1. MYRRH
A hop is a relationship, R, hopping from one entity, E, to another entity, F.
Strong Rule Mining (SRM) finds all frequent and confident rules, AC.
R(E,F) 1 2 3 4 E F 5
Frequency can lower bound the antecedent, consequent or both (ARM: A,CE,
Its justification is the elimination of insignificant cases. Its purpose is the tractability of SRM.
ct(&eACRe)  mnsp)
Confidence lower bounds the frequency of both over the antecedent frequency: ct(&eARe &eCRe) / ct(&eARe)mncf
The crux of SRM is frequencies. To compare frequencies meaningfully, they must be on the same entity (the focus entity).
SRMs are categorized by the number of hops, k, whether transitive or non-transitive and by the focus entity. ARM is 1-hop, non-transitive (A,CE), F-focused SRM (1nF) (note: non-transitivity is difficult to define in multi-hop SRM??)
1-hop, transitive (AE,CF), F-focused SRM (1tF)
APRIORI:
ct(&eARe)  mnsp
ct(&eARe &PC) / ct(&eARe)  mncf
1. (antecedent downward closure)
If A is frequent, all of its subsets are frequent. Or, if A is infrequent, then so are all of its supersets.
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 we demonstrate throughout this section is: For transitive (a+c)-hop Apriori 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 then one can use downward/upward closure on that step in the mining of strong (frequent and confident) rules.
In this case A is 1-hop from F (odd, use downward closure).
C is 0-hops from F (even, use upward closure).
We will be checking more examples to see if the Odddownward Evenupward theorem seems to hold.
1-hop, transitive, E-focused rule, AC SRM (1tE)
|A|=ct(PA)  mnsp
ct(PA&fCRf) / ct(PA)  mncf
1. (antecedent upward closure) If A is infrequent, then so are all of its subsets.
2. (consequent downward closure) If AC is non-confident, then so is AD for all supersets, D, of C.
In this case A is 0-hops from E (even, use upward closure).
C is 1-hop from E (odd, use downward closure).

2. ct(&eARe &gCSg) / ct(&eARe)  mncf
2-hop transitive F-focused (focus on middle entity, F)
AC strong if:
ct(&eARe)  mnsp and C G
S(F,G)
ct(&eARe &gCSg) / ct(&eARe)  mncf 1 4 1 3
1. (antecedent downward closure) If A is infrequent, so are all of its supersets. 1 2 1 1
2. (consequent downward closure) If AC is non-confident, so is AD for all supersets, D.
3. Apriori for 2-hops: Find all freq antecedents, A, using downward closure. For each:
find C1G, the set of g's s.t. A{g} is confident.
Find C2G, the set of C1G pairs that are confident consequents for antecedent, A.
Find C3G, the set of triples (from C2G) s.t. all subpairs are in C2G (ala Apriori), etc.
1,1 odd so down, down correct. 2 3 4 5 F
2,0 even so up,up is correct.
0,2 even so up,up is correct. 4 1 3 1 2 1
Does standard ARM give the same info after collapsing R and S into T via, 1 2 3 4 G
T(E,G) E
A  C
T(e,g) = 1 iff (&eARe)g = 1
NO! Standard ARM has A and C both subsets of E (or G)
If G=E and S(F,E)=transposeR(E,F) then it is standard ARM?
What if G=E=F and T is collapsed? Think about the facebook case (where G=E=people, R and S are directional friendships...?
ct(&flist&eAReSf)mnsp
 mncf
ct(&flist&eAReSf & PC)
/ &flist&eAReSf 1 1
A  E
R(E,F)
2-hop trans G-foc
ct(&fl&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. (consequent upward closure) If AC is non-confident, so is AD for all subsets, D.
2-hop trans E-foc
ct(PA)mnsp
 mncf
ct(PA&f&gCSgRf )
/ ct(PA)
1. (antecedent upward closure) If A is infrequent, so are all subsets.
2. (consequent upward closure) If AC is non-confident, so is AD for all subsets, D.
We can for multi-hop relationships from the cards of 1-hop relationships
Does this open up a new area of text mining?
TDP=2
TDP=1 1 2 3 4 D T 5
A  C
TDP=2
TDP=1 1 2 3 4 T D 5
A  C 1 2 … 9
Term 3 D
DTPe k=1..7 TDRolodexCd 1 2 … 7
Pos 3 D
DTPe k=1..9 PDCd 1 2 … 7
Pos 9 T
DTPe k=1..3 PTCd

3. RoloDex Model: Relational Model: |0 0| 1 |0 1| 1 |1 0| 1 |2 0| 2
2 Entities many relationships 1
customer rates movie as 5 card
One can form multi-hops with any of these cards.
Are there any that provide and interesting setting for ARM data mining?
 Customer 1 2 3 4
Item
cust item card 1 5
people 2 3 4
items
terms
DataCube Model for 3 entities, items, people and terms. 1 2 3 4
Author 5 6 7
People 
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)
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:

4. RoloDex Model: Relational Model: |0 0| 1 |0 1| 1 |1 0| 1 |2 0| 2
2 Entities many relationships
Supp(A) =
CusFreq(ItemSet)
Conf(AB) =Supp(AB)/Supp(A) 5 6 16
ItemSet
ItemSet
antecedent 1 2 3 4 5 6 16
itemset itemset card
 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 1 2 3 4
Author 5 6 7
People 
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 1 3
Exp 7 6 5 4 3 2 t 1
termterm card (share stem?)
expPI card
expgene card
genegene card (ppi)
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:

5. ct(&eARe ct(&eARe &g&hCThSg) / ct(&eARe ct(&f&eAReSf) &hCTh)
3-hop
Collapse T: TC≡ {gG|T(g,h) hC} That's just 2-hop case w TCG replacing C. ( can be replaced by  or any other quantifier. The choice of quantifier should match that intended for C.).
Collapse T and S: STC≡{fF |S(f,g) gTC} Then it's 1-hop w STC replacing C.
 mnsup
ct(&eARe
S(F,G)
R(E,F) 1 2 3 4 E F 5 G A C
T(G,H) H
Focus on F
 mncnf
ct(&eARe
&g&hCThSg)
/ ct(&eARe
antecedent downward closure: A infreq. implies supersets infreq. A 1-hop from F (down
consequent upward closure: AC noncnf implies AD noncnf. DC. C 2-hops (up
 mnsp
ct(&f&eAReSf)
Focus on G
 mncnf
&hCTh)
ct(&f&eAReSf
/ ct(&f&eAReSf)
antecedent upward closure: A infreq. implies all subsets infreq. A 2-hop from G (up)
consequent downward closure: AC noncnf impl AD noncnf. DC. C 1-hops (down)
ct( &g=1,3,4 Sg ) /ct(1001)
ct( &1001&1000&1100) / 2
ct( ) / = 1/2
Focus on F Are they different? Yes, because the confidences can be different numbers Focus on G.
ct(&eARe
&glist&hCThSg )
/ct(&eARe
&hCTh)
ct(&flist&eAReSf
/ ct(&flist&eAReSf)
ct(&f=2,5Sf
&1101 )
/ ct(&f=2,5Sf
ct(1101 & &
/ ct(1101 & )
ct(0001 )
/ ct(0001) = 1/1 =1
ct(PA & Rf)
f&g&hCThSg
/ ct(PA)  mncnf
 mnsup
ct(PA)
Focus on E
antecedent upward closure: A infreq. implies subsets infreq. A 0-hops from E (up)
consequent downward closure: AC noncnf implies AD noncnf. DC. C 3-hops (down)
Focus on H
antecedent downward closure: A infreq. implies all subsets infreq. A 3-hops from G (down)
consequent upward closure: AC noncnf impl AD noncnf. DC. C 0-hops (up)
ct(& Tg & PC)
g&f&eAReSf
mncnf
/ct(& Tg)
ct(& Tg)
mnsp

6. * ct(&iCUi) ) ct(&f&eAReSf) ct( &f&eAReSf &h&iCUiTh )
4-hop
S(F,G)
R(E,F) 1 2 3 4 E F 5 G A C
T(G,H) H
U(H,I) I
Focus on G? Replace C by UC; A by RA as above (not different from 2 hop?)
Focus on H (RA for A, use 3-hop) or focus on F (UC for C, use 3-hop).
Another focus on G (the main way)
 mnsup
ct(&f&eAReSf)
 mncnf
ct( &f&eAReSf
&h&iCUiTh )
/ ct(&f&eAReSf)
...
R(E,G) 1 2 3 4 E G 5 A
Sn(G,G)
S1(G,G)
U(G,I) C I
F=G=H=genes and S,T=gene-gene intereactions.
More than 3, S1, ..., Sn?
&iCUi))+
(ct(S1(&eARe
mncnf
/ ( (ct(&eARe))n
* ct(&iCUi) )
&iCUi))+...
ct(S2(&eARe
&iCUi)) )
ct(Sn(&eARe
If the S cube can be implemented so counts can be can be made of the 3-rectangle in blue directly, calculation of confidence would be fast.
4-hop APRIORI focus on G:
 mnsup
ct(&f&eAReSf)
 mncnf
ct(&f&eAReSf
&h&iCUiTh)
/ ct(&f&eAReSf)
1. (antecedent upward closure) If A is infrequent, then so are all of its subsets (the "list" will be larger, so the AND over the list will produce fewer ones) Frequency involves only A, so mine all qualifying antecedents using upward closure.
2. (consequent upward closure) If AC is non-confident, then so is AD for all subsets, D, of C (the "list" will be larger, so the AND over the list will produce fewer ones) So  frequent antecedent, A, use upward closure to mine out all confident consequents, C.

7. ct(&f&eAReSf) 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:
 mnsup
ct(&f&eAReSf)
ct( &f&eAReSf
&h(& )UiTh ) /
ct(&f&eAReSf)
 mncnf
i(&jCVj)
5-hop APRIORI focus on G:
1. (antecedent upward closure) If A is infrequent, then so are all of its subsets (the "list" will be larger, so the AND over the list will produce fewer ones) Frequency involves only A, so mine all qualifying antecedents using upward closure.
2. (consequent downward closure) If AC is non-confident, then so is AD for all supersets, D, of C. So  frequent antecedent, A, use downward closure to mine out all confident consequents, C.

8. ct( &f(& )ReSf) ct( &f(& )ReSf &h(& )UiTh) / ct( &f(& )ReSf )
6-hop
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 D
Q(D,E)
The conclusion we have demonstrated (but not proven) is: for (a+c)-hop transitive Apriori ARM with focus the 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 that step in the mining of strong (frequent and confident) rules.
Focus on G:
ct(
&f(& )ReSf)
 mnsup
e(&dDQd)
ct(
&f(& )ReSf
&h(& )UiTh) /
e(&dDQd)
i(&jCVj)
ct(
&f(& )ReSf )
 mncnf
e(&dDQd)
6-hop APRIORI:
1. (antecedent downward closure) If A is infrequent, then so are all of its supersetsbsets. Frequency involves only A, so mine all qualifying antecedents using downward closure.
2. (consequent downward closure) If AC is non-confident, then so is AD for all supersets, D, of C. So  frequent antecedent, A, use downward closure to mine out all confident consequents, C.

9. Given any 1-hop labeled relationship (e. g
Given any 1-hop labeled relationship (e.g., cells have values from {1,2,…,n} then there is:
1. a natural n-hop transitive relationship, A implies D, by alternating entities for each specific label value relationship.
2. cards for each entity consisting of the bitslices of cell values.
E.g., in netflix, Rating(Cust,Movie) has label set {0,1,2,3,4,5}, so in 1. it generates a bonafide 6-hop transitive relationship.
In 2. an alternative is to bitmap each label value (rather than bitslicing them). Below Rn-i can be bitslices or bitmaps
R3(C,M)
R2(M,C) 1 2 3 4 M C 5 A D
R4(M,C)
R5(C,M)
R0(M,C)
R1(C,M)
R0(E,F)
Rn-2(E,F)
Rn-1(E,F) F 2 3 4 5 1 E
A 
... D
E.g., equity trading on a given day, QuantityBought(Cust,Stock) w labels {0,1,2,3,4,5} (where n means n thousand shares) so that generates a bonafide 6-hop transitive relationship:
E.g., equity trading - moved similarly, (define moved similarly on a day --> StockStock(#DaysMovedSimilarlyOfLast10)
E.g., equity trading - moved similarly2, (define moved similarly to mean that stock2 moved similarly to what stock1 did the previous day.Define relationship StockStock(#DaysMovedSimilarlyOfLast10)
E.g., Gene-Experiment, Label values could be "expression level". Intervalize and go!
Has Strong Transitive Rule Mining (STRM) been done? Are their downward and upward closure theorems already for it?
Is it useful? That is, are there good examples of use: stocks, gene-experiment, MBR, Netflix predictor,...

10. ct(&iABBi &tDBt) / ct(&iABBi)  mncf
Let Types be an entity which clusters Items (moves Items up the semantic hierarchy),
E.g., in a store, Types might include; dairy, hardware, household, canned, snacks, baking, meats, produce, bakery, automotive, electronics, toddler, boys, girls, women, men, pharmacy, garden, toys, farm).
Let A be an ItemSet wholly of one Type, TA, and l
et D by a TypesSet which does not include TA.
Then: 1
Buys(C,T)
BoughtBy(I,C,)
Items
Customers 2 3 4 5
Types (of Items)
A  D 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
AD might mean If iA s.t. BB(i,c) then tT, B(c,t)
AD might mean If iA s.t. BB(i,c) then tT, B(c,t)
AD might mean If iA s.t. BB(i,c) then tT, B(c,t)
AD might mean If iA s.t. BB(i,c) then tT, B(c,t)
AD confident might mean
ct(&iABBi &tDBt) / ct(&iABBi)  mncf
ct(&iABBi | tDBt) / ct(&iABBi)  mncf
ct( | iABBi | tDBt) / ct( | iABBi)  mncf
ct( | iABBi &tDBt) / ct( | iABBi)  mncf
AD frequent might mean
ct(&iABBi)  mnsp
ct( | iABBi)  mnsp
ct(&tDBt)  mnsp
ct( | tDBt)  mnsp
ct(&iABBi
&tDBt)  mnsp, etc.

11. ct(&iABBi &tDBt) / ct(&iABBi)  mncf
Let Types be an entity which clusters Items (moves Items up the semantic hierarchy),
E.g., in a store, Types might include; dairy, hardware, household, canned, snacks, baking, meats, produce, bakery, automotive, electronics, toddler, boys, girls, women, men, pharmacy, garden, toys, farm).
Let A be an ItemSet wholly of one Type, TA, and l
et D by a TypesSet which does not include TA.
Then: 1
Buys(C,T)
BoughtBy(I,C,)
Items
Customers 2 3 4 5
Types (of Items)
A  D 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
AD might mean If iA s.t. BB(i,c) then tT, B(c,t)
AD might mean If iA s.t. BB(i,c) then tT, B(c,t)
AD might mean If iA s.t. BB(i,c) then tT, B(c,t)
AD might mean If iA s.t. BB(i,c) then tT, B(c,t)
AD confident might mean
ct(&iABBi &tDBt) / ct(&iABBi)  mncf
ct(&iABBi | tDBt) / ct(&iABBi)  mncf
ct( | iABBi | tDBt) / ct( | iABBi)  mncf
ct( | iABBi &tDBt) / ct( | iABBi)  mncf
AD frequent might mean
ct(&iABBi)  mnsp
ct( | iABBi)  mnsp
ct(&tDBt)  mnsp
ct( | tDBt)  mnsp
ct(&iABBi
&tDBt)  mnsp, etc.