1. In taking the inner product of 32 bitwidth Scalar pTreeSets (e. g
In taking the inner product of 32 bitwidth Scalar pTreeSets (e.g., for Oblique or Hull Classification) we want line segments to be tight against the Training Class, but not too tight (because Training Classes are almost always only estimates of the actual classes). 
I.e., we may want to leave room between the Training Class and the bordering line segment, because of the approximate-ness of the Training Classes
We can do that as follows:
For the segment on the Minimum side (the segment perpendicular to the unit vector, d, through minimum{d dot x | xTraining Class}, set the 24 LoBits to 0 (only the 8 HiBits are then used in inner product). This moves the bordering line segment away from that Training Class on that side.
For the segment on the Maximum side set the 24 LoBits to 1 (Better yet, add 1 to Hi 8th bits, i.e., set the 24 LoBits=0, add 1 to the 8th HiBit (which is almost the same as setting the 24 LoBits to all 1s but gives a much faster inner product calculation). This moves the bordering line segment away from that Training Class on the other side.
This approach is is a win-win: it places the line segments better for Classification, and it lowers inner product costs  (to 8 bit-width costs, instead of 32 bit-width costs????).
The split of 32 into 24 and 8 could be varied and could depend on expected Training Set accuracy.  For accurate Training Sets, use, e.g., 12 HiBits (a very tight Hull), else use 4 HiBits only (a very loose Hull).
When might we judge that the Training Set is very approximate? - When there are few Training points.
Remember, for example, that the main criticism of most cancer prediction systems is that they are based on too few expert opinions or experimental cases because each is very expensive to obtain (i.e., we usually settle for just a few training points).
While we’re at it, a new algorithm, Oblique-Hull, might only place hull segments when there is a gap between a pair of classes (separate segment pairs for each class pair).
Continue to include new unit vectors until each class pair has been separated. The gap placement can use the 1st k HiBits value that produces a gap, k=1,2… (1st k HiBit value between the min inner product for one class and the max inner product for the other class).
It seems like Mohammad’s 2’s complement procedure works that way anyway???? (proceeding one bit slice at a time from the high side? Or is it the low side?), so we can continually check for a gap and early exit as soon as one appears????
So for pair of Training Classes, we might use the unit vector between class means, then project the two classes using k HiBits only, k=1,2… (i.e., until a gap appears between the k HiBit min of the origin (of unit vector) class and the k HiBit max (plus 1) of the destination class …

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
Actually it is not true that 1 must be a BC, since there could be a different expansion for each of those 4, intersecting in C.
In that case, we get each of those different expansions as an EBCT, but then the other operator will give us C (we will AND those expansions yielding the core leaf but OR the singletons giving the correct Part of C.
=C a MaxClique.Then 1 of
must be a BC, say
Expanding it must give 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
Thm: Every Maximal Clique is an Expanded Base Clique.
Pf: Let be any MaxClique, C. Then some leaf expansion of each of … is a BCT. After we apply
a..aoa..a with o in each but the last position, we will
have an EBCT with the upper Parts of C and a leaf
that covers the leaf of C. However, the leaf of that
EBCT cannot strictly cover the leaf of C lest it be a
MaxClique that strictly covers C.
Thus, that EBCT=C 1 . 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. Rotate! Base Clique Motifs For Bipartitie graph, G9.1:
Investors(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)
For Unipartite G1.1: Proteins(1,2,3,4,5) interacting 1 2 3 4 5 1 3 4 2
SI Expanded
Base CTrees 3 2 4 1 5
Edge Tbl B A C D E 1 3 2 4
SI Base CTrees S I 5 1 2 3 4 5 A B C D E 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 for Map
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 2 1 3 4 5
3Level Stride=5 NPZ pTrees for SIRE Map
Lev=2
Lev=1
Lev=0 1 4 2 3 5
IS BCTs I S B A C D E
IS EBCTs 1 2 4 5 3
Create EBCTs
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 BCTs are induced subgraphs (also cliques), EBCTs are maximal 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
3L St=5 NPZ pTree for ISRE Map
Lev=2
Lev=1
Lev=0 2 1 3 4 5

4. APPENDIX: 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

5. 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

6. 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)

7. 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)

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.
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     

9. 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     

10. 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 

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

12. In 32 bitwidth Scalar pTreeSets, for Oblique or Hull Classification, we want tight TrainSetClass line-segments, but not too tight (because TrainingSetClasses are only estimates of the actual classes).  I.e., we may want to leave some room between the TrainingSetClass and the bordering line segment, because of the approximatness of the TrainingClasses.  We can do that as follows:
For the Minimum-Segment (the segment perpendicular to the unit vector, d, through the minimum{d dot x | xTrainingClass}, set the 24 LoBits to 0 (only the 8 HiBits are then used in inner product). This moves the bordering line segment away from that TrainingClass on its one side.
For the Maximum-Segment set the 24 LoBits to 1 (Better, add 1 to 8th bit, i.e., set the 24 LoBits=0, add 1 to the 8th HiBit (which is almost the same as setting the 24 LoBits to all 1s but is a much faster calculation). This moves the bordering line segment away from that TS on the other side.
This approach is is a win-win: it places the line segments better for Classification, and it lowers inner product costs  (to 8 bit-width costs, instead of 32 bit-width costs????). The 24-8 split could depend on TrainingSet accuracy.  For accurate TS, use 12 HiBits (very tight Hull), else 4 HiBits only.
When might we fear that the TS is very approximate? - When there are few Training points (remember, for example, that the main criticism of most cancer prediction systems is that they are based on too few expert opinions or experimental cases because they are very expensive to produce (i.e., too few training points).
While we’re at it, a new algorithm, Oblique-Hull, might only place hull segments when there is a gap between a pair of classes (separate segment pairs for each class pair). Continue to include new unit vectors until each class pair has been separated. The gap placement can use the 1st k HiBits value that produces a gap, k=1,2… (1st k HiBit value between the min inner product for one class and the max inner product for the other class).
So for each TS class pair we might use the unit vector between class means, then project the two classes using k HiBits only, k=1,2… (i.e., until a gap appears between the k HiBit min of the origin (of unit vector) class and the k HiBit max (plus 1) of the destination class …
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: CLQkkCliqueSet, CCLQk+1Candk+1Cliques. By SGE, CLQk+1= all s of CLQk pairs w k-1 common vertices. Let CCCLQk+1 be a union of 2 kcliques 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.
Breadth1st Clique Alg: Find CLQ3 w CS0. A Kclique and 3clique sharing an edge form a (K+1)clique iff all K-2 edges between non-shared vertex exist. Find CLQ4 then CLQ5..
Depth1st 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).
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
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.
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.
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).
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
Cliques, Kplexes. Kcores are subgraphs (communities) defined using internal edge count. A Motif is a subgraph defined using external “isomorphism count. A motif must occur (isomorphically) > expected # od times. Criticism:[62] motif structure 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?
All KPaths are isomorphic (alway a Kpath motif). ShortestKPath is an Induced subgraph. What does seq FG(1PathMotif)=|V|, FG(2PathMotif),…tell us? Seq FG(Shortest1Path), FG(Shortest2Path)..? Seq FG(MaxShortest1Path), FG(MaxShortest2Path)… tells us? where MaxS2P is not part of a S3P. Extend to HyperEdges? Path in a 3HyperGraph?
Or?
Both? 2HGInterface3HyperGraphPath. 1HGI3HGP.(hHGIkHGP, where 0<h<k) At the other extreme (all SPs are length=1:
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)