1. Rotate! Base Clique Motifs for bipartite graph G9.1
Investors(1,2,3,4,5) recommending Stocks(A,B,C,D,E)  1 2 3 4 5 A B C D E 2 1 3 4 5 I
A B C D E S
Adjacency Matrix 1 2 4 3 A B C D E
SI-Raster Edge Table
(Traditional) 1 3 4 2
ExpBase
SI cTrees B A C D E 1 3 2 4
Base SI cTrees S I 5
All bipartite graph BcTrees are induced subgraphs (and cliques).
All bipartite graph EBcTrees are maximal cliques.
How do we mine for other motifs?
Is motif mining even useful in the Investor-Stock case? (Maybe it would be useful to know that the 3-3 motif occurs many times (3 investors recommending 3 stock).
Motifs seem to be of greatest interest in the context of Gene-Gene or Protein-Protein interaction graphs in which the two label sets are the same and therefore there is just one Base cTreeSet and one EBcTreeSet to create (easier) and the h-k motifs are not distinct from the k-h motifs.
A key question is: in PPI graphs, would the counts of Expanded Base Clique Motifs provide important information? 1 2 3 4 5 A B C D E
SI-Raster Edge Map
ExpBase
IS cTrees 1 2 4 5 3 1 4 2 3 5
Base IS cTrees I S B A C D E 2 1 3 4 5
3Lev Stride=5 NPZ SI pTrees
Lev=2
Lev=1
Lev=0
Create Expanded Base cTrees
Rotate!
The number of isomorphic copies of an ExpandedBaseClique Motif (EBCM) can be counted by analyzing cTree counts: I
Adjacency Matrix B A C D E S 1 B A C D E 1 2 3 4
IS-Raster EdgeTbl
Thus for this bipartite graph there are:
2 3,3 SI EBC Motifs
2 4,2 SI EBC Motifs
1 2,4 SI EBC Motif
1 5,1 SI EBC Motif
In addition, we have:
2 1,4 SI BC Motifs
2 4,1 SI BC Motifs
2 1,3 SI BC Motifs
1 3,1 SI BC Motif
1 1,2 SI BC Motif 1 B A C D E 2 3 4 5
IS-Raster EdgeMap
3Lev Stride=5 NPZ IS pTrees
Lev=2
Lev=1
Lev=0 2 1 3 4 5

2. Base Clique Motifs for unipartite graph G1
Base Clique Motifs for unipartite graph G1.1 Proteins(1,2,3,4,5) interactions 1 2 1 3 4 5
Adjacency Matrix 3 2 4 1 5
Edge Table 2 1 3 4 5
Base cTrees 2 3 4 5 2 1 3 4 5
3Lev Stride=5 NPZ pTrees
Lev=2
Lev=1
Lev=0 1 2 3 4 5
Edge Map 1 2 4
EB cTrees (oa)
Create Expanded Base cTrees
The number of isomorphic copies of an EBC Motif can be counted by analyzing cTree counts:
Thus for this unipartite graph there are:
1 2,2 EBC Motif
1 4,1 EBC Motif
In addition:
,3 BC Motifs
11 1,2 BC Motifs

3. Stock Day Investor cTrees Day Stock Investor cTrees Stock Investor
Base CliqueTrees for 3HG2 TriEdgeTable (S,D,I,R) has 6 key sort orders, SDI,DSI,SID,ISD,DIS,IDS. The Adjacency Matrix (data cube) has 1 for each existing TriEdge (that Investor recommended that stock on that day). There are 6 Base cTreeSets and 1 operator, aoa, to generate Expanded Base cliqueTrees. S A B … D α      1 I 2 3 4 5 R C E
CtI
Stock
Day
Investor cTrees
S (1st sort dim)
D (2nd)
I (3rd)
4Level Stride=5 rasterSDI NPZ pTrees
Lev=3
Lev=2
Lev=1
Lev=0 1 3 4 2 5
4Level Stride=5 rasterDSI NPZ pTrees
Lev=3
Lev=2
Lev=1
Lev=0 1 3 4 2 5 D   … A S B C E 1 I 2 3 4 5 R
Day
Stock
Investor cTrees   
CtI
4Level Stride=5 rasterSID NPZ pTrees
Lev=3
Lev=2
Lev=1
Lev=0 1 5 2 4 3 B A C D E      2 1 3 4 5
CtD
Stock
Investor
Day cTrees S … I α R

4. Investor Stock Day cTrees Day Investor Stock cTrees Investor Day
Base CliqueTrees for 3HG2 TriEdgeTable (S,D,I,R) last 3 key sort orders, ISD,DIS,IDS.
4Level Stride=5 rasterISD NPZ pTrees (same as SID on pevious slide)
Lev=3
Lev=2
Lev=1
Lev=0 1 5 2 4 3 B A C D E      2 1 3 4 5
CtD
Investor
Stock
Day cTrees … α
ISDR
4Level Stride=5 rasterDIS NPZ pTrees
Lev=3
Lev=2
Lev=1
Lev=0 1 4 5 3 1 4 B A C D E      2 3 5
Day
Investor
Stock cTrees
CtS …
DISR 1 4 B A C D E      2 3 5
Investor
Day
Stock cTrees
CtS … α
IDSR
4Level Stride=5 rasterIDS NPZ pTrees
Lev=3
Lev=2
Lev=1
Lev=0 1 4 5 3

5. Stock Day Investor Base cTrees Day Stock Investor Base cTrees Stock
Maximal Base CliqueTrees for 3HG2
Stock
Day
Investor Base cTrees B A C D E      2 1 3 4 5
CtI 1 4 3 2 5 1 4 3 2 5
aoa
oaa (all of these will be Max Cliques)
We can count the S=1 D=1 I=4 motifs?
COMBO(5,4)=5 = 11
113? 10+6C(4,3)+C(5,3) = 54
112? C3,2+6C4,2+C5,2 = 83
Day
Stock
Investor Base cTrees B A C D E      2 1 3 4 5
CtI 1 3 4 2 5 1 3 4 2 5
aoa
oaa (all of these Max Cliques, only 3 new ones)
Stock
Investor
Day Base cTrees B A C D E      2 1 3 4 5
CtD 1 2 5 3 4 1 2 5 3 4
aoa
oaa (all of these Max Cliques, only 3 new ones)

6. Investor Stock Day cTrees Day Investor Stock cTrees Investor Day
Base CliqueTrees for 3HG2 last 3.
Investor
Stock
Day cTrees 2 1 3 4 5 1 5 2 4 3 1 2 5 4 3 1 2 5 4 3 B A C D E     
aoa
oaa (all of these will be Max Cliques)
Day
Investor
Stock cTrees      1 4 5 3 1 2 4 5 3 1 3 2 4 5 2 1 3 4 5 B A C D E
aoa
oaa (all of these will be Max Cliques)
Investor
Day
Stock cTrees 2 1 3 4 5 1 4 5 3 1 3 4 5 2 1 2 3 4 5      B A C D E
aoa
oaa (all of these will be Max Cliques)

7. Maximal Base CliqueTrees for 3HG2
aoa then oaa on the 6 cTrees (removing duplicates - no covers since aoa then oaa gives Maximal Cliques only). We get 34 MCs below.
Theorem: These 34 MCs are the only Maxmal Cliques. Proof: Let
Let C be MaxClique, v1Part1(C), w1Part2(C), {z1..zn}=Part3(C).
Apply aoa to that BaseClique, B. aoa(B)={v1,w1..wm,z1..zn} is a clique
W={w1..wm}Part2(C) else C is not max.
oaa(aao(B))={v1..vk,W,Z} is clique. V={v1..vk}Part1(C) else C not mx.
Thus {V,W,Z} is a MaxClique  C and therefore {V,W,Z}=C.
Thus C is one of the Expanded Base Cliques under aoa then oaa.
General thm: {a..ao(a..oa(…oa..a(B)|B=BaseClique} is the MaxCliqueSet.
Thus, for a bipartite graph, ao(B) is MCS. 1 2 5 3 4 B A C D E      B A C D E 1 4 3 2 5      2 1 3 4 5 1 2 5 4 3 B A C D E      1 3 4 2 5 B A C D E      1 5 2 4 3 B A C D E      1 2 3 4 5 B A C D E      1 4 3 2 5 B A C D E      1 5 2 4 3 B A C D E      1 3 2 4 5 B A C D E      1 2 5 3 4 B A C D E      1 2 5 3 4 B A C D E     

8. Stock-Day-Investor BaseCliqueTrees (leaves Inv)
Base CliqueTrees for 3PART HyperGraph, 3PHG2
{12345}=Investors recommending Stocks={ABCDE} on Days={,,,,}, 74 recommendations
ACD  124
ABCDE  1234
ABCDE  124
AE  124
A  123
ABCD  12
B 
ABCD  12
ABE  14
ABCDE  2345
ABCDE  12
CD  1234
CD  1234
CDE  234
CDE  234
oaa results
E 
E 
E 
E 
A 
B 
C 
C 
C 
D 
D 
D 
D  2
aoa results
Stock-Day-Investor BaseCliqueTrees (leaves Inv)
ACD 
ABCDE 
ABCDE 
AE 
ABCD 
ABCD 
ABE 
ABCDE 
ABCDE 
CD 
CD 
CDE 
CDE 
ACDE 
ABCDE 
oaa
ABCD 
aoa on these
CD 
D  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
ABCDE 
CD 
CD 
A 
B 
C 
D 
oaa
ABCD  124
ABC  124
ABCD 
ABCD  124
ABCE 
aoa on these
ACD 
AE 
CD  124
C  124
D  124
ACD 
ABCDE 
AE 
CD 
CD 
E 
A 
B 
C 
D 
D 
C 
E 
E 
E 
A 
B 
C 
C 
D 
D 
D 
C  12
C 
E 
Stock-Investor-Day BaseCTrees (leaves Days)
AC  1
ABCDE 
oaa results
ACDE  2
AE  4
ABC  1
ABCDE  2
ABCDE 
ABCDE 
CDE 
ACDE 
ABCD 
ABCE 
aao results
C  12
C 
E  2
E  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
AC  1
ABCDE 
ACDE  2
AE  4
ABC  1
ABCDE  2
ABCDE 
ABCDE 
CDE 
ACDE 
ABCD 
ABCE 
ABCDE 
ABE 
Inv-Day-Stock BaseClTrees (leaves Stocks)
aoa results
ABCDE 
ABE 
aao results
ACDE 
ABCDE  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
aao
ABCDE 
ABCDE  23
ABCDE  124
aoa on these
ABCDE 
ABCDE  12
ABCDE 
ABCDE  2
ABCDE  1
oaa
AB 
ABC  12
ABCDE  12
aao on these
AC  12
A 
B 
C  12
AC  1
ACDE  2
ABCDE 
ABCDE 
ABCDE 
ABCDE 
ABCDE 
ABCDE 
ABCDE  2
ABCDE 
ABCDE 
ABCDE 

9. 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. If (x,y)E and Count(NewPtSet(v,w,x,y)CLQ3pTree(v,w)&CLQ3pTree(x,y)): 0, 4 v’s form a max4Clique (i.e., v,w,x,y). 1, 5 v’s form a max5Clique (i.e., v,w,x,y,NewPt) 2, 6 v’s form max6Clique if NewPairE, else form 2 max5Cliques. 3, 7 v’s form max7Clique if each NewPairE, elseif 1 or 2 NewPairsE each 6VertexSets (vwxy + 2 EdgeEndpts) form Max6Clique, elseif 0 NewPairsE, each 5VertexSet (vwxy + 1 NewVertex) forms 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).
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 )
2graph=2hypergraph.
Bipartite Clique Mining finds MaxCliques at cost of pairwise &s. Each LETpTreeMCLQ unless  pairwise & with same count.A&B, B w Ct(A&B)=Ct(A) is a MCLQ.
 potential for a k-plex [k-core] mining alg here. Instead of Ct(A&B)=Ct(A), 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.
HyperClique Mining: A 3hyperGraph has 3 vertex PARTS and each edge is a planar triangle (vertex triple), one from each PART. Stock recommender is 3PARThyperGraph (Investors, Stocks, Days) A triangular "edge" connects Investor #k, Stock X, and Day n if k recommended X on day n. A 3PARThyperClique is a community s.t. all the investors in the clique recommend all the stocks in the clique on each of the days in the clique (A strong signal?) Tweet example: PART1=tweeters; PART2=hashtags; PART3=tweets.
Conjecture: KmultiCliques and KhyperCliques in 1-1 corresp. (K vertex set)? So, one of the mining processes only? Represent these common objects w cliqueTrees (cTrees).
Cliques, Kplexes. Kcores are subgraphs (communities) defined using internal edge count. A Motif is a subgraph defined using external “isomorphisms in the graph” counting. A motif must occur (isomorphically) in the graph more times than “expected”. Criticism: Some authors argue[62] motif structure does not necessarily determine function. Recent research[64] shows the connections of a motif to the network, is too important to draw function inferences just from local structure.[65] Research shows certain topological features of biological networks naturally give rise to canonical motifs,.[66]
Are Stock-Inv or Stock-Inv-Day Motifs useful? Some questions/theorems/thoughts:
All K-Paths are isomorphic (thus, there’s alway a Kpath motif) A ShortestKPath is an Induced subgraph.
What does sequence FG(1PathMotif)=|V|, FG(2PathMotif),…tell us?
Sequence of FG(Shortest1Path), FG(Shortest2Path), …?
Sequence FG(MaxShortest1Path), FG(MaxShortest2Path)… tell us? where a MaxS2P is not part of a S3P.
Extend to HyperEdges? What is a path in, e.g., a 3HyperGraph?
Both?
2HGInterface3HyperGraphPath. 1HGI3HGP.
(In general, hHGIkHGP, where 0<h<k)
At the other extreme (all SPs are length=1:
Or?
I’ll bet most important motifs, M(V’,E’) in G are “Shortest Path Motifs”: x,yV’,  a G-ShortestPath in M running from x to y. I.e., M is made up of G-SPs.
A Clique is a SPMotif (made up entirely of Shortest1Paths)

10. MOTIFs: Cliques, k-plexes, k-cores and other communities are subgraphs defined by internal edge count. A Motif is a subgraph defined by isomorphism count(external). Wikipedia: motifs are recurrent and statistically significant sub-graphs or patterns. They may reflect functional properties. Motif detection is computationally challenging. Most find induced Motifs. A graph, G′, is a subgraph of G (G′⊆G) if V′⊆V and E′⊆E∩(V′×V′). If G′⊆G and G′ contains all ‹u,v›∈E with u,v∈V′, G′ is induced sub-graph. G′ and G are isomorphic (G′↔G), if  a bijection f:V′→V with ‹u,v›∈E′⇔‹f(u),f(v)›∈E u,v∈V′. G″⊂G and  an isomorphism between G″ and G′, G′ appears in G). The number of appearances G′ in G is the frequency FG of G′ in G, FG(G’). G is recurrent or frequent in G, when FG(G’)>threshold (pattern=frequent subgraph). Motif discovery includes exact counting, sampling, pattern growth. Motif discovery has 2 steps: calculate the # of occurrences; evaluating the significance.
Mfinder implements full enumeration and sampling. Brute force exact counting (Milo et al.[3], was computationally feasible only for small motifs of size < 5 vertices.
Kashtan et al [9] edge sampling NM alg, estimate concentrations of induced subgraphs for directed or undirected networks starting from an edge (subgraph size 2) then continues choosing random nbr edges until subgraph size=n. Finaly the subgraph is expanded to include all of the edges that exist in the network between these n nodes. It finds motifs up to size=6 and thus, most significant motifs. mfinderSampling: Es=set of picked edges. Vs= set of all nodes that are touched by the edges in E. Initilize Vs and Es=. 1. Pick random edge, e1=(vi,vj). Update Es={e1}, Vs={vi,vj} 2. Make list L of all nbr edges of Es. Omit from L all edges between vertices in Vs Pick random edge e= {vk,vl} from L. Update Es=Es⋃{e}, Vs=Vs⋃{vk,vl}. 4. Repeat 2-3 until |Vs|=n. 5. Calculate the probability to sample the picked n-node subgraph. Apply to G9 below: 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 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