1. Rotate! Base Clique Motifs Bipartitie graph, G9.1:
Inv(1,2,3,4,5)
recommend
Stock(A,B,C,D,E) 1 2 4 3 A B C D E
SI-Raster Edge Table
(Traditional)
Unipartite G1.1: Proteins(1,2,3,4,5) interactions 1 3 4 2
ExpBase
SI cTrees 3 2 4 1 5
Edge Tbl B A C D E 1 3 2 4
Base SI cTrees S I 5 1 2 3 4 5 A B C D E 1 2 3 4 5 2 1 3 4 5
Adj Matrix 2 1 3 4 5 I
A B C D E S
Adjacency Matrix 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 3 4 5 A B C D E
SI-Raster Edge Map
ExpBase
IS cTrees 1 2 4 5 3 2 1 3 4 5
3Lev Stride=5 NPZ SI pTrees
Lev=2
Lev=1
Lev=0 1 4 2 3 5
Base IS cTrees I S B A C D E
Create EBcTrees
Isomorphic EBCMs counted from cTree counts:
Rotate!
2 1,4 SI BCMotifs
2 4,1 SI BCMotifs
2 1,3 SI BCMotifs
1 3,1 SI BCMotif
1 1,2 SI BCMotif
Create Expanded Base cTrees
2 3,3 SI EBCMotifs
2 4,2 SI EBCMotifs
1 2,4 SI EBCMotif
1 5,1 SI EBCMotif 2 1 3 4 5
Base cTrees B A C D E 1 2 3 4
IS-Raster EdgeTbl
The number of isomorphic copies of an EBC Motif can be counted by analyzing cTree counts: I
Adjacency Matrix B A C D E S 1
Bipartite BcTrees are induced subgraphs (also cliques), EBcTrees are max cliques.
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 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.
Question: in PPI graphs, would the counts of Expanded Base Clique Motifs provide important information?
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 1 2 4
EB cTrees (oa) 1 B A C D E 2 3 4 5
IS-Raster EdgeMap
3Lev St=5 NPZ IS pTs
Lev=2
Lev=1
Lev=0 2 1 3 4 5

2. cliqueTrees Stock BCTs I S Investor BCTs S I Stock EBCTs I S Inv EBGTs
Bipart G11: Inv(12345) rec Stk(ABCDE)
Stock BCTs I S 1 4 2 3 5 B A C E D
NPZpTr st=5
L=2
L=1
L=0 1 3 2 4
Investor BCTs S I 1 3 2 4 5 A C B D E
G11 Stock EBCTs I S 1 2 4 3 5 B A C E D 1 2 A B C    3
H1 Stock EBCTs 1 B A C D E 2 3 4 5
EdgeMap
EdgeTbl
Adj Matrix
Graph
Traditional data structures 1 A B C D E 2 3 4 5
New DSs:
NPZpT st=5
L=2
L=1
L=0 1 4 5 3
oa
oa
Stock EBCTs I S 1 4 2 3 5 A B D C E
Inv EBGTs 1 3 2 5 4 B A C E D
=C a MaxClique.Then 1 of
must be a BC, say
Expanding it gives C.
Thus, for Bipartite Graphs,
every MaxClique is an EBCT. 1
H1: On
Day()
I(123)
recommend
S(ABC)
NPZ pTree (stride=3)
L=3
L=2
L=1
L=0 1 2 3
=C a MaxClique.Then 1 of
must be a BC, say
Expanding it gives C.
Thus, for Tripartite Graphs,
every MaxClique is an EBCT. 1 1 2 A B C    3
DI StockBaseCliqueTrees D I S 1 2 A B C    3
DI StockBaseCliqueTrees D I S
oaa
aoa 1 2 3 A B C    1 2 3 A B C   
aoa
oaa 1 2 3 A B C    1 2 A B C    3

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 1 4 3 2 5
Stock
Day
Investor Base cTrees B A C D E      2 1 3 4 5
CtI 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 5 4 2 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 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, the MCS is {ao(B) | B a BaseClique}.
(Seems to say that only one of the 6 cTrees will generatea all of MCS?) 1 4 3 2 5 1 2 5 3 4 B A C D E      B A C D E      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. 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.
Thm: The 34 MCs are only MaxCliques. Pf: C=MaxClique={V,W,Z}.
aoa{v,w,Z})={v,W’,Z}, WW’. oaa(aoa{v,w,Z})={V’,W’,Z}, V’V
If w’W’-W then v’V-V’ (w’C, v’C) and then
aoa{v’,w’,Z}={v’,W”,Z}, {w’,W} 1 4 3 2 5 1 2 5 3 4 B A C D E      B A C D E      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     

9. 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 

10. 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)

11. 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