N G 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 2 N H 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 N I 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 N J 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 N K 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 N L 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 N M 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 A I 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 B E 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 B F 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 B G 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 B H 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 B I 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 J F 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 J G 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 3 J H 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 4 H I 1 0 1 0 0 0 0 1 1 1 1 1 1 0 0 1 0 0 9 J L 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 5 B 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 C 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 6 D 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 E 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 8 F 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 8 G 0 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 a H 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 e I 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 c J 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 5 K 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 4 L 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 6 M 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 N 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 A 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 2 3 4 5 6 7 8 9 a b v d e f g h i For bipartite graphsthe clique Trees (cTrees) would have the NUMBERpTree on top and the LETTERpTree on the bottom: The cTreeSet of a bipartite graph is closed under AND (i.e., ANDing 2 cTrees is a cTree and we will call it aa). The OR of 2cTrees with a substantial AND might reveal interesting communities. The cTreeSet of a bipartite graph is closed under OR-AND in which we OR the PART1 and AND PART2pTrees(called oa) and AND-OR (called ao. All Bipartite Max Clique Mining algs involve oa. Max Base CLQs (MBCLQs). Thms: Every MCLQ is generated, using oa, from the base CLQs, YES! Every MCLQ is generated, using oa, from the maximized base CLQs (1st round only)? oaapplied to two MBCLQs gives a MCLQ. No.. Counterexample?. Find all MCLQ(A)? There doesn’t seem to be an algebraic method. Probably it will have to be a POSET method? Commutative monoids (lack inverses) (CLQ,oa) (CLQ,ao) (CLQ,aa). (BCS.oa) generates (CLQ,oa) MBCLQ(X) X =? BCS={32 base cliques}; ops, and-and (aa), or-and (oa), and-or (ao), each forming a comm. Monoid. B 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 C 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 6 1 D 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 4 1 E 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 8 1 F 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 8 1 G 0 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 a 1 H 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 e 1 I 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 c 1 J 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 5 1 K 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 4 1 L 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 6 1 M 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 3 1 N 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 1 8 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 1 7 3 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 8 4 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 7 5 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 4 6 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 1 4 7 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 4 8 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 3 9 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 1 4 a 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 1 4 b 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 4 c 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0 1 6 d 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 0 1 7 e 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1 0 1 8 f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 1 5 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 2 h 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 2 i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 2 A 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 A B C D E F G H I J K L M N 124-ACEFH 124-ACEFH Usually, the ops, XOR, AND make {0,1) into a field (the Galois Field GF(2) ). It’d be useful if we could make the cTreeSet into a field (with an XOR and AND operation). However, on cTrees, the op, XOR over AND (xa) has ID=01 and a cTree, xy, has XOR inverseof x but no AND inverse. 1 2 3 4 5 6 7 8 9 a b v d e f g h i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 C# CL 12-ABCEFH 123-BCEFH 123456-CE 14-ACDEFH 1345-CDE 12345679-E (1234678,14)-F 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17,18 1-ABCDEFHI MBCLQ(X) X =? 24-ACEFGH (234579,10,13,14,15)-G 2-ABCEFGH (12346789.10,11,12,13,15,16)-H 3-BCDEFGHI 1-ACEFHI (1389.10,11,12,13,14,16,17,18)-I 4-ACDEFGH (11,12,13,14,15)-JL 345-CDEG (14.15.17.18)-K 12346-CEFH (10,11,12,13,14,15)-L 2347-EFGH (12,13,14)-MIJLN 78-EFG A B C D E F G H I J K L M N 359-EGHI (10,13)-GHIL (11,12,13)-HIJL (12,13)-HIJLMN 13-GHIJLMN 14-FGIJKLMN 15-GHJKL (1,3,8,9,10,11,12,13,16)-HI (14,17,18)-IK

  4. Base clTrees I S D R 3PART HyperGraphs 3-Level pTrees with strides, 14 (Stocks) 5 (Days) 18 (Inv) 1 2 3 4 5 6 7 8 9 a b v d e f g h i A B C D E F G H I J K L M N CtI CtD CtS A B C D E F G H I J K L M N 1 2 3 4 5 6 7 8 9 a b v d e f g h i A B C D E F G H I J K L M N stock, we mine BMCLQs by &ing day-A pTrees then combining all other day-X pTrees with the same count (This is oaa on Stocks, Days, Investors). … 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 0 1 1 0 0 0 0 0 1 1 8 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 A A A A A A A A A A A A A A A A A A B B B B B B B B B B B B B B B B B B C C C C C C C C C C C C C C C C C C D D D D D D D D D D D D D D D D D D E E E E E E E E E E E E E E E E E E F F F F α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α … … {123..}=Investors recommending Stocks={ABC..} on Days={,,,,}. 1 We might want communities of type: • BC12, which tells Stocks B,C have been recommended every Day by Investors 1,2. The operation here is oaa (or Stocks, and Days, and Investors). This is a clique. • BCEFH(DayCt2)12, which tells us Stocks B,C,E,F,H have been recommended 2 Days by Investors 1,2. The op here is oCt2 where Ct2 operator applies to the 5 Stock=A,Day=? pTrees and produces a Investor mask pTree showing those Investors who have recommended BCEFH at least 2 Days out of the 5. This is not a clique. Md? Of course we can implement this operator as 5 SPTS additions (where each SPTS is 1 bit wide), followed by an EINring type operator on that column of sums that masks to 1, those investors whose sum2. But might there be a single operation to produce that 5 way sum? Or even one operator that produces the Investor mask pTree directly from the 5 input pTrees? It would be an EINring type operator, but instead of treating the input SPTS as bitslicepTrees, it would treat them as individual mask pTrees Base cTrees A B C D E… (Stocks) Base cTrees        (Days)   1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 2 3 4 5 6 7 8 9 a b v d e f g h i 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3      1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 … (Investors) 1 2 3 4 5 6 7 8 9 a b v d e f g h i ctS ctD ctI      The Base Cliques are the 1-1-many cliques like those above (1 Stock, 1 Day, many Investors. And it doesn’t matter whether Stocks or Days is on top). Other Base Cliques are (1 Investor, 1 Day, many Stocks) and (1 Stock, 1 Investor, many Days): 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0      CtI CtS CtD 3-Level pTrees with strides, 18 (Investors) 5 (Days) 14 (Stocks) 3-Level pTrees with strides, 18 (Investors) 14 (Stocks) 5 (Days) 1 1 Is there a fast MaxClique mining algorithm for tripartite graphs? 1 2 3 4 5 … (Investors) 1 2 3 4 5 … (Investors) 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0        (Days) A B C D E … ABCDE… ABCDE… ABCDE… (Stocks) ABCDE…        1 1 1 1 1 1 0 1 1 0 0 0 0 0 8 1 1 1 1 1 1 0 1 0 0 0 0 0 0 7 1 1 0 0 1 1 0 1 1 0 0 0 0 0 6 1 1 1 1 1 1 0 1 1 0 0 0 0 0 8 0 0 0 1 1 1 0 1 1 0 0 0 0 0 5 1 0 1 1 0 3 1 0 1 0 0 2 0 0 1 1 0 2 1 0 1 1 0 3 1 0 1 0 0 2 A B C D E F G H I J K L M N … (Days) … (Stocks) 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1

  5. 3PART HyperGraph, HG1. Base cliqueTrees (Stocks, Days, Investors) A B C D E F G H I J K L M N 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 5 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 4 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 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0 0 1 0 0 0 1 0 0 0 0 0 1 1 2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 0 0 1 1 4 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 4 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1      1 2 3 4 5 6 7 8 9 a b v d e f g h i ctS ctD ctI oa with A oa with A oa with A oa with A oa with A On these we do aoa On these do aoa again Do aoa again ACD124 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 4 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 4 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 4 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 4 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 A124 ACD124 ABCDE1234 A124 A124 ACD124 ACD124 ACD124 A124 ABCDEHI124 A124 ACD124 Thus, this initial round of 3PHG clique mining yields 8 cliques containing stock=A: A124 ACD124 A124 AE124 A124 A124 ABCDEHI  124 ABCDE1234 ACD124 ACD  124 ABCDEHI124 A124 ACD  124 ABCDE124 A124 A  124 ABCDE1234 A124 ABCDE  124 AE124 A124 ACD  124 AE124 A124 A  124 ABCDEHI124 AE124 AE  124 AE124 2 and3 are contained in 5. 4 is contained in 7. So there are only 5. AE124 The same 3 Investors, 124, are doing all the recommendations. Stock=A alone is recommended on 4 days by the 3 investors. Is that’s the strongest signal? ACD124 AE124 ABCDE1234 ABCDEHI124

  6. J C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A B 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 A C 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 A D 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 A E 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 A F 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 A G 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 A H 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 D E 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 D F 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 D G 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 3 D H 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 D I 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 G H 0 1 1 1 0 0 1 0 1 1 0 0 1 0 1 0 0 0 8 G I 0 0 1 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 5 B C 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 B D 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 C E 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 6 C F 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 5 C G 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 C H 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 C I 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 E F 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 6 E G 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 6 E H 1 1 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 7 E I 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 3 J L 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 5 A I 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 A J 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A K 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A M 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A N 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B A 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 B J 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B K 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B M 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B N 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D C 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 C L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D J 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D K 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 F E 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 6 F G 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 4 H G 0 1 1 1 0 0 1 0 1 1 0 0 1 0 1 0 0 0 8 K C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 L C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 K D 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 L E 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 K E 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 L F 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 J E 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 K F 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 L G 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 4 J F 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 K G 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 2 L H 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 5 J G 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 3 K H 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 L I 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 5 J H 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 4 K I 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 3 M A-F 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 N A-F 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 J I 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 K J 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 2 K L 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 2 M G 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 2 N G 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 2 N H 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 M H 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 M I 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 N I 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 M J 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 N J 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 M K 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 N K 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 M L 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 N L 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 N M 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 M N 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 B E 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 B F 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 B G 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 B H 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 B I 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 F H 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 7 F I 1 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 4 H I 1 0 1 0 0 0 0 1 1 1 1 1 1 0 0 1 0 0 9 APPENDIX Breadth First Bipartite Max Clique Mine on G9 Each LETTERpTree is a MCLQ unless there are pairwise ANDs with the same count. I H 1 0 1 0 0 0 0 1 1 1 1 1 1 0 0 1 0 0 9 A 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 B 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 C 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 6 D 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 E 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 8 F 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 8 G 0 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 a H 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 e I 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 c J 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 5 K 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 4 L 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 6 M 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 N 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 1 2 3 4 5 6 7 8 9 a b v d e f g h i 124-ACEFH is a (3,5)MCLQ 359-EGHI is a (3,4)MCLQ (1,3,8,9,10,11,12,13,16)-HI is a (9,2)MCLQ 345-CDEG is a (3,4)MCLQ 123-BCEFH is a (3,5)MCLQ (10,13)-GHIL is a (2,4)MCLQ 12346-CEFH is a (5,4)MCLQ (14,17,18)-IK is a (3,2)MCLQ 123456-CE is a (6,2)MCLQ 1345-CDE is a (4,3)MCLQ (11,12,13)-HIJL is a (3,4)MCLQ 2347-EFGH is a (4,4)MCLQ (11,12,13,14,15)-JL is a (5,2)MCLQ (12,13)-HIJLMN is a (2,6)MCLQ 78-EFG is a (2,3)MCLQ (12,13,14)-MIJLN is a (3,5)MCLQ The LETpTrees that are (many,1)MCLQs: E,F,G,H,I,K,L The #pTrees that are (1,many)MCLQs: 1,2,3,4,13,14,15 2 1 1 1 1 0 1 1 0 1 0 0 0 0 0 0 6 2 3 0 1 1 0 1 1 1 1 0 0 0 0 0 0 6 2 4 1 0 1 0 1 1 1 1 0 0 0 0 0 0 6 2 d 0 0 0 0 0 0 1 1 0 0 0 0 0 0 2 2 e 0 0 0 0 0 1 1 0 0 0 0 0 0 0 2 1 e 0 0 0 0 0 1 0 0 1 0 0 0 0 0 2 3 1 0 1 1 1 1 1 0 1 1 0 0 0 0 0 7 3 e 0 0 0 0 0 1 1 0 1 0 0 0 0 0 3 4 1 1 0 1 1 1 1 0 1 0 0 0 0 0 0 6 4 3 0 0 1 1 1 1 1 1 0 0 0 0 0 0 6 5 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 3 4 e 0 0 0 0 0 1 1 0 0 0 0 0 0 0 2 5 2 0 0 1 0 1 0 1 0 0 0 0 0 0 0 3 5 3 0 0 1 1 1 0 1 0 0 0 0 0 0 0 4 5 4 0 0 1 1 1 0 1 0 0 0 0 0 0 0 4 5 6 0 0 1 0 1 0 0 0 0 0 0 0 0 0 2 5 7 0 0 0 0 1 0 1 0 0 0 0 0 0 0 2 5 9 0 0 0 0 1 0 1 0 0 0 0 0 0 0 2 5 a 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 5 b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 d 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 5 e 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 5 f 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 6 1 0 0 1 0 1 1 0 1 0 0 0 0 0 0 4 6 2 0 0 1 0 1 1 0 1 0 0 0 0 0 0 4 6 3 0 0 1 0 1 1 0 1 0 0 0 0 0 0 4 6 4 0 0 1 0 1 1 0 1 0 0 0 0 0 0 4 6 5 0 0 1 0 1 0 0 0 0 0 0 0 0 0 2 6 7 0 0 0 0 1 1 0 1 0 0 0 0 0 0 3 6 9 0 0 0 0 1 0 0 1 0 0 0 0 0 0 2 6 a 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 6 b 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 6 c 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 6 d 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 6 e 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 6 f 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 3 0 1 1 1 1 1 0 1 1 0 0 0 0 0 7 7 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 3 7 2 0 0 0 0 1 1 1 1 0 0 0 0 0 0 4 7 3 0 0 0 0 1 1 1 1 0 0 0 0 0 0 4 7 4 1 0 0 0 1 1 1 1 0 0 0 0 0 0 4 7 5 0 0 0 0 1 0 1 0 0 0 0 0 0 0 2 7 6 0 0 0 0 1 1 0 1 0 0 0 0 0 0 3 7 9 0 0 0 0 1 0 1 1 0 0 0 0 0 0 3 7 a 0 0 0 0 0 0 1 1 0 0 0 0 0 0 2 7 b 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 7 c 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 7 d 0 0 0 0 0 0 1 1 0 0 0 0 0 0 2 7 e 0 0 0 0 0 1 1 0 0 0 0 0 0 0 2 7 f 0 0 0 0 0 0 1 1 0 0 0 0 0 0 2 Each #pTree is a MCLQ unless  pairwise ANDs with same Ct, then those numbers together with common letters form a MCLQ 4 2 1 0 1 0 1 1 1 1 0 0 0 0 0 0 6 4 d 0 0 0 0 0 0 1 1 0 0 0 0 0 0 2 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 8 2 1 1 1 0 1 1 1 1 0 0 0 0 0 0 7 3 0 1 1 1 1 1 1 1 1 0 0 0 0 0 8 4 1 0 1 1 1 1 1 1 0 0 0 0 0 0 7 5 0 0 1 1 1 0 1 0 0 0 0 0 0 0 4 6 0 0 1 0 1 1 0 1 0 0 0 0 0 0 4 7 0 0 0 0 1 1 1 1 0 0 0 0 0 0 4 8 0 0 0 0 0 1 0 1 1 0 0 0 0 0 3 9 0 0 0 0 1 0 1 1 1 0 0 0 0 0 4 a 0 0 0 0 0 0 1 1 1 0 0 1 0 0 4 b 0 0 0 0 0 0 0 1 1 1 0 1 0 0 4 c 0 0 0 0 0 0 0 1 1 1 0 1 1 1 6 d 0 0 0 0 0 0 1 1 1 1 0 1 1 1 7 e 0 0 0 0 0 1 1 0 1 1 1 1 1 1 8 f 0 0 0 0 0 0 1 1 0 1 1 1 0 0 5 g 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 h 0 0 0 0 0 0 0 0 1 0 1 0 0 0 2 i 0 0 0 0 0 0 0 0 1 0 1 0 0 0 2 A B C D E F G H I J K L M N h or i 9 1 0 0 0 0 1 0 0 1 1 0 0 0 0 0 3 9 2 0 0 0 0 1 0 1 1 0 0 0 0 0 0 3 9 3 0 0 0 0 1 0 1 1 1 0 0 0 0 0 4 9 4 1 0 0 0 1 0 1 1 0 0 0 0 0 0 3 9 5 0 0 0 0 1 0 1 0 0 0 0 0 0 0 4 9 6 0 0 0 0 1 0 0 1 0 0 0 0 0 0 2 9 7 0 0 0 0 1 0 1 1 0 0 0 0 0 0 3 9 a 0 0 0 0 0 0 1 1 1 0 0 0 0 0 3 9 b 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 9 c 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 9 d 0 0 0 0 0 0 1 1 1 0 0 0 0 0 3 9 e 0 0 0 0 0 0 1 0 1 0 0 0 0 0 2 9 f 0 0 0 0 0 0 1 1 0 0 0 0 0 0 2 8 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 2 8 2 0 0 0 0 0 1 0 1 0 0 0 0 0 0 2 8 3 0 0 0 0 0 1 0 1 0 0 0 0 0 0 2 8 4 0 0 0 0 0 1 0 1 0 0 0 0 0 0 2 8 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 6 0 0 0 0 0 1 0 1 0 0 0 0 0 0 2 8 7 0 0 0 0 0 1 0 1 0 0 0 0 0 0 3 8 9 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 8 a 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 8 b 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 8 c 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 a 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 8 d 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 a 2 0 0 0 0 0 0 1 1 0 0 0 0 0 0 2 8 e 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 a 3 0 0 0 0 0 0 1 1 1 0 0 0 0 0 3 8 f 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 a 4 0 0 0 0 0 0 1 1 0 0 0 0 0 0 2 a 5 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 a 6 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 a 7 0 0 0 0 0 0 1 1 0 0 0 0 0 0 2 a 9 0 0 0 0 0 0 1 1 1 0 0 0 0 0 3 c 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 c 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 c 3 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 c 4 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 c d 0 0 0 0 0 0 0 1 1 1 0 1 1 1 6 c e 0 0 0 0 0 0 0 0 1 1 0 1 1 1 5 g 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 g 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 g 3 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 g 4 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 g 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g 6 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 g 7 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 g 8 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 g 9 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 g a 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 g b 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 g c 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 g d 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 g e 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 g f 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 g h 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 g i 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 d 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 d 2 0 0 0 0 0 0 1 1 0 0 0 0 0 0 2 d 3 0 0 0 0 0 0 1 1 1 0 0 0 0 0 3 d 4 0 0 0 0 0 0 1 1 0 0 0 0 0 0 2 d e 0 0 0 0 0 0 1 0 1 1 0 1 1 1 6 f 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 f 2 0 0 0 0 0 0 1 1 0 0 0 0 0 0 2 f 3 0 0 0 0 0 0 1 1 0 0 0 0 0 0 2 f 4 0 0 0 0 0 0 1 1 0 0 0 0 0 0 2 f c 0 0 0 0 0 0 0 1 0 1 0 1 0 0 3 f d 0 0 0 0 0 0 1 1 0 1 0 1 0 0 4 f e 0 0 0 0 0 0 1 0 0 1 1 1 0 0 3 a b 0 0 0 0 0 0 0 1 1 0 0 1 0 0 3 a c 0 0 0 0 0 0 0 1 1 0 0 1 0 0 3 a d 0 0 0 0 0 0 1 1 1 0 0 1 0 0 4 a e 0 0 0 0 0 0 1 0 1 0 0 1 0 0 3 a f 0 0 0 0 0 0 1 1 0 0 0 1 0 0 3 a g 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 a h 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 a i 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 e 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 2 e 3 0 0 0 0 0 1 1 0 1 0 0 0 0 0 3 e 4 0 0 0 0 0 1 1 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 9 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 a 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 b 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 c 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 d 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 e 0 0 0 0 0 0 0 0 1 0 1 0 0 0 2 f 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 g 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 h 0 0 0 0 0 0 0 0 1 0 1 0 0 0 2 i 0 0 0 0 0 0 0 0 1 0 1 0 0 0 2 b c 0 0 0 0 0 0 0 1 1 1 0 1 0 0 4 b d 0 0 0 0 0 0 0 1 1 1 0 1 0 0 4 b e 0 0 0 0 0 0 0 0 1 1 0 1 0 0 3 b f 0 0 0 0 0 0 0 1 0 1 0 1 0 0 3 b g 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 b h 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 b i 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 b 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 b 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 2 b 3 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 b 4 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 b 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b 6 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 b 7 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 b 9 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 b a 0 0 0 0 0 0 0 1 1 0 0 1 0 0 3

  7. Depth First Bipartite Max Clique Mine on G9 J C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D E 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 D F 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 D G 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 3 D H 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 D I 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 G H 0 1 1 1 0 0 1 0 1 1 0 0 1 0 1 0 0 0 8 G I 0 0 1 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 5 B C 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 B D 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 C E 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 6 C F 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 5 C G 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 C H 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 C I 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 E F 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 6 E G 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 6 E H 1 1 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 7 E I 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 3 J L 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 5 B A 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 B J 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B K 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B M 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B N 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D C 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 C L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D J 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D K 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 F E 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 6 F G 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 4 H G 0 1 1 1 0 0 1 0 1 1 0 0 1 0 1 0 0 0 8 K C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 L C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 K D 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 L E 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 K E 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 L F 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 J E 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 K F 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 L G 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 4 J F 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 K G 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 2 L H 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 5 J G 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 3 K H 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 J H 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 4 L I 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 5 K I 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 3 M A-F 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 N A-F 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 J I 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 K J 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 2 K L 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 2 M G 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 2 N G 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 2 N H 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 M H 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 M I 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 N I 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 N J 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 M J 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 M K 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 N K 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 N L 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 M L 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 N M 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 M N 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 B E 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 B F 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 B G 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 B H 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 B I 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 F H 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 7 F I 1 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 4 H I 1 0 1 0 0 0 0 1 1 1 1 1 1 0 0 1 0 0 9 Find all MCLQ(A) A B 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 A C 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 A D 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 A E 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 A F 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 A G 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 A H 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 I H 1 0 1 0 0 0 0 1 1 1 1 1 1 0 0 1 0 0 9 A I 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 A J 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A K 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A M 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A N 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 B 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 C 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 6 D 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 E 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 8 F 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 8 G 0 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 a H 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 e I 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 c J 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 5 K 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 4 L 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 6 M 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 N 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 The ApTreeMCLQ(A) unless there are A&XpTrees with Ct(A&X)=Ct(A). Ct(A)=Ct(AC)=Ct(AE)=Ct(AF)=Ct(AH)=3, so ACEFH-124MCLQ(A). Next check for MCLQs with Ct=Ct(A)-1=2. We have 2# CLQs: ACEFH-12, ACEFH-14 and ACEFH-24. Each Ct(A&XpTree)=2=CtA-1 expands one of these 3. (namely AB-12 AD-14 AG-24), Each expanded CLQ is maximal. We get new 2# MCLQ(A): ABCEFH-12 ACDEFH-14 ACEFGH-24 Next check MCLQs with Ct=Ct(A)-2=1. When we reach Ct=1 we simply check the #pTrees containing A (e.g., 1,2,4) for maximal with the 1st step. A B C D E F H I 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 A B C E F G H 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 A C D E F G H 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 The BpTreeMCLQ(B) unless Ct(B&X)=Ct(B). 3=Ct(B)=Ct(BC)=Ct(BE)=Ct(BF)=Ct(BH), so BCEFH-123MCLQ(B). A B C E F H 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 A C D E F H 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 A C E F G H 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 Next check MCLQs w Ct=Ct(B)-1=2. We have 2# CLQs: BCEFH-12, BCEFH-13, BCEFH-23 . Each Ct(B&X)=2 expands one of these 3. (namely BA-12 BD-13 BG-23 ), Each expanded CLQ is max. We get 2# MCLQ(B): ABCEFH-12 BCDEFH-13 BCEFGH-23, but ABCEFH-12 not new. Next MCLQs w Ct=Ct(B)-2=1. Check #pTreesB (e.g., 1,2,3) w 1st step (only 3 is new). CMCLQ(C) unless Ct(C&X)=Ct(C). 6=Ct(C)=Ct(CE) so CE-123456MCLQ(C). Next check MCLQs w Ct=Ct(C)-1=5. We have 5# CLQs: CE-12345 CE-12346 CE-12356 CE-12456 CE-13456 CE-23456 Each Ct(C&X)=5 expands one of these, namely CF-12346 (expands to CEF-12346) The Ct(C&X)=4 are CG-2345 CH-1234 may expand MCLQ(C)s CEF-12346 CE-123456 (check in this order) CG-2345 expands CE-123456 to CEG-2345 and CH-1234 expands CEF-12346 to CEFH-1234) B C D E F G H I 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 B C D E F H 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 B C E F G H 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 C D E F H I 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 The Ct(C&X)=2 is CI-13 may expand MCLQ(C)s CEFH-1234 CEG-2345 CEF-12346 CE-123456 (check in this order). CI-13 expands CEFH-1234 to CEFHI-13 C E F H I 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 C D E F H 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 A C E F H 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 B C E F H 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 C D E G 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 3 C E F H 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 C D E 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 C E G 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 C E F 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 5 C E 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 6 DMCLQ(D) unless Ct(D&X)=Ct(D). 4=Ct(D)=Ct(CD) =Ct(DE) so CDE-1345MCLQ(D). Next Ct=Ct(D)-1=3. 3# CLQ(D)s: DF-134 DG-345 DH-134, so we have DFH-134 and DG-345 Each expands a maximal: DFH-134 expands CDE-1345 to CDEFH-134. DG-345 expands CDE-1345 to CDEG-345 The Ct(D&X)=2 is DI-13 which expands CDEFH-134 to CDEFHI-13

  8. Depth First Bipartite Max Clique Mine on G9 J C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D E 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 D F 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 D G 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 3 D H 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 D I 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 G H 0 1 1 1 0 0 1 0 1 1 0 0 1 0 1 0 0 0 8 G I 0 0 1 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 5 B C 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 B D 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 C E 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 6 C F 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 5 C G 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 C H 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 C I 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 E F 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 6 E G 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 6 E H 1 1 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 7 E I 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 3 J L 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 5 B A 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 B J 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B K 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B M 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 B N 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D C 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 C L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D J 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D K 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 F E 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 6 F G 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 4 H G 0 1 1 1 0 0 1 0 1 1 0 0 1 0 1 0 0 0 8 K C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 L C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 K D 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 L E 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 K E 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 L F 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 J E 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 K F 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 L G 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 4 J F 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 K G 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 2 L H 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 5 J G 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 3 K H 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 L I 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 5 J H 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 4 K I 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 3 M A-F 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 N A-F 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 J I 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 K J 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 2 K L 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 2 N G 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 2 M G 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 2 M H 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 N H 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 M I 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 N I 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 M J 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 N J 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 M K 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 N K 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 N L 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 M L 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 N M 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 M N 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 B E 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 B F 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 B G 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 B H 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 B I 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 F H 1 1 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 7 F I 1 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 4 H I 1 0 1 0 0 0 0 1 1 1 1 1 1 0 0 1 0 0 9 Find all MCLQs containing A: The ApTree is a MCLQ unless there are pairwise ANDs with same count. A B 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 A C 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 A D 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 A E 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 A F 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 A G 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 A H 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 I H 1 0 1 0 0 0 0 1 1 1 1 1 1 0 0 1 0 0 9 A I 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 A J 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A K 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A L 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A M 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A N 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 B 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 C 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 6 D 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 4 E 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 8 F 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 8 G 0 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 a H 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 e I 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 c J 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 5 K 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 4 L 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 6 M 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 N 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 G H I J L M N 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 H I J L M N 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 I J L M 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 H I J L M N 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 2 I J L M N 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 The EpTree is a MCLQ(E) unless there are pairwise ANDs with that EpTree having Ct=Ct(E)=8. None, so E-12345679MCLQ(E). The MpTree is a MCLQ(M) unless there are pairwise M-ANDs with it having Ct=Ct(M)=3. Then Ct(MI)=Ct(MJ)=Ct)ML)=Ct(MN)=3. Next check for MCLQs with Ct=Ct(M)-1=2. Combine the LetSets of those with the same NumSet. Then combine their LetSets with the LetSet of the , Ct(MG)=Ct(MH)=2 are non-Maximal-CLQs, because they expand IJLMN(12,13,14) to GIJLMN(13,14) and HIJLMN(12,13). Next check for MCLQs with Ct=Ct(E)-1=7. E1234567 E1234569 E1234579 E1234679 E1235679 E1245679 E1345679 E2345679 are 7 non-Maximal-CLQs. Each Ct(EXpTree)=7=CtE-1, expands one of the 7 into a MCLQ(E). We first check if any pairs give have the same #set by pairwise ANDing them: None give the same #set (There is just one E-AND, EH1234679) b 0 0 0 0 0 0 0 1 1 1 0 1 0 0 4 c 0 0 0 0 0 0 0 1 1 1 0 1 1 1 6 d 0 0 0 0 0 0 1 1 1 1 0 1 1 1 7 When Ct=1 check #pTrees containing M(12,13,14). 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 8 2 1 1 1 0 1 1 1 1 0 0 0 0 0 0 7 3 0 1 1 1 1 1 1 1 1 0 0 0 0 0 8 4 1 0 1 1 1 1 1 1 0 0 0 0 0 0 7 5 0 0 1 1 1 0 1 0 0 0 0 0 0 0 4 6 0 0 1 0 1 1 0 1 0 0 0 0 0 0 4 7 0 0 0 0 1 1 1 1 0 0 0 0 0 0 4 9 0 0 0 0 1 0 1 1 1 0 0 0 0 0 4 A B C D E F G H I J K L M N A B C D E F H I 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 B C D E F G H I 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 A B C E F G H 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 A C D E F G H 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 E I 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 3 G I J L M N 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 2 …Each Ct(EXpTree)=3=Ct(E)-5 (i.e., EI) expands one into a MCLQ(E). Each Ct(EXpTree)=6=Ct(E)-2 (EF, EG) expands one. 1st ANDs, EF&EG= EFG-2347 C D E G 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 C E F H 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 C E F H 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 E G H I 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 E F G 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 4 E G 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 6 E F 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 6 E H 1 1 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 7 E 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 8 When Ct=1 check the #pTrees containing E (12345679) for maximal using 1st step above.

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