1. GRAPH (linear edges, 2 vertices) kHYPERGRAPH (edges=k vertices)
Edge Count Clique Thms Graph C is a clique iff |EC||PUC|=COMB(|VC|,2)|VC|!/((|VC|-2)!2!)
(VC,EC) is a k-clique iff  induced k-1 subgraph, (VD,ED) is a (k-1)-clique.
Apriori Clique Mining Alg Uses an ARM-Apriori-like downward closure property: CSkkCliqueSet, CCSk+1Candidatek+1CliqueSet. By SGE, CCSk+1= all s of CSk pairs w k-1 common vertices. Let CCCSk+1 be a union of 2 k-cliques w k-1 common vertices. Let v,w be the kth vertices (different) of the w k-cliques: CCSk+1 iff (PE)(v,w)=1.
Breadth-1st Clique Alg: CLQK=all Kcliques. Find CLQ3 w CS0. A Kclique and 3clique sharing 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 MaxClique 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 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…
GRAPH (linear edges, 2 vertices) kHYPERGRAPH (edges=k vertices)
kPARTITE GRAPH
(V=!Vi i=1..k (x,y)Ex,ysame Vi )
kPARTITE HYPERGRAPH
(V=!Vi i=1..k (x1..xk)Exj,xjsame Vi )
Note, for k=2 there is no difference between a 2graph and a 2hypergraph.
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 (1 edge). Taking any MCLQ 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? In a Tripartite Graph edges must start and end in different vertex parts. E.g., 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 Graphs Bipartite, Tripartite (have 2,3 PARTs resp.) … The rule is that no edge can start and end in the same PART.
Hyperpartite Clique Mining: A 3hyperpartite graph has 3 vertex PARTS and each edge is a 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.). Another example might involve tweets: PART1=tweeters; PART2=hashtags; PART3=tweets.
Conjecture: KmultiPARTITEcliques and KhyperPARTITEcliques 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? We also represent these common objects using cliqueTrees (cTrees).

2. Bipartite Graph cTrees (If there are only 2PARTs
Bipartite Graph cTrees (If there are only 2PARTs. Multipartite and Hyperpartite collapse to the same thing. 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 A I 1 A B 1 2 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 E C 1 6 F C 1 5 G C 1 4 H C 1 4 I C 1 2 C D 1 4 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 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 F 1 G J 1 3 H J 1 4 L J 1 5 K F 1 G K 1 2 K H 1 I K 1 3 J K 1 2 L K 1 2 L F 1 G L 1 4 H L 1 5 I L 1 5 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 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 1 2 3 4 5 6 7 8 9 a b v d e f g h i
This bipartite graph clique TreeSet has NUMpTree on top and LETpTree at bottom (e.g., Nums=Investors, Lets=Socks):
(10,13)-GHIL
345-CDEG
12346-CEFH
2347-EFGH
78-EFG
359-EGHI
(11,12,13)-HIJL
(12,13)-HIJLMN
(1,3,8,9,10,11,12,13,16)-HI
(14,17,18)-IK
1-ABCDEFHI
2-ABCEFGH
3-BCDEFGHI
4-ACDEFGH
13-GHIJLMN
14-FGIJKLMN
15-GHJKL
MBCLQ(X)
X =? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
17,18
It is closed under AND (call it aa). The OR of 2 cTrees with a substantial AND might reveal interesting communities.
It is closed under OR-AND (called oa) and AND-OR (called ao).
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)? oa applied 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?
(BCS.oa) generates (CLQ,oa)
Commutative monoids (lack inverses) (CLQ,oa) (CLQ,ao) (CLQ,aa). B A 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 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 c
BCS={32 base cliques}; ops, and-and (aa), or-and (oa), and-or (ao), each forming a comm. Monoid.
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 inverse of x but no AND inverse.
124-ACEFH
123-BCEFH CE
1345-CDE
(12,13,14)-MIJLN
(11,12,13,14,15)-JL E
( ,14)-F
(234579,10,13,14,15)-G
( ,11,12,13,15,16)-H
( ,11,12,13,14,16,17,18)-I
( )-K
(10,11,12,13,14,15)-L
12-ABCEFH
14-ACDEFH
24-ACEFGH
1-ACEFHI
MBCLQ(X) X =? A B C D E F G H I J K L M N

3. is connected to each final PART so to the OR of those PARTs.
The Base CliqueTrees for a 3PartiteHyperGraph: {123..}=Investors recommending Stocks={ABC..} on Days={,,,,}. (Investors are the leaf pTrees) B A 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
CtS
CtD
CtI 1 3 1 3 1 4 1 3 1 3 1 2 1 2 1 5 1 3 1 2 1 4 1 2 1 4 1 3 1 2 1 4 1 2 1 4 1 3 1 1 3 1 2 1 4 1 3 1 3 1 3 1 2 1 4 1 3 1 3 1 3 1 4 1 4 1 3 1 3 1 3 1 3 1 4 1 6 1 3 1 3 1 3 1 4 1 6 1 3 1 3 1 2 1 4 1 5 1 3 1 2 1 2 1 2 1 1 3 1 2 1 2 1 4 1 4 1 3 1 2 1 2 1 4 1 4 1 3 1 2 1 2 1 1 2 1 1 2 3
oaa w A 1 2
oaa w A 3 1 4
oaa A 1 3 2
oaa with A 1 2 3
oaa w A
ABCDEHI 
ABCDE 
ABCDE 
A  124
AE 
ACD 
On these we do aoa
ACD124
ABCDE1234
ACD124
ACD124
ABCDEHI124
A124
AE124
ABCDE124
AE124
AE124
On these do aoa again
Do aoa again
ACD124
A124
A124
A124
ACD124
A124
A124
2, 3  5.
4  7.
1st rnd: 6 cliques w A:
ABCDEHI 
ABCDE 
ACD 
ACD 
A 
ABCDE 
A  124
AE 
ACD 
A124
A124
A124
Investors, 124, do all recommends. Stock=A alone recommended on 4 days by 3 investors. Strong signal?
ACD124
A124
A124
A124
A124
A124
AE124
THM: For a khypergraph, an op of k-1 ANDs, (e.g., aao, aoa, aaao) result
in a clique. Pf: When we AND all but 1 PART each resulting (k-1)VertexPolyhedrons
is connected to each final PART so to the OR of those PARTs.
ACD124
ABCDE1234
ABCDEHI124
AE124

4. Stocks-Days (leaf pTrees are over Investors);
The Base CliqueTrees for a 3PartiteHyperGraph: {123..}=Investors recommending Stocks={ABC..} on Days={,,,,}. (Investors are the leaf pTrees) B A 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
CtS
CtD
CtI 1 3 1 3 1 4 1 3 1 3 1 2 1 2 1 5 1 3 1 2 1 4 1 2 1 4 1 3 1 2 1 4 1 2 1 4 1 3 1 1 3 1 2 1 4 1 3 1 3 1 3 1 2 1 4 1 3 1 3 1 3 1 4 1 4 1 3 1 3 1 3 1 3 1 4 1 6 1 3 1 3 1 3 1 4 1 6 1 3 1 3 1 2 1 4 1 5 1 3 1 2 1 2 1 2 1 1 3 1 2 1 2 1 4 1 4 1 3 1 2 1 2 1 4 1 4 1 3 1 2 1 2 1 1 2 1 1 2
oaa w B 1 2
oaa w B 1 4 5
oaa w B
aoa w B 1 2 5 3
… Note oaa w B yields BaseMaxClique B  only, .i.e., the orig BMC is not expanded by this alg. Also, a Clique that doesn’t get IDed using cTrees w CtS=4 is ACDE  Next slide: we cover this clique w ABCDE  Using the alg on the Base Cliques for Investors and Days (rather than those used here which are for Stocks and Days).
Conjecture: get all (interesting?) MaxCliques from this alg applied to all 3 BaseCliques,
Stocks-Days (leaf pTrees are over Investors);
Investors-Days (leaf pTrees are over Stocks);
Stocks-Investors (leaf pTrees are over Days).
The 3 Base cTreeSets for this 3hyperpartite Graph, Investors-Stocks-Days provide us with all the Many-to-One(pair) Cliques. The alg just expands those that can be expanded by examining counts of the pairwise ANDs. B A 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
CtS
CtD
CtI
ABCD 
B 
etc.
ABCDE 
B 
etc.
Thus, for a khypergraph,
we AND leaf pTrees, so we can
OR any 1 of the k-1 other PARTs,
for each of the k possible
leaf PARTs.
We always AND the leaf pTrees, so we can ao or we can oa nonleaves.
Thus, there are k(k-1) ways of applying the Expanded Base Clique Mining alg., each giving more MaxCliques (with some redundancy too)
For 3hgraphs, there are 6 ways of applying the theorem.

5. Investor-Day-Stock cTrees (leaf pTrees on Stocks).
ABCD  12
aao
w 1 1 d
aao
w 1 1 4
Etc.
A 
aao w 1 1 4
aao w 1 1 5
aao w 1 1 5
ABCDE  B A C D E F G H I J K L M N      1 2 3 4 5 6 7 8 9 a b c d e f g h i
CtS
CtD
CtI 1 4 1 4 1 1 3 1 1 1 1 1 1 1 1 1 7 1 1 1 8 1 7 1
ABCDEFGHIJLMN 
ABCE 
Do pairwise aoa
A  12
ABCD  12
ABCD  1
ABCE  1
A 
A  1
A  1
ABCDE  1
ABCE  1
ABCE  1
Do pairwise aoa
A  12
A  1
ABCD  1
ABC   1
ABCE  1 B A C D E F G H I J K L M N      1 2 3 4 5 6 7 8 9 a b c d e f g h i
CtS
CtD
CtI 1 4 1 5 1 3 1 4 1 1 9 1 6 1 1 1 1 1 7 1 1 1 1 1 1 1 1 5 1 5 1 1 1 1 4 1 1 8 1 1 1 1 8 1 1 1 1 1 1 5 1 5 1 5 1 5 1 1 8 1 1 1 7 1 7 1 1 8 1 1 1 1 1 1 1 d 1 b 1 2 1 7 1 1 1 1 1 1 1 1 7 1 1 7 1 1 5 1 1

6. Stock-Investor-Day cTrees (leaf pTrees on Days).B A C D E F G H I J K L M N      1 2 3 4 5 6 7 8 9 a b c d e f g h i
CtS
CtD
CtI 1 5 3 4 1 4 2 1 5 3 1 3 5 1 3 5 4 1 2 1 2 5 1 2 3 1 2 3
A  12
aao A1 1 5
aao B1 1 4 1 5
aao C1 1 3
aao D1
B  1
C  12
D  B A C D E F G H I J K L M N      1 2 3 4 5 6 7 8 9 a b c d e f g h i
CtS
CtD
CtI 1 2 3 1 2 1 2 3 1 2 1 2
E 
F  ceg
G  ceg
H  d
I  c
J  c
K 
L  ce
M  ce
N 

7. Stock-Day-Investor BaseCliqueTrees (leaves Inv)
Base CliqueTrees for 3PART HyperGraph, 3PHG2
{12345}=Investors recommending Stocks={ABCDE} on Days={,,,,}, 74 recommendations
Stock-Day-Investor BaseCliqueTrees (leaves Inv) 1 3 1 3 1 4 1 3 1 3 1 2 1 2 1 5 1 3 1 2 1 4 1 2 1 4 1 3 1 2 1 4 1 2 1 4 1 3 1 1 3 1 2 1 4 1 3 1 3
Stock-Investor-Day BaseCliqueTrees (leaves Days) B A C D E      1 2 3 4 5
CtS
CtD
CtI 1 5 1 5 1 2 1 4 1 4 1 4 1 2 1 2 1 1 5 1 5 1 3 1 3 1 3 1 5 1 3 1 3 1 3 1 5 1 3 1 4
Investor-Day-Stock BaseCliqueTrees (leaves Stocks) B A C D E      1 2 3 4 5
CtS
CtD
CtI 1 4 1 5 1 3 1 4 1 1 5 1 5 1 5 1 5 1 5 1 5 1 1 5 1 5 1 5 1 4 1 4 1 3

8. Stock-Day-Investor BaseCliqueTrees (leaves Inv)
Base CliqueTrees for 3PART HyperGraph, 3PHG2
oaa with:
aoa with:
Stock-Day-Investor BaseCliqueTrees (leaves Inv) 1 3  A 5 4   2   1 4 2  B  5  3   1 2 4  C  5  3   1 2 4  D  5  3   1 3  E  5 4   2  1 2 3  A 5 4     1 3 2  B  5    1 2 4  C 3     1 2 4  D 3    5  1 2 3  E  4    1 3 1 3 1 4 1 3 1 3 1 2 1 2 1 5 1 3 1 2 1 4 1 2 1 4 1 3 1 2 1 4 1 2 1 4 1 3 1 1 3 1 2 1 4 1 3 1 3
Then aoa on these: B A C D E      1 2 3 4 5
CtS
CtD
CtI
ACD  124
ACD  124
ACD  12
ACD  124
ABCDE  1234
ABCDE  124
AE  124
ABCD  12
B 
ABE  14
ABCD  12
CD  1234
ABCDE  2345
ABCDE  12
CD  1234
ACDE  2
CDE  234
CDE  234
A  123
A 
A 
A  1234
B 
B 
B 
B 
B 
C 
C  23
C  124
C  12
D 
D  23
D 
D 
D  2
E 
E 
E 
E 
ACD  12
ACD  14
ACD  124

9. Breadth First Bipartite Max Clique Mine on G9
APPENDIX
Breadth First 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 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
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
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
78-EFG is a (2,3)MCLQ
(12,13)-HIJLMN is a (2,6)MCLQ
(11,12,13,14,15)-JL is a (5,2)MCLQ
(12,13,14)-MIJLN is a (3,5)MCLQ
The #pTrees that are (1,many)MCLQs: 1,2,3,4,13,14,15
The LETpTrees that are (many,1)MCLQs: E,F,G,H,I,K,L
Each #pTree is a MCLQ unless  pairwise ANDs with same Ct, then those numbers together with common letters form a MCLQ 1 3 7 e 2 2 1 6 3 4 d e 3 1 7 e 4 1 6 3 e 2 d 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 7 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 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 1 2 3 4 5 6 7 8 9 a b c d e f g h i
h or i

10. Depth First Bipartite Max Clique Mine on G9 Find all MCLQ(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 A I 1 A J A K A L A M A N E B 1 3 F G 2 H I A J K L N M C D 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
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 AD AG-24),
Each expanded CLQ is maximal. We get new 2# MCLQ(A): ABCEFH-12 ACDEFH 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.
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).
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 (namely BA BD BG-23 ),
Each expanded CLQ is max. We get 2# MCLQ(B): ABCEFH-12 BCDEFH 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 MCLQ(C).
Next check MCLQs w Ct=Ct(C)-1=5. We have 5# CLQs: CE CE CE CE CE CE-23456
Each Ct(C&X)=5 expands one of these, namely CF (expands to CEF-12346)
The Ct(C&X)=4 are CG-2345 CH-1234 may expand MCLQ(C)s CEF CE (check in this order)
CG-2345 expands CE to CEG-2345 and CH-1234 expands CEF to CEFH-1234) B C D E F G H I 1 A B C D E F H I 1 G
The Ct(C&X)=2 is CI-13 may expand MCLQ(C)s CEFH CEG CEF CE
(check in this order). CI-13 expands CEFH-1234 to CEFHI-13 C D E F H I 1 2 B C D E F H 1 2 G A B C E F H 1 2 D G C D E F H 1 3 C E F H I 1 2 B C E F H 1 3 A C E F H 1 3 C D E G 1 3 C E F H 1 4 D C E 1 4 E C G 1 4 E C F 1 5 E C 1 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 DG-345 expands CDE-1345 to CDEG-345
The Ct(D&X)=2 is DI-13 which expands CDEFH-134 to CDEFHI-13

11. Depth First Bipartite Max Clique Mine on G9
Find all MCLQs containing A: The ApTree is a MCLQ unless there are pairwise ANDs with same count. 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 A I 1 A J A K A L A M A N 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
The EpTree is a MCLQ(E) unless there are pairwise ANDs with that EpTree having Ct=Ct(E)=8. None, so E MCLQ(E). E 1 8
Next check for MCLQs with Ct=Ct(E)-1=7. E E E E E E E E 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, EH ) H E 1 7
When Ct=1 check the #pTrees containing E ( ) for maximal using 1st step above. 1 8 2 7 3 4 5 6 9 A B C D E F H I G
When Ct=1 check
#pTrees
containing
M(12,13,14). J I L M 1 H N G b 4 c 6 d 7 A B C D E F K
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). H I J L M N 1 2 G I J L M N 1 2
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. I J L M N 1 3
Each Ct(EXpTree)=6=Ct(E)-2 (EF, EG) expands one. 1st ANDs, EF&EG= EFG-2347 G E 1 6 F 4
…Each Ct(EXpTree)=3=Ct(E)-5 (i.e., EI) expands one into a MCLQ(E). I E 1 3

12. 3PART HyperGraphs Base clTrees …1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 A B C D E F
I S D R α
3PART HyperGraphs 1 3
Base
clTrees B A C D E F G H I J K L M N      2 4 5 6 7 8 9 a b v d e f g h i
ctS
ctD
ctI …
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).
{123..}=Investors recommending
Stocks={ABC..} on Days={,,,,}.
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 bitslice pTrees, it would treat them as individual mask pTrees
(Days)
A B C D E… (Stocks)
3-Level pTrees with strides,
14 (Stocks)
5 (Days)
18 (Inv) 1
 …
     3 2 4 5 6 7 8 9 a b v d e f g h i
(Investors)
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 8
Base
cTrees 2 3 4 5 6 7 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      … 1 3
Base cTrees 2 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
CtS
CtD …
3-Level pTrees with strides,
18 (Investors)
14 (Stocks)
5 (Days) 1
… (Investors) …
(Stocks)
(Days)
A B C D E …      3 2
ABCDE…
3-Level pTrees with strides,
18 (Investors)
5 (Days)
14 (Stocks) 1
… (Investors)
 …
(Days)
(Stocks)
     B A C D E F G H I J K L M N 8 7 6 5
Is there a fast MaxClique
mining algorithm for
tripartite graphs?

13. Considering all pairwise ANDs for A and then for B, etc
Considering all pairwise ANDs for A and then for B, etc. (over all Days as well as over all Stocks) B A 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
CtS
CtD
CtI 1 3 1 3 1 4 1 3 1 3 1 2 1 2 1 5 1 3 1 2 1 4 1 2 1 4 1 3 1 2 1 4 1 2 1 4 1 3 1 1 3 1 2 1 4 1 3 1 3 1 3 1 2 1 4 1 3 1 3 1 3 1 4 1 4 1 3 1 3 1 3 1 3 1 4 1 6 1 3 1 3 1 3 1 4 1 6 1 3 1 3 1 2 1 4 1 5 1 3 1 2 1 2 1 2 1 1 3 1 2 1 2 1 4 1 4 1 3 1 2 1 2 1 4 1 4 1 3 1 2 1 2 1 1 2 1
oaa w A Ct=3
ABCDE  1 3 1 4 1 3 1 3 1 5 1 3 1 4 1 4 1 3 1 4 1 4 1 3 1 4 1 3 1 3 1 6 1 6

14. Stock-Investor-Day cTrees (leaf pTrees on Days).B A C D E F G H I J K L M N      1 2 3 4 5 6 7 8 9 a b c d e f g h i
CtS
CtD
CtI 1 4 1 4 1 1 3 1 1 1 1 1 1 1 1 1 7 1 1 1 8 1 7 1 1 2 3 1 2 1 4 1 5 1 3 1 4 1 1 9 1 6 1 1 1 1 1 7 1 1 1 1 1 1 B A C D E F G H I J K L M N      1 2 3 4 5 6 7 8 9 a b c d e f g h i
CtS
CtD
CtI 1 4 1 5 1 3 1 4 1 1 9 1 6 1 1 1 1 1 7 1 1 1 1 1 1 1 1 5 1 5 1 1 1 1 4 1 1 8 1 1 1 1 8 1 1 1 1 1 1 5 1 5 1 5 1 5 1 1 8 1 1 1 7 1 7 1 1 8 1 1 1 1 1 1 1 d 1 b 1 2 1 7 1 1 1 1 1 1 1 1 7 1 1 7 1 1 5 1 1

15. Stock-Day-Investor BaseCliqueTrees (leaves Inv)
Base CliqueTrees for 3PART HyperGraph, 3PHG2
oaa
aoa
Stock-Day-Investor BaseCliqueTrees (leaves Inv) 1 3  A 5 4   2   1 4 2  B  5  3   1 2 4  C  5  3   1 2 4  D  5  3   1 3  E  5 4   2  1 2 3  A 5 4     1 3 2  B  5    1 2 4  C 3     1 2 4  D 3    5  1 2 3  E  4    1 3 1 3 1 4 1 3 1 3 1 2 1 2 1 5 1 3 1 2 1 4 1 2 1 4 1 3 1 2 1 4 1 2 1 4 1 3 1 1 3 1 2 1 4 1 3 1 3
{12345}=Investors recommending Stocks={ABCDE} on Days={,,,,}, 74 recommendations
Stock-Investor-Day BaseCliqueTrees (leaves Days) B A C D E      1 2 3 4 5
CtS
CtD
CtI 1 5 1 5 1 2 1 4 1 4 1 4 1 2 1 2 1 1 5 1 5 1 3 1 3 1 3 1 5 1 3 1 3 1 3 1 5 1 3 1 4
ACD  124
ABCDE  1234
ABCDE  124
AE  124
ACDE  12
B 
ABE  14
ACDE  12
CD  1234
ABCDE  2345
ABCDE  12
CD  1234
ACDE  2
CDE  234
CDE  234
A  123
A 
A 
A  1234
B 
B 
B 
B 
B 
C 
C  23
C  124
C  12
D 
D  23
D 
D 
D  2
E 
E 
E 
E 
Investor-Day-Stock BaseCliqueTrees (leaves Stocks) B A C D E      1 2 3 4 5
CtS
CtD
CtI 1 4 1 5 1 3 1 4 1 1 5 1 5 1 5 1 5 1 5 1 5 1 1 5 1 5 1 5 1 4 1 4 1 3