ICDM 2021 THyMe+: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting

1. THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
ICDM 2021
THyMe+
: Temporal Hypergraph Motifs
and Fast Algorithms for Exact Counting
Kijung Shin
Geon Lee

2. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
I 2021 2
ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
Hypergraphs are Everywhere
• Hypergraphs consist of nodes and hyperedges.
• Each hyperedge is a subset of any number of nodes.
Collaborations of Researchers Co-purchases of Items Joint Interactions of Proteins
Conclusion
Introduction Algorithms
Observations
Backgrounds Concepts

3. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
Hypergraphs Evolve Over Time
• In many real-world scenarios, hypergraphs evolve over time.
• Temporal hypergraphs consist of temporal hyperedges.
t1=7
1
4
5 1
2
4
5
3
t2=10
1
2
4
5 6
3
1
2
4
5 6 8
7
3
t3=11 t4=12 t5=14 t5=17
1
2
4
5 6 8
7
3
1
2
4
5 6 8
7
3
Conclusion
Introduction Algorithms
Observations
Backgrounds Concepts

4. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
Our Question
What are local structural & temporal properties of real-world hypergraphs?
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4
5
1
2
4
5
3
1
2
4
5 6
3
1
2
4
5 6 8
7
3
1
2
4
5 6 8
7
3
1
2
4
5 6 8
7
3
How do temporal hyperedges appear in a short period of time?
Temporal perspective
How do three hyperedges overlap each other?
Structural perspective
Conclusion
Introduction Algorithms
Observations
Backgrounds Concepts

5. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
Roadmap
1. Backgrounds
2. Concepts
3. Observations
4. Algorithms
5. Conclusion

6. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
Hypergraph Motifs
• Hypergraph motifs (h-motifs) describe connectivity patterns of three
connected hyperedges in static hypergraphs.
• H-motifs describe the connectivity pattern of hyperedges 𝑒𝑖, 𝑒𝑗, and 𝑒𝑘 by the
emptiness of seven subsets.
1
2 3
4 6
5
7
(1) ei ej ek
(2) ej ek ei
(3) ek ei ej
(4) ei ∩ ej ek
(5) ej ∩ ek ei
(6) ek ∩ ei ej
(7) ei ∩ ej ∩ ek
ei
ej ek
Backgrounds Conclusion
Introduction Concepts Algorithms
Observations

7. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
Hypergraph Motifs (cont.)
• While there can exist 27 h-motifs, 26 h-motifs remain once we exclude:
1. symmetric ones
2. those with duplicated hyperedges
3. those cannot be obtained from connected hyperedges
h-motif 1 h-motif 2 h-motif 3 h-motif 4 h-motif 5 h-motif 6 h-motif 7 h-motif 8 h-motif 9 h-motif 10 h-motif 11 h-motif 12 h-motif 13
h-motif 14 h-motif 15 h-motif 16 h-motif 17 h-motif 18 h-motif 19 h-motif 20 h-motif 21 h-motif 22 h-motif 23 h-motif 24 h-motif 25 h-motif 26
Backgrounds Conclusion
Introduction Concepts Algorithms
Observations

8. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
Hypergraph Motifs (cont.)
• Hypergraph motifs describe connectivity patterns of three connected hyperedges.
• For example:
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2
4
5 6 8
7
3
Static Hypergraph
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2
4
5 6
3
1
4
5 6 8
7
2
4
5 6 8
7
3
1
2
5
4
3
6
7
1
2
5
4
3
6
7
1
2
5
4
3
6
7
h-motif 6 h-motif 21 h-motif 22
෤
e2
෤
e1
෤
e3
෤
e4
෤
e4
෤
e4
෤
e1
෤
e1
෤
e2 ෤
e2
෤
e3
෤
e3
෤
e3
Backgrounds Conclusion
Introduction Concepts Algorithms
Observations

9. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
Roadmap
1. Backgrounds
2. Concepts
3. Observations
4. Algorithms
5. Conclusion

10. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
Temporal Hypergraph Motifs
• How can we define motifs in temporal hypergraph?
• We define temporal hypergraph motifs (TH-motifs) that describe structural and
temporal patterns in sequences of three connected temporal hyperedges.
1
e1 = (෤
e1, t1=7)
2
4
5 6 8
7
3
e2 = (෤
e2, t2=10)
e3 = (෤
e3, t3=11)
e4 = (෤
e4, t4=12)
Set of nodes Timestamp
Conclusion
Introduction Concepts Algorithms
Observations
Backgrounds

11. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
Temporal Hypergraph Motifs (cont.)
Q1. How can we capture structural properties of hypergraphs?
A1. We consider the emptiness of seven subsets of three temporal hyperedges.
1
2 3
4
5
6
7
e1 = (෤
e1, t1=7)
e2 = (෤
e2, t2=10) e3 = (෤
e3, t3=11)
Is it empty?
Conclusion
Introduction Concepts Algorithms
Observations
Backgrounds

12. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
Temporal Hypergraph Motifs (cont.)
Q2. How can we capture temporal properties of hypergraphs?
1
2 3
4
5
6
7
e1 = (෤
e1, t1=7)
e2 = (෤
e2, t2=10) e3 = (෤
e3, t3=11)
A2-1. The three temporal hyperedges should arrive within 𝜹 time.
max t1, t2, t3 − min t1, t2, t3 ≤ 𝛿
A2-2. The order of the three temporal hyperedges is considered.
e1 → e2 → e3
Conclusion
Introduction Concepts Algorithms
Observations
Backgrounds

13. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
Temporal Hypergraph Motifs (cont.)
• For example, let 𝜹 = 3.
2
4
5 6 8
7
3
e2 = (෤
e2, t2=10)
e3 = (෤
e3, t3=11)
e4 = (෤
e4, t4=12)
1
e1 = (෤
e1={1,4,5}, t1=7)
2
4
5 6 8
7
3
e2 = (෤
e2={2,3,4}, t2=10)
e3 = (෤
e3={4,5,6}, t3=11)
e6 = (෤
e6={4,5,6}, t3=17)
e4 = (෤
e4={6,7,8}, t4=12)
e5 = (෤
e5={6,7,8}, t5=14)
Arrived first
Arrived second Arrived third
1
4
5 6 8
7
e1 = (෤
e1, t1=7)
e3 = (෤
e3, t3=11)
e4 = (෤
e4, t4=12)
Does not arrive
within 𝜹 time units!
Conclusion
Introduction Concepts Algorithms
Observations
Backgrounds

14. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
Temporal Hypergraph Motifs (cont.)
• We define 96 temporal hypergraph motifs (TH-motifs).
Conclusion
Introduction Concepts Algorithms
Observations
Backgrounds

15. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
Roadmap
1. Backgrounds
2. Concepts
3. Observations
4. Algorithms
5. Conclusion

16. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
Observations: Real Hypergraphs are Not Random
Obs1. Real hypergraphs are clearly distinguished from randomized hypergraphs.
Backgrounds Conclusion
Introduction Concepts Observations
email-Eu
Real-world Hypergraph RandomHypergraph
contact-primary threads-math
tags-ubuntu coauth-DBLP
Algorithms

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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
Observations: TH-motifs Distinguish Domains
Obs2. TH-motifs play a key role in capturing structural & temporal patterns.
Email
domain
Contact
domain
Coauthorship
domain
Tags
domain
Threads
domain
Characteristic Profile (CP): Relative significance of each TH-motif.
𝐶𝑃𝑡 ≔
∆𝑡
σ𝑡=1
96 ∆𝑡
2
where ∆𝑡≔
𝑀 𝑡 −𝑀𝑟𝑎𝑛𝑑 𝑡
𝑀 𝑡 +𝑀𝑟𝑎𝑛𝑑 𝑡 +𝜖
Backgrounds Conclusion
Introduction Concepts Observations Algorithms

18. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
Observations: TH-motifs Distinguish Domains (cont.)
Obs2. TH-motifs play a key role in capturing structural & temporal patterns.
Co-authorship
Contact
Email
Tags
Threads
Temporal Hypergraph Motif
• Within-domain: 0.900
• Between-domain: 0.141
Gap: 0.759
Times: 6.38X
Static Hypergraph Motif
• Within-domain: 0.951
• Between-domain: 0.434
Gap: 0.517
Times: 2.19X
Backgrounds Conclusion
Introduction Concepts Observations Algorithms

19. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
Observations: TH-motifs Help Predict Future Hyperedges
Obs3. TH-motifs can be used as powerful features for predicting future hyperedges.
THM96 SHM26
THM26
email-Enron email-Eu contact-primary contact-high
# of each TH-motifs’ instances
that each hyperedge is contained.
# of each static h-motifs’ instances
that each hyperedge is contained.
Backgrounds Conclusion
Introduction Concepts Observations Algorithms

20. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
Roadmap
1. Backgrounds
2. Concepts
3. Observations
4. Algorithms
5. Conclusion

21. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
I 2021 21
ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
DP: Naï
ve Approach Using Dynamic Programming
• DP enumerates the instances of static h-motifs in the induced static hypergraph.
1
e1 = (෤
e1={1,4,5}, t1=7)
2
4
5 6 8
7
3
e2 = (෤
e2={2,3,4}, t2=10)
e3 = (෤
e3={4,5,6}, t3=11)
e6 = (෤
e6={4,5,6}, t3=17)
e4 = (෤
e4={6,7,8}, t4=12)
e5 = (෤
e5={6,7,8}, t5=14)
1
2
4
5 6 8
7
3
෤
e1
෤
e2
෤
e4
෤
e3
Temporal hypergraph Induced static hypergraph Instances of static h-motifs
(≤ 3 hyperedges)
{෤
e1, ෤
e1, ෤
e1}
{෤
e2, ෤
e2, ෤
e2}
{෤
e3, ෤
e3, ෤
e3}
{෤
e4, ෤
e4, ෤
e4}
{෤
e1, ෤
e2, ෤
e2}
{෤
e1, ෤
e3, ෤
e3}
{෤
e2, ෤
e3, ෤
e3}
{෤
e3, ෤
e4, ෤
e4}
{෤
e1, ෤
e2, ෤
e3}
{෤
e1, ෤
e3, ෤
e4}
{෤
e2, ෤
e3, ෤
e4}
Backgrounds Conclusion
Introduction Concepts Algorithms
Observations

22. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
DP: Naï
ve Approach Using Dynamic Programming (cont.)
• DP counts the instances of TH-motifs from instances of static h-motifs using
dynamic programming.
{e3, e4, e5}
{e6, e4, e5}
{e1, e2, e3}
{e1, e2, e3}
{e1, e3, e4}
{e2, e3, e4}
{e2, e3, e5}
Instances of static h-motifs
(≤ 3 hyperedges)
{෤
e1, ෤
e1, ෤
e1}
{෤
e2, ෤
e2, ෤
e2}
{෤
e3, ෤
e3, ෤
e3}
{෤
e4, ෤
e4, ෤
e4}
{෤
e1, ෤
e2, ෤
e2}
{෤
e1, ෤
e3, ෤
e3}
{෤
e2, ෤
e3, ෤
e3}
{෤
e3, ෤
e4, ෤
e4}
{෤
e1, ෤
e2, ෤
e3}
{෤
e1, ෤
e3, ෤
e4}
{෤
e2, ෤
e3, ෤
e4}
Instances of temporal h-motifs
(𝛿 = 5)
1
e1 = (෤
e1={1,4,5}, t1=7)
2
4
5 6 8
7
3
e2 = (෤
e2={2,3,4}, t2=10)
e3 = (෤
e3={4,5,6}, t3=11)
e6 = (෤
e6={4,5,6}, t3=17)
e4 = (෤
e4={6,7,8}, t4=12)
e5 = (෤
e5={6,7,8}, t5=14)
Backgrounds Conclusion
Introduction Concepts Algorithms
Observations

23. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
DP: Naï
ve Approach Using Dynamic Programming (cont.)
• DP enumerates all the instances of static h-motifs in the induced static hypergraph.
• However, only a small fraction of them are induced by any valid instance of TH-motifs.
email contact threads
Backgrounds Conclusion
Introduction Concepts Algorithms
Observations

24. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
THyMe: Preliminary Version of Counting TH-Motifs
• THyMe exhaustively enumerates the instances of TH-motifs.
• THyMe incrementally maintains the projected graph 𝑃 = 𝑉𝑃, 𝐸𝑃 where each
temporal hyperedge is represented as a node.
1
e1 = (෤
e1={1,4,5}, t1=7)
2
4
5 6 8
7
3
e2 = (෤
e2={2,3,4}, t2=10)
e3 = (෤
e3={4,5,6}, t3=11)
e6 = (෤
e6={4,5,6}, t3=17)
e4 = (෤
e4={6,7,8}, t4=12)
e5 = (෤
e5={6,7,8}, t5=14)
e2
e1
e3
e4
e5
e6
Projected graph 𝑃
Backgrounds Conclusion
Introduction Concepts Algorithms
Observations

25. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
THyMe: Preliminary Version of Counting TH-Motifs (cont.)
• THyMe incrementally maintains the projected graph 𝑃 = 𝑉𝑃, 𝐸𝑃 (e.g., 𝛿 = 4).
e1 = (෤
e1={1,4,5}, t1=7)
1
4
5 1
2
4
5
3
e2 = (෤
e2={2,3,4}, t2=10)
1
2
4
5 6
3 2
4
5 6 8
7
3
e3 = (෤
e3={4,5,6}, t3=11) e4 = (෤
e4={6,7,8}, t4=12) e5 = (෤
e5={6,7,8}, t5=14) e6 = (෤
e6={4,5,6}, t5=17)
1
2
4
5 6 8
7
3
1
4
5 6 8
7
e1
e2
e1
e2
e1
e3
e2
e3
e4
e2
e3
e4
e5
e4
e5
e6
e1 is added e2 is added e3 is added e1 is expired
e4 is added
e5 is added e2 , e3 are expired
e6 is added
Backgrounds Conclusion
Introduction Concepts Algorithms
Observations

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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
THyMe: Preliminary Version of Counting TH-Motifs (cont.)
Obs4. Duplicated temporal hyperedges are common.
→ There can exist multiple identical nodes in the projected graph 𝑃.
email-Eu threads-math coauth-DBLP
The number of repetitions follow
a near power-law distribution.
Backgrounds Conclusion
Introduction Concepts Algorithms
Observations

27. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
THyMe: Preliminary Version of Counting TH-Motifs (cont.)
Obs5. Future temporal hyperedges are more likely to repeat recent hyperedges.
→ Identical nodes are more likely to exist within the window in the projected graph 𝑃.
email-Eu threads-math coauth-DBLP
The time intervals of N
consecutive duplicated
temporal hyperedges.
Backgrounds Conclusion
Introduction Concepts Algorithms
Observations

28. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
THyMe+
: Advanced Version of Counting TH-Motifs
• THyMe+ incrementally maintains the projected graph 𝑄 = 𝑉𝑄, 𝐸𝑄, 𝑡𝑄 where
each induced static hyperedge is represented as a node.
{2,3,4}
{1,4,5}
{4,5,6}
{6,7,8}
t={10} t={11,17}
t={7} t={12,14}
1
e1 = (෤
e1={1,4,5}, t1=7)
2
4
5 6 8
7
3
e2 = (෤
e2={2,3,4}, t2=10)
e3 = (෤
e3={4,5,6}, t3=11)
e6 = (෤
e6={4,5,6}, t3=17)
e4 = (෤
e4={6,7,8}, t4=12)
e5 = (෤
e5={6,7,8}, t5=14)
Projected graph 𝑄
Backgrounds Conclusion
Introduction Concepts Algorithms
Observations

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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
THyMe+
: Advanced Version of Counting TH-Motifs (cont.)
• THyMe+ incrementally maintains the projected 𝑄 = 𝑉𝑄, 𝐸𝑄, 𝑡𝑄 (e.g., 𝛿 = 4).
e1 = (෤
e1={1,4,5}, t1=7)
1
4
5 1
2
4
5
3
e2 = (෤
e2={2,3,4}, t2=10)
1
2
4
5 6
3 2
4
5 6 8
7
3
e3 = (෤
e3={4,5,6}, t3=11) e4 = (෤
e4={6,7,8}, t4=12) e5 = (෤
e5={6,7,8}, t5=14) e6 = (෤
e6={4,5,6}, t5=17)
1
2
4
5 6 8
7
3
1
4
5 6 8
7
{1,4,5} is added {2,3,4} is added {1,4,5} is expired
{6,7,8} is added
no changes
{1,4,5}
{1,4,5}
{2,3,4}
{4,5,6} is added
{1,4,5}
{2,3,4} {4,5,6} {2,3,4} {4,5,6}
{6,7,8}
{2,3,4} {4,5,6}
{6,7,8}
{2,3,4} is expired
{4,5,6}
{6,7,8}
Backgrounds Conclusion
Introduction Concepts Algorithms
Observations

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: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
THyMe+
: Advanced Version of Counting TH-Motifs (cont.)
• THyMe+ avoids the exhaustive enumeration of instances of TH-motifs by
counting them based on the timestamps of the nodes 𝑉𝑄 of 𝑄.
{2,3,4} {4,5,6}
{6,7,8}
t={10} t={17}
t={12,14}
{{4,5,6}, {6,7,8}}
An instance detected from 𝑄
Projected graph 𝑄
1 ⋅ 1 ⋅ 2
Number of induced TH-motifs
{{2,3,4}, {4,5,6}, {6,7,8}}
1 ⋅ 2
Backgrounds Conclusion
Introduction Concepts Algorithms
Observations

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: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
Speed and Efficiency of THyMe+
• THyMe+ is faster and more space efficient than DP and THyMe.
email-Eu contact-primary threads-math tags-ubuntu coauth-DBLP
Backgrounds Conclusion
Introduction Concepts Algorithms
Observations

32. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
Roadmap
1. Backgrounds
2. Concepts
3. Observations
4. Algorithms
5. Conclusion

33. Simple Epidemic Models with Segmentation Can Be Better than Complex Ones
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ICDM 2021 THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
Conclusion
• We propose temporal hypergraph motifs (TH-motifs) for describing structural
and temporal patterns of real-world temporal hypergraphs.
Code & datasets: https://github.com/geonlee0325/THyMe
Our contributions are:
✓ New Concept: We define 96 temporal hypergraph motifs.
✓ Fast & Exact Algorithm: We develop THyMe+
for counting instances of TH-motifs.
✓ Empirical Discoveries: TH-motifs reveal interesting structural & temporal patterns.
Backgrounds Conclusion
Introduction Concepts Observations Algorithms

34. THyMe+
: Temporal Hypergraph Motifs and Fast Algorithms for Exact Counting
ICDM 2021
THyMe+
: Temporal Hypergraph Motifs
and Fast Algorithms for Exact Counting
Kijung Shin
Geon Lee