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.). More simply put:  pTree, A, the AND of all pairwise ANDs. A&B, B with Ct(A&B)=Ct(A) is a MCLQ (including 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

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

G9 H I F E C K A L D M J B G N 124-ACEFH is a (3,5)MCLQ 345-CDEG is a (3,4)MCLQ 359-EGHI is a (3,4)MCLQ (1,3,8,9,10,11,12,13,16)-HI is a (9,2)MCLQ 123-BCEFH is a (3,5)MCLQ 12346-CEFH is a (5,4)MCLQ (10,13)-GHIL is a (2,4)MCLQ (14,17,18)-IK is a (3,2)MCLQ 123456-CE is a (6,2)MCLQ 2347-EFGH is a (4,4)MCLQ (11,12,13)-HIJL is a (3,4)MCLQ 1345-CDE is a (4,3)MCLQ (11,12,13,14,15)-JL is a (5,2)MCLQ 78-EFG is a (2,3)MCLQ (12,13)-HIJLMN is a (2,6)MCLQ (12,13,14)-MIJLN is a (3,5)MCLQ G9 A B C D E F G H I J K L M N Bipartite graph of the Southern Women Event Participation. Women are numbers, events are letters. Or Investors are numbers and stocks are letters.

Bipartite Max Clique Mine on G9-2 What we have NOT yet found is MCLQs that do not include a full pTree A C E F H 1 3 B C E F H 1 3 D C E 1 4 B A 1 2 D A 1 2 G A 1 2 A I 1 A J A K A L A M A N A B 1 2 D B 1 2 G B 1 2 I B 1 2 B J B K B L B M B N E C 1 6 F C 1 5 G C 1 4 H C 1 4 I C 1 2 C L F D 1 3 G D 1 3 H D 1 3 I D 1 2 D J D K D L F E 1 6 G E 1 6 H E 1 7 I E 1 3 E F 1 6 G F 1 4 H F 1 7 I F 1 4 H G 1 8 I G 1 5 G H 1 8 I H 1 9 H I 1 9 J C J E J F 1 G J 1 3 H J 1 4 J I 1 L J 1 5 K C K D K E K F 1 G K 1 2 K H 1 I K 1 3 J K 1 2 L K 1 2 L C L E L F 1 G L 1 4 H L 1 5 I L 1 5 M A-F G M 1 2 H M 1 2 I M 1 3 J M 1 3 M K 1 L M 1 3 N M 1 3 N A-F G N 1 2 H N 1 2 I N 1 3 J N 1 3 N K 1 L N 1 3 M N 1 3 A 1 3 B 1 3 C 1 6 D 1 4 E 1 8 F 1 8 G 1 a H 1 e I 1 c J 1 5 K 1 4 L 1 6 M 1 3 N 1 3 A C E F H D G 1 2 B A B C E F H 1 2 D G I B C D E F H 1 2 G I B C E F H 1 2 I C D E F H 1 3 G E C F 1 5 14-ABCEFGH is a (2,7)MCLQ 12-ABCEFH is a (2,6)MCLQ 13-BCDEFHI is a (2,7)MCLQ 12346-CEF is a (5,3)MCLQ 134-CDEFH is a (3,5)MCLQ 345-CDEFG is a (3,5)MCLQ

Breadth 1st Bipartite Clique Thm on G9 (LETpTrees; exhaustive search; elim if Ct=0|1 AAC; BBC; CCE; DCD; MIM; NIN; B A 1 2 C A 1 3 D A 1 2 E A 1 3 F A 1 3 G A 1 2 H A 1 3 C B 1 3 D B 1 2 E B 1 3 F B 1 3 G B 1 2 H B 1 3 I B 1 2 D C 1 4 E C 1 6 F C 1 5 G C 1 4 H C 1 4 I C 1 2 E D 1 4 F D 1 3 G D 1 3 H D 1 3 I D 1 2 F E 1 6 G E 1 6 H E 1 7 I E 1 3 G F 1 4 H F 1 7 I F 1 4 H G 1 8 I G 1 5 J G 1 3 K G 1 2 L G 1 4 M G 1 2 N G 1 2 I H 1 9 J H 1 4 L H 1 5 M H 1 2 N H 1 2 J I 1 4 K I 1 3 L I 1 5 M I 1 3 N I 1 3 K J 1 2 L J 1 2 L K 1 2 M L 1 3 N L 1 3 N M 1 3 A 1 3 B 1 3 C 1 6 D 1 4 E 1 8 F 1 8 G 1 a H 1 e I 1 c J 1 5 K 1 4 L 1 6 M 1 3 N 1 3 Cliques: ABCEFH12; ADEFH13; BDEFHI13; GIJLMN(13,14); HLMN(12,13); ILMN(12,13,14); B A C 1 2 B A E 1 2 B A F 1 2 B A H 1 2 C A D 1 2 C A E 1 3 C A F 1 3 C A G 1 2 C A H 1 3 D A E 1 2 D A F 1 2 D A H 1 2 E A F 1 3 E A G 1 2 E A H 1 3 F A G 1 2 F A H 1 3 G A H 1 2 C B D 1 2 C B E 1 3 C B F 1 3 C B G 1 2 C B H 1 3 C B I 1 2 D B E 1 2 D B F 1 2 D B H 1 2 D B I 1 2 E B F 1 3 E B G 1 2 E B H 1 3 E B I 1 2 F B G 1 2 F B H 1 3 F B I 1 2 G B H 1 2 H B I 1 2 D C E 1 4 D C F 1 3 D C G 1 3 D C H 1 3 D C I 1 2 E C F 1 5 E C G 1 4 E C H 1 4 E C I 1 2 F C G 1 3 F C H 1 4 F C I 1 2 G C H 1 3 H C I 1 2 E D F 1 3 E D G 1 3 E D H 1 3 E D I 1 2 F D G 1 2 F D H 1 3 F D I 1 2 G D H 1 2 H D I 1 2 F E G 1 4 F E H 1 6 F E I 1 2 G E H 1 5 G E I 1 2 H E I 1 3 G F H 1 4 H F I 1 3 A B C E F H 1 2 G I J L M N 1 2 A D E F H 1 2 B D E F H 1 2 A C D E 1 2 A C D F 1 2 A C D H 1 2 A C E F 1 3 A C E G 1 2 A C E H 1 3 A C F G 1 2 A C F H 1 3 A C G H 1 2 I L M N 1 2 H G I 1 4 H G J 1 2 H G L 1 3 I G J 1 2 I G L 1 2 I G M 1 2 I G N 1 2 J G K 1 2 J G L 1 3 J G M 1 2 J G N 1 2 K G L 1 2 L G M 1 2 L G N 1 2 M G N 1 2 I H J 1 3 I H L 1 4 I H M 1 2 I H N 1 2 J H L 1 4 J H M 1 2 J H N 1 2 L H M 1 2 L H N 1 2 M H N 1 2 J I L 1 5 J I M 1 3 J I N 1 3 L I M 1 3 L I N 1 3 M I N 1 3 K J L 1 2 M L N 1 3

G9 Bipartite Clique Thm on G9 (#pTrees) H I F E C K A L D M J B G N Maximal Cliques containing 1 2 1 3 5 4 6 7 8 9 a b c e d f g h i 1 2 6 3 7 4 5 8 9 a b c d e f g h i W 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 8 B A C D E F G H I J K L M N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 B A C D E F G H I J K L M N B A C D E F G H I J K L M N 8 7 8 8 4 4 4 3 4 4 4 6 7 8 5 2 2 2 Next I x out those that are contained in the next level (or have Ct=0 which means there is no subgraph). Each WpTree is itself a 1-many clique, maximal iff no other contains it, Only 3 4 14 qualify based on Ct but none contains (“& with 1” count<8). Many of these3-many are contained in uncomputed also. 13a,13b,13c,13d,13g13abcde and 13h,13i13hi Each 2W pTree is a 2-many clique, maximal iff no 3-many contains it (check later) 2 1 3 4 5 6 7 8 9 a b c e d f g There are 2-many cliques that are contained in 3-many which weren’t computed, such as 15135, 18 138, 19139, 1a13a, 1b13b, 1c13c, 1e13e, 1d13d, 1g13g, 1h13h, 1i13i. 2 1 3 4 5 6 7 8 9 a b c e d f g G9 A B C D E F G H I J K L M N Bipartite graph of the Southern Women Event Participation. Women are numbers, events are letters. B A C D E F G H I J K L M N 1 2 3 4 5 6 7 9 1 3 a b c d e 2 1 2 3 4 5 6 8 9 a b c e d f g 7 3 1 h i 1 3 5 1 3 8 1 3 9 B A C D E F G H I J K L M N B A C D E F G H I J K L M N If numbers=investors and letters=stock (recommends relationship), would we like a clique with many investors and many stocks? eg, MaxClique 12346CEFH (5 investors, 4 stocks). Is 12345679E (8 investor, 1 stock) better? A K-1 clique  an original pTree (e.g., 12345679E  LETTERpTree (E) (actually =). Thus, we can remove if Ct= 0 or 1.

Bipartite Clique Thm on G9 (LETpTrees; &ing w highest Ct; elim if Ct=0|1) I H G B 1 C D E 2 F J L M N I H A 1 B 2 C D E 3 F G 4 J L M N A H 1 3 B C 5 D E 7 F G 8 I 9 J 4 K L M 2 N A 1 3 B 1 3 C 1 6 D 1 4 E 1 8 F 1 8 G 1 a H 1 e I 1 c J 1 5 K 1 4 L 1 6 M 1 3 N 1 3 1 2 3 4 5 6 7 8 9 a b c d e f g h i Using LETpTrees; & w lowest Ct; elim Ct=0|1) 6-2 CLQ: ABCEFH12 Of course we should do an exhaustive search! B A C 1 2 D E F G H B A 1 2 C 3 D E 4 F G H I J L M N K 5 6 7 8 9 a b c d e f g h i A B C D E F G H I J K L M N G9 Bipartite graph of Southern Women Event Participation. Women=#s, events=letters. Or a recommender: Analyst=#s, stocks=letters