1. The Multi-hop closure theorem for the Rolodex Model using pTrees
Arijit Chatterjee, Arjun G. Roy, Mohammad Hossain William Perrizo Computer Science Department North Dakota State University

2. Data Mining
Reference :

3. But it is pure (pure0) so this branch ends
Vertical Structuring into predicate Trees (pTrees): project attributes (4 files)
pTrees
then vertically slice off each bit position (12 files)
Given a table of Horizontal records.
(traditionally Vertically Processed, so VPHD )
then compress each bit slice into a pTree
e.g., compress R11 into P11: =2
Determine the number of occurences of
We use Horizontal Processing of Vertical Data,
HPVD, to find the number of occurences of ?
R(A1 A2 A3 A4)
R[A1] R[A2] R[A3] R[A4]
Base 10
Base 2 =
for Horizontally structured
records
Scan vertically
R11 1
R11 R12 R13 R21 R22 R23 R31 R32 R33 R41 R42 R43
pure1? false=0
pure1? true=1
pure1? false=0
pure1? false=0
pure1? false=0
Record the truth of the predicate pure1 (all 1-bits) in a tree recursively on halves until the half is pure.
1. Whole is pure1? false  0
0 0
0 1
P11
P11 P12 P13 P21 P22 P23 P31 P32 P33 P41 P42 P43
0 0
0 1 1
1 0
1 0 01
0 1
0 0 ^
2. Left half pure1? false  0
3. Right half pure1? false  0
0 0
P11
4. Left half of rt half ? false0
0 0
5. Rt half of right half? true1
0 0
0 1
To count occurrences of 7,0,1,4 use P11^P12^P13^P’21^P’22^P’23^P’31^P’32^P33^P41^P’42^P’43
But it is pure (pure0) so this branch ends

4. Multi-relationships RoloDex Model: and the RoloDex Model
2 Entity Many relationship cards
axes
Multi-relationships
and the RoloDex Model
 Customer 1 2 3 4
Item 1
customer rates movie as 5 card 1 5
people 2 3 4
items
terms
DataCube Model for 3 entities, items, people and terms.
cust item card 5 6 7
People  1 2 3 4
Author
movie 2 3 1 5 4
customer rates movie card 2 3 4 5 PI 2 3 4 5 PI 4 3 2 1
Course
Enrollments 1
Doc
termdoc card
authordoc card 1 3 2
Doc 1 2 3 4
Gene
genegene card (ppi)
docdoc
People 
term  7 6 5 4 3
Gene 1 2 3 4 G 5 6 7 6 5 4 3 2 t 1
termterm card (share stem?) 1 3
Exp
expPI card
expgene card
genegene card (ppi)
The bottom line point of this slides is that all entities are interrelated through multiple relationships (multiple "hops")
Items: i1 i2 i3 i4 i5
|0 001|0 |0 11|
|1 001|0 |1 01|
|2 010|1 |0 10|
People: p1 p p3 p4
|0 100|A|M|
|1 001|T|M|
|2 010|S|F|
|3 011|B|F|
|4 100|C|M|
Terms: t1 t2 t3 t4 t5 t6
|1 010|1 101|2 11|
|2 001|0 000|3 11|
|3 011|1 001|3 11|
|4 011|3 001|0 00|
Relationship: p1 i1 t1
|0 0| 1
|0 1| 1
|1 0| 1
|2 0| 2
|3 0| 2
|4 1| 2
|5 1|_2
Relational Model:
How can we data mine those multi-relationships?

5. Multi-hop rule mining using the RoloDex model:
A hop is a relationship, R, hopping from one entity, E to another, F.
Given a high value "relationship" data set, we pay the one-time cost of creating pTrees both ways. 2 3 4 5 F 4 1 3 1 2 1 1 1
R, as a matrix, has both E-pTrees (horizontal bit slices, Re) and F-pTrees (vertical bit slices, Rf) E
R(E,F)
Standard ARM finds strong (frequent, confident) single-entity rules. ( i.e., both A and C  E Counts are on the other entity, F).
ct(&eARe)  mnsp
ct(&eARe &eCRe) / ct(&eARe)  mncf
What about multi-entity rules? ( i.e., AE, CF ).
A 1-hop (AE and CF) F-focused rule is strong if: ( "focused on" refers to where the counts are taken)
ct(&eARe)  mnsp
ct(&eARe & PC) / ct(&eARe)  mncf
1-hop, F-focused strong rules can be mined efficiently because of
1. (antecedent downward closure)
If A is frequent, then all of its subsets are frequent. Or, if A is infrequent, then all its supersets are infrequent.
Since frequency involves only A, we can mine for all qualifying antecedents efficiently using downward closure.
2. (consequent upward closure) If AC is non-confident, then so is AD for all subsets, D, of C.
So  frequent antecedent, A, use upward closure to mine for all of its' confident consequents.
The theorem suggested here is:
For (a+c)-hop strong rule mining with a focus entity which is
a hops from the antecedent and
c hops from the consequent,
if a {c} is odd [even], use downward [upward] closure on the frequency {confidence} step in the mining.
In this case A is 1-hop from F (1 is odd, use downward closure).
C is 0-hops from F (0 is even, use upward closure).
A 1-hop (AE and CF) E-focused rule is strong if:
ct( PA)  mnsp
ct( PA &fC SC ) / ct( PA )  mncf

6. ct(&eARe &gCSg) / ct(&eARe)  mncf
2-hop F-focused (The focus is on middle entity, F) C G
S(F,G) 1 4 1 3
AC strong if:
ct(&eARe)  mnsp
ct(&eARe &gCSg) / ct(&eARe)  mncf 1 2 1 1
1. (antecedent downward closure) If A is infrequent, then so are all of its supersets. 2 3 4 5 F
2. (consequent downward closure) If AC is non-confident, so is AD for all supersets, D. 4 1
1, down, down 3 1 2 1 1 1
A  E
R(E,F)
2-hop G-focused
ct(&f&eAReSf)mnsp
 mncf
ct(&f&eAReSf & PC)
/ &f&eAReSf
1. (antecedent upward closure) If A is infrequent, then so for are all subsets.
2,0 up, up
2. (consequent upward closure) If AC is non-confident, so is AD for all subsets, D.
2-hop E-focused
ct(PA)mnsp
 mncf
ct(PA&f&gCSgRf )
/ ct(PA)
0,2 up,up
1. (antecedent upward closure) If A is infrequent, then so for are all subsets.
2. (consequent upward closure) If AC is non-confident, so is AD for all subsets, D.
It was 2-hop F-focus that generated the interest in multi-hop rule mining, in particular: R = a "friends" relationship (e.g., from Facebook) S = a "buys" relationship between people and items.
Is it a strong rule that friends of those who bought a set of items, also buy those items?

7. ct( &f&eAReSf &h(& )UiTh ) / ct(&f&eAReSf)
S(F,G)
R(E,F) 1 2 3 4 E F 5 G A C
T(G,H) H
U(H,I) I
V(I,J) J
5-hop
Focus on G: (if yellow then green)
 mnsp
ct( &f&eAReSf
&h(& )UiTh ) /
ct(&f&eAReSf)
 mncnf
i(&jCVj)
5-hop focus on G:
1. (antecedent has upward closure)
2. (consequent has downward closure)

8. Multi-hop closure property theorem
“For transitive (a+c)-hop strong rule mining with a focus entity which is
‘a’ hops from the antecedent and
‘c’ hops from the consequent,
if a [or c] is
odd or even then one can use
downward or upward closure respectively on that step“

9. &elist(&clist(&aDXa)Yc)We
The Multi-hop Closure Theorem
A condition is downward [upward] closed: If when it is true of A, it is true for all subsets [supersets], D, of A.
Given an (a+c)-hop multi-relationship, where the focus entity is a hops from the antecedent and c hops from the consequent, if a [or c] is odd/even then downward/upward closure applies.
A pTree, X, is said to be "covered by" a pTree, Y, if  one-bit in X, there is a one-bit at that same position in Y (the list corresponding to the bitmap, Y, is a superset of the list corresponding to the bitmap, X)
Lemma-0: For any two pTrees, X, Y; X&Y is covered by X and thus ct(X&Y)  ct(X) and list(X&Y)list(X)
Proof-0: ANDing with Y may zero some of X's ones but it will never change any zeros to ones.
Lemma-1: Let AD, &aAXa covers &aDXa (ANDing over a superset always covers)
Lemma-2: Let AD, &clist(&aDXa)Yc covers
&clist(&aAXa)Yc
Proof-1&2: Let Z=&aD-AXa then &aDXa =Z&(&aAXa). lemma-1 now follows from lemma-0, as does
D'=list(&aAXa)
A'=list(&aDXa) 
so by lemma-1, we get lemma-2:
Lemma-2: Let AD, &clist(&aDXa)Yc covers
&clist(&aAXa)Yc
Lemma-3: AD, &elist(&clist(&aAXa)Yc)We covers
&elist(&clist(&aDXa)Yc)We
Proof-3: lemma-3 follows in the same way from lemma-1 and lemma-2. Continuing this establishes: If there are an odd number of nested &'s then the expression with D is covered by the expression with A. Therefore the count with D  with A. Thus, if the frequent expression and the confidence expression are > threshold for A then the same is true for D. This establishes downward closure.
Exactly analogously, if there are an even number of nested &'s we get the upward closures.