1. Structural Patterns and
Generative Models of
Real-world Hypergraphs
Manh Tuan Do, Se-eun Yoon, Bryan Hooi, Kijung Shin
26th ACM SIGKDD International Conference on
Knowledge Discovery and Data Mining
KDD 2020
Contact: Manh Tuan Do (manh.it97@kaist.ac.kr)

2. Roadmap
1. Introduction <<
2. Decomposition
3. Structural Patterns
4. Generators
5. Conclusions
2/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)

3. 3/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)
Coauthorship Email Tags
Common patterns Underlying mechanisms
Motivation Structural Patterns GeneratorsDecomposition Conclusion
Hypergraphs

4. Roadmap
1. Motivation
2. Decomposition <<
3. Structural Patterns
4. Generators
5. Conclusions
4/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)

5. • Hypergraphs: not straightforward to analyze
o complex representation
o lack tools
5/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)
Only interactions
at the level of
nodes
1
2 3
4 5
7
6 1
2 3
4
7
5
6
Motivation for a New Tool
Motivation Structural Patterns GeneratorsDecomposition Conclusion
• Projection
o information loss
o no high-order level information

6. • Multi-level decomposition:
◦ representation by pair-wise graphs
◦ leveraging existing tools & measurements
◦ no information loss: original hypergraph is reconstructible
6/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)
Our Tool: Decomposition
Motivation Structural Patterns GeneratorsDecomposition Conclusion
1 7
2 42 3
1 4 3 4
1 21 3
5 6
4 5
3 5
3 4
5
1 2
3
1 2
4
1 3
4
2 3
4
1
2 3
4
7
5
6

7. 7/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)
1
2 3
4 5
7
6
{1, 2, 3, 4}
{3, 4, 5}
{5, 6}
{1, 7}
1
2 3
4
7
5
6
3 4
5
1 2
3
1 2
4
1 3
4
2 3
4
1 2
3 4
Node level
Triangle level 4clique level
Our Tool: Decomposition
2 42 3
1 4 3 4
1 21 31 7
5 6
4 5
3 5
Edge level
Motivation Structural Patterns GeneratorsDecomposition Conclusion

8. 8/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)
{1, 2, 3, 4}
{3, 4, 5}
{5, 6}
{1, 7}
Maximum
hyperedge size: 𝑁
Our Tool: Decomposition
Hypergraph Decomposed graphs
Node-level
Edge-level
…..
(𝑁-1)-clique level
Decomposition
Reconstruction
1 7
2 42 3
1 4 3 4
1 21 3
5 6
4 5
3 5
3 4
5
1 2
3
1 2
4
1 3
4
2 3
4
1
2 3
4
7
5
6
1
2 3
4 5
7
6
Motivation Structural Patterns GeneratorsDecomposition Conclusion

9. Roadmap
1. Motivation
2. Decomposition
3. Structural Patterns <<
4. Generators
5. Conclusions
9/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)

10. • 13 datasets from 6 domains
◦ Email: recipient addresses of an email
◦ Drug components: classes or substances within a single drug, listed
in the National Drug Code Directory
◦ Drug use: drugs used by a patient, reported to the Drug Abuse
Warning Network, before an emergency visit
◦ Online tags: tags in a question in Stack Exchange forums
◦ Online threads: users answering a question in Stack Exchange forums
◦ Coauthorship: coauthors of a publications
10/26Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)
Motivation Structural Patterns GeneratorsDecomposition Conclusion
Real-word Datasets

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Structural Patterns
P1. Degree distribution: heavy-tailed
P2. Connected component: giant
P3. Clustering coefficient: high
P4. Effective diameter: small
P5. Singular value distribution: heavy-tailed
Motivation Structural Patterns GeneratorsDecomposition Conclusion

12. 12/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)
P1+P5. Heavy-tailed Distributions
Abundant low-
degree nodes
A few high-
degree nodes
Degree Singular values
Motivation Structural Patterns GeneratorsDecomposition Conclusion
Degree and singular-value distributions are heavy-tailed
J. Leskovec, J. Kleinberg, Cornell, C. Faloutsos . 2005. Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations. In KDD

13. 13/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)
Motivation Structural Patterns GeneratorsDecomposition Conclusion
Statistical tests: to confirm heavy-tailed distributions
Lilliefors Test(1)
• 𝐻0: distribution is exponential
• 𝐻1: distribution is not exponential
𝐻0 rejected at 2.5% significance level
Log likelihood ratio(2)
𝑟 = 𝑙𝑜𝑔
𝐿1
𝐿0
• 𝐿1 : likelihood of a heavy-tailed
distribution (power-law, log
normal)
• 𝐿0 : likelihood of the exponential
distribution
If 𝑟 > 0 : the distribution is more
likely to be heavy-tailed.
P1+P5. Heavy-tailed Distributions
(1) Hubert W Lilliefors. 1969. On the Kolmogorov-Smirnov test for the exponential distribution with mean unknown. Journal of American Statististical
Association 64 (1969), 387–389.
(2) Jeff Alstott and Dietmar Plenz Bullmore. 2014. powerlaw: a Python package for analysis of heavy-tailed distributions. PloS one 9, 1 (2014)

14. 14/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)
P2. Giant Connected Component
A large proportion of nodes are connected
Connected components
Proportionofnodes
Motivation Structural Patterns GeneratorsDecomposition Conclusion
J. Leskovec, J. Kleinberg, Cornell, C. Faloutsos . 2005. Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations. In KDD

15. 15/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)
P3. High Clustering Coefficient
Local clustering coefficient:
𝐶𝑖 =
2 ∗ 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠 𝑎𝑡 𝑖
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑒𝑑𝑔𝑒𝑠 𝑎𝑡 𝑖
Clustering coefficient:
𝐶 =
1
|𝑉|
𝑖∈𝑉
𝐶𝑖
High likelihood of having links between “friends of friends”
Motivation Structural Patterns GeneratorsDecomposition Conclusion
Wedge at 𝑖: open triangle
𝑖

16. 16/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)
P4. Small Effective Diameter
d = 8
90%
of
pairs
Most pairs of connected nodes: reachable within a small distance
Motivation Structural Patterns GeneratorsDecomposition Conclusion
https://web.stanford.edu/class/cs224w/handouts/02-gnp-smallworld.pdf

17. 17/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)
Structural Patterns: Intuition
Real-world
graphs
P1. Degree distribution: heavy-tailed
P2. Connected component: giant
P3. Clustering coefficient: high
P4. Effective diameter: small
P5. Singular value distribution: heavy-tailed
Hypergraph Decomposed
graphs
All decomposition
levels
Real-world graph
Decomposition
Motivation Structural Patterns GeneratorsDecomposition Conclusion
J. Leskovec, J. Kleinberg, Cornell, C. Faloutsos. 2005. Graphs over Time:
Densification Laws, Shrinking Diameters and Possible Explanations. In KDD

18. 18/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)
Structural Patterns
Degree distributions: heavy-tailed
Node level Edge level Triangle level 4clique level
Motivation Structural Patterns GeneratorsDecomposition Conclusion

19. 19/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)
Structural Patterns
Singular-value distributions: heavy-tailed
Node level Edge level Triangle level 4clique level
Motivation Structural Patterns GeneratorsDecomposition Conclusion

20. 20/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)
Structural Patterns
Giant connected components vary among datasets
Motivation Structural Patterns GeneratorsDecomposition Conclusion
If there is a giant connected component
• High Clustering Coefficient
• Small Effective Diameter
Proportion of nodes in the
largest connected component
Small numbers indicate the
absence of a giant connected
component

21. Roadmap
1. Motivation
2. Decomposition
3. Structural Patterns
4. Generators <<
5. Conclusions
21/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)

22. 22/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)
Our Model: HyperPA
1
2
3
A new node: 4
1: 2 2: 2 3: 1
(1,2): 2 (1,3): 1 (2,3): 1
(1,2,3): 1
2 hyperedges
1st hyperedge: size 2 {2,4}
2nd hyperedge: size 3 {1,3,4}
{1,2}
{1,2,3}
Motivation Structural Patterns GeneratorsDecomposition Conclusion
Main idea: “Subsets get rich together”

23. 23/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)
Our Model: HyperPA
Main idea: “Subsets get rich together”
Introduce a new node
Add the newly formed hyperedges
Update subset degrees
Repeat the process
1
2
3
1: 3 2: 3 3: 2
(1,2): 2 (1,3): 2 (2,3): 1
(1,2,3): 1
4
4: 2
(1,3,4): 1
(1,4): 1 (2,4): 1
{1,2}
{1,2,3}
(3,4): 1
A new node: 5
Motivation Structural Patterns GeneratorsDecomposition Conclusion
{2,4}
{1,3,4}

24. 24/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)
Node level Edge level Triangle level 4clique level
Motivation Structural Patterns GeneratorsDecomposition Conclusion
HyperPA
Real HyperPA: considers degrees of groups of nodes
Our Model: HyperPA

25. 25/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)
Node level Edge level Triangle level 4clique level
NaivePA
Real
Motivation Structural Patterns GeneratorsDecomposition Conclusion
NaivePA: considers node degrees individually
Baseline: NaïvePA

26. Roadmap
1. Motivation
2. Decomposition
3. Structural Patterns
4. Generators
5. Conclusions <<
26/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)

27. 27/28Structural Patterns and Generative Models of Real-world Hypergraphs (by Manh Tuan Do)
Conclusion
Motivation Structural Patterns GeneratorsDecomposition Conclusion
• Our contributions in this work:
◦ Decomposition Tool: convenient analysis of hypergraphs
◦ Structural Patterns: 5 patterns across decomposition levels
◦ HyperPA: generator reproducing the 5 structural patterns
P1. Degree distribution: heavy-tailed
P2. Connected component: giant
P3. Clustering coefficient: high
P4. Effective diameter: small
P5. Singular-value distribution: heavy-tailed

28. Structural Patterns and
Generative Models of
Real-world Hypergraphs
Manh Tuan Do, Se-eun Yoon, Bryan Hooi, Kijung Shin
26th ACM SIGKDD International Conference on
Knowledge Discovery and Data Mining
KDD 2020
Contact: Manh Tuan Do (manh.it97@kaist.ac.kr)