Nowadays, more and more data is collected in large amounts, thus the need of studying it both efficiently and profitably is arising; we want to achieve new and significant informations that weren’t known before the analysis. At this time many graph mining algorithms have been developed, but an algebra that could systematically define how to generalize such operations is missing. In order to propel the development of a such automatic analysis of an algebra, We propose for the first time (to the best of my knowledge) some primitive operators that may be the prelude to the systematical definition of a hypergraph algebra in this regard.

1. Hypergraph Mining for Social Networks
Giacomo Bergami
giacomo90@libero.it
Università di Bologna
July 17, 2014
Giacomo Bergamigiacomo90@libero.it (Università di Bologna)Hypergraph Mining for Social Networks July 17, 2014 1 / 19

2. Contents
1 Goals & State of the Art
2 Why Hypergraphs?
3 Data Mining Algebra
4 gSpan
The Original Algorithm
Specialization Proposal
5 Conclusions & Future work
Giacomo Bergamigiacomo90@libero.it (Università di Bologna)Hypergraph Mining for Social Networks July 17, 2014 2 / 19

3. Goals & . . .
Define an hypergraph data structure for both data and relations.
Define some algebra operators.
Open question: how to automate the mining process.
Evaluations: online social network (OSN) data representation.
Experiments: Graph clustering of Iris data set.
Giacomo Bergamigiacomo90@libero.it (Università di Bologna)Hypergraph Mining for Social Networks July 17, 2014 3 / 19

4. . . . & State of the Art
Toon Calders’s Data Mining (Relational) Algebra.
Data structures that have already been studied in detail:.
Data mining operations have been developed in both graphs and
relational graphs.
Well known algorithms (optimal computational cost) and operators.
While hypergraphs may be an intuitive representation of higher order
similarities, it seems (anecdotally at least) that graphs lie at the heart
of this problem.
Giacomo Bergamigiacomo90@libero.it (Università di Bologna)Hypergraph Mining for Social Networks July 17, 2014 4 / 19

5. Why Hypergraphs?
tommy32
Tom S.
(lat1,long1)
Tom S. (lat1,long1) tommy32
Giacomo Bergamigiacomo90@libero.it (Università di Bologna)Hypergraph Mining for Social Networks July 17, 2014 5 / 19

6. Data Structures: ED as data entities’ representation
tommy32
Tom S.
(lat1,long1)
sam
Adeola S.
(lat2,long2)
47561
10231
75234
80235
11230
UserID UserLocation UserName w ϕ
tommy23 (lat1,long1) Tom Sawyer 1 1000
sam (lat2,long2) Adeola Samuel 1 10002



Entities(ED)
PostID PostLocation PostContent UserID w ϕ
47561 (lat3,long3) Have a nice day! tommy32 1 1
80235 (lat4,long4) Some great news... sam 1 2
10231 (NULL,NULL) J.S. Bach and ... tommy32 1 3
75234 (lat5,long5) Telemann & Xenakis... tommy32 1 4
11230 (NULL,NULL) Yet another plot! sam 1 5



Entities(ED)
Database with Uncertain Data
Users<:Object
Posts<:Object
Collection
Attributes
Primary key Foreign key
Giacomo Bergamigiacomo90@libero.it (Università di Bologna)Hypergraph Mining for Social Networks July 17, 2014 6 / 19

7. Data Structures: EE as binary relationships’ representation
tommy32
Tom S.
(lat1,long1)
sam
Adeola S.
(lat2,long2)
47561
10231
75234
80235
11230
tommy32
10231
75234
47561
sam
80235
10231follows
follows
The information’s atomization allows to automatically identify relationships
between data (users’ posts)
If we keep binary EE relations between ED, we could retain a linear time
complexity on the size of a binary graph (O(|G|)), where ED are mapped as
G’s vertices.
Giacomo Bergamigiacomo90@libero.it (Università di Bologna)Hypergraph Mining for Social Networks July 17, 2014 7 / 19

8. DHImp, the final step. (DHImp = (db, T))
Relational Database will express data relations (ED) while tensors will
express data correlations (EE )
This definition permits to define also tensors τi ∈ T for non binary
relations (OSN relations are mainly binary).
Permits to separate the operations for data and the operations over
relations.
Giacomo Bergamigiacomo90@libero.it (Università di Bologna)Hypergraph Mining for Social Networks July 17, 2014 8 / 19

9. (Relational) Data Mining Algebra
D E
Pop
I
πA πRDA
κ
λ
Toons’ DMA Identifies different “data” categories: Data World,
Intensional World, Extensional World.
Some internal world operations and external world operations are
defined.
Data Mining algorithms could be described by this algebra.
Is it possible to map these worlds over our hypergraphs? Yes
Is it possible to define an algebra for (weighted and indexed)
hypergraphs? Yes
Giacomo Bergamigiacomo90@libero.it (Università di Bologna)Hypergraph Mining for Social Networks July 17, 2014 9 / 19

10. Is it possible to map these worlds over our hypergraphs?
DHImp (as Hypergraph)
euser
epos
eid
fuser
fpos
fid
Pure I-Hypergraph
{e}
{f }
{e, f }
HDM = (h, H, EL)
Giacomo Bergamigiacomo90@libero.it (Università di Bologna)Hypergraph Mining for Social Networks July 17, 2014 10 / 19

11. Is it possible to define an algebra for (weighted and indexed) hypergraphs?
Definition (Database operations)
˙(DB) = { (t) | t ∈ DB }
DB1 ˙ DB2 = { t1 t2 | t1 ∈ DB1 ∧ t2 ∈ DB2 }
Definition (Index-consistency)
A database unary operation ˙ is said to be index-consistent iff. for all the
tables of the current database, the indices among the tables are kept
distinct. (Similarly for binary ones).
Relational algebra operators over DWorld should be redefined for
weight update and indexing properties.
The reindexing over the tables obtained as a result of algebraic
operations is performed via dovetailing.
All the relational algebra operations have to be proved as
index-consistent.
Giacomo Bergamigiacomo90@libero.it (Università di Bologna)Hypergraph Mining for Social Networks July 17, 2014 11 / 19

12. Hypergraph Data Mining Example: H. Clustering via Binary Graph Clustering
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Real data (left) vs. Clustered (right) values.
Giacomo Bergamigiacomo90@libero.it (Università di Bologna)Hypergraph Mining for Social Networks July 17, 2014 12 / 19

13. Hypergraph Data Operators Example: 2nd Neighbours and Node Degree
<gyankos,Jack Bergus>
<jsbach,Johann Sebastian Bach>
<handel,G F Haendel>
<mozart,W A Mozart>
<faux,P.D.Q. Bach>
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Giacomo Bergamigiacomo90@libero.it (Università di Bologna)Hypergraph Mining for Social Networks July 17, 2014 13 / 19

14. G ←
Calcf (x)=1 as Count(ρ{NScreenName←ScreenName,NUserId←UserId}(G))
<Jack Bergus,gyankos,1>
<Johann Sebastian Bach,jsbach,1>
<G F Haendel,handel,1>
<W A Mozart,mozart,1>
<P.D.Q. Bach,faux,1>
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Giacomo Bergamigiacomo90@libero.it (Università di Bologna)Hypergraph Mining for Social Networks July 17, 2014 14 / 19

15. G ← σP
e→w(e)≥0.5(G x,y→ϕ(x)=ϕ(y)∨T[x,y,hasFriend]=0 G )
<jsbach,Johann Sebastian Bach,G F Haendel,handel,1>
<jsbach,Johann Sebastian Bach,P.D.Q. Bach,faux,1>
<gyankos,Jack Bergus,Jack Bergus,gyankos,1><handel,G F Haendel,G F Haendel,handel,1>
<mozart,W A Mozart,W A Mozart,mozart,1>
<gyankos,Jack Bergus,P.D.Q. Bach,faux,1>
<jsbach,Johann Sebastian Bach,Johann Sebastian Bach,jsbach,1>
<mozart,W A Mozart,G F Haendel,handel,1><faux,P.D.Q. Bach,Johann Sebastian Bach,jsbach,1>
<faux,P.D.Q. Bach,P.D.Q. Bach,faux,1>
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Giacomo Bergamigiacomo90@libero.it (Università di Bologna)Hypergraph Mining for Social Networks July 17, 2014 15 / 19

16. G ← σP
e→w(e)≥0.25(Γsum(Count)-1 as Count
ScreenName,UserId (G ))
<W A Mozart,mozart,1>
<Jack Bergus,gyankos,1>
<P.D.Q. Bach,faux,1>
<Johann Sebastian Bach,jsbach,2>
<G F Haendel,handel,0>
0.75
0.62
0.25
0.38
0.58
0.25
0.5
0.28 0.67
Giacomo Bergamigiacomo90@libero.it (Università di Bologna)Hypergraph Mining for Social Networks July 17, 2014 16 / 19

17. gSpan
The Original Algorithm
gSpan is a frequent subgraph mining algorithm that works in the following
way:
§
gSpan(DB, minsupp, Solution) {
Ξ ←sort(FrequentEdges(DB,minsupp))
Solution ← Ξ;
NStack ← Solution
5 while (g ← NStack.pop()) {
if g = minDfsCode(g) continue;
Solution ← g
∀e ∈ Ξ. if (e re g)⇒ { // if e re g is a rightmost expansion of g by e
if GS(e re g, DB) ≥ minsupp
0 NStack.push(e re g)
}
}
}
¦ ¥
Listing 1 : gSpan
Giacomo Bergamigiacomo90@libero.it (Università di Bologna)Hypergraph Mining for Social Networks July 17, 2014 17 / 19

18. gSpan
Specialization Proposal
https://github.com/jackbergus/gSpanExtended
Suppose to see DHImp = (db, T) as a directed (binary) graph:
({ϕ(e)|t ∈ db ∧ e ∈ t}, {(e, f )|∃e, f , k.T[e, f , k] = 0})
We suppose that our HDM contains only one vertex per entity
instance. Hence, each vertex will have a unique label, that is its data
representation (D(e)) or index (ϕ(e)).
Suppose that relation label λ(e) = λ(v, w) = (ϕ(v), T(e), ϕ(w)),
each edge has an unique label because each vertex is unique.
gSpan algorithm guarantees that the minimal dfs code is unique for
each graph. This algorithm strengthens the fact.
Subgraph isomorphism over distinct edges and vertices reduces to an
approximated ordered-subset test that could be implemented in linear
time.
The overall original algorithm has a polynomial time complexity.
Giacomo Bergamigiacomo90@libero.it (Università di Bologna)Hypergraph Mining for Social Networks July 17, 2014 18 / 19

19. Conclusions & Future work
Conclusion
We used R for experimenting data mining operations. Some Java
implementations of the algebra were provided
(https://github.com/jackbergus/hypergraphalgebra/)
Hypergraph express n-ary relations, and change both data and
relations.
Hypergraph problems could be reduced to graph ones.
Future Work
We could study the time complexity of each data mining operators for
hypergraphs and complete the algebra definition.
We could set an environment where execute such hypergraph algebric
operations with distributed or parallel algorithms.
Giacomo Bergamigiacomo90@libero.it (Università di Bologna)Hypergraph Mining for Social Networks July 17, 2014 19 / 19