Community Detection: Connection-Based Approach

1. Lecture 6-1 Community Detection • Weili Wu Ding-Zhu Du • University of Texas at Dallas lidong.wu@utdallas.edu

2. Outline • Community Structure • Connection-Based Detection • LP-formulation

3. Community • People in a same community share common interests in • - clothes, music, beliefs, • movies, food, etc. • Influence each other strongly.

4. Community Structure Community with overlap Community without overlap * same color, same community

5. Community Structure In the same community, • two nodes can reach each other in three steps. • A few of tied key persons: C, D • Member A reaches Member B via A-C-D-B

6. Community Structure For different communities, • Two nodes may have distance more than three.

7. Community Structure For two overlapping communities, • Two nodes can reach each other by at most six steps. A B C

8. Question ? How to find a Community? The definition is ambiguous. So, we can only do model-based detection.

9. Model-Based Detection Community Detection Accurate or not? Formulation (Model) Solve formulated problem

10. Model-Based Physics The Real World Accurate or not? Newton Model Solve physics problem

11. No Satisfied Community Model !

12. Question ? How to find a Community? • A simplest way is • Connection-Based Detection

13. Outline • Community Structure • Connection-Based Detection • LP-formulation

14. Based Fact • More connections inside each community. • Less connections between different communities. • There are several ways to understand this property.

15. Connection-Based Condition 1 (Radicchi et al. 2004) • Each community has more connections inside • than connections to outside.

16. Connection-Based Condition 1 Inside red > outside blue + outside green • Each community has more connections inside • than connections to outside.

17. Connection-Based Condition 2 (Hu et al. 2008) (2) Each community has more connections inside than connections to any other community.

18. Connection-Based Condition 2 Inside red > outside blue Inside red > outside green (2) Each community has more connections inside than connections to any other community.

19. Connection-Based Condition 3 (3) Each node in a community has more connections Inside than connections to outside.

20. Connection-Based Condition 3 At each red node Inside red > outside blue + outside green (3) Each node in a community has more connections Inside than connections to outside.

21. Connection-Based Condition 4 (4) Each node in a community has more connections Inside than connections to any other community.

22. Connection-Based Condition 4 At each red node Inside red > outside blue Inside red > outside green (4) Each node in a community has more connections Inside than connections to any other community.

23. Relationship of Conditions (3) (4) (1) (2) Weak sense Most weak sense

24. Max Community Partition Theorem (Lu et al. 2013)

25. Qualified Cut Approx. for Max Community Partition

26. Outline • Community Structure • Connection-Based Detection • LP-formulation

27. Indicator For example

29. Linear Constraints

30. Linear Constraints

33. References 1 2

34. THANK YOU!