lidong.wu@utdallas

1. Lecture 6-2 Modularity Maximization • Ding-Zhu Du • University of Texas at Dallas lidong.wu@utdallas.edu

2. Model-Based Detections • Connection-based detection • Modularity maximization • Influence-based detection • Overlapping community detection • Hierarchy community detection

3. Model-Based Detection Modularity Maximization Is the most popular one

4. Outline • Modularity Function • Greedy • Spectral Method and MP • Hybrid Method

5. Modularity Function (Newman 2006)

7. Modularity Function (Newman 2006)

8. Newman 2006 • M.E. J. Newman: Modularity and community structure in networks, Proceedings of the National Academy of Sciences, vol 103 no 23 (2006) pp. 8577-8582.

9. Modularity Function

10. Modularity Function (Newman 2006)

11. Modularity Function (digraph)

12. Why call Modularity? • Module = community in some complex networks • The function describes the quality of modules.

13. Modularity Max is NP-hard • U. Brandes, D. Delling, M. Gaertler, R. Gorke, M. Hoefer, Z. Nikoloski, and D. Wagner: On modularity clustering, IEEE Transactions on Knowledge and Data Engineering (TKDE), vol 20, no 2 (2008) pp 172-188

14. Outline • Modularity Function • Greedy • Spectral Method • Hybrid Method

15. Increment

16. Greedy Algorithm

17. Outline • Modularity Function • Greedy • Spectral Method and MP • Hybrid Method

18. Qualified Cut Community Partition

19. Quadratic Form

20. Spectral Method

21. Linear Program

22. Vector Program Semi-definite Program

23. Outline • Modularity Function • Greedy • Spectral Method and MP • Hybrid Method

24. Resolution limit • Misidentification: some derived communities do not satisfy the weak community definition or even the most weak community definition • In other words, obtained communities may have sparser connection within them than between them.

25. Hybrid Detection: a Possible Research Direction

26. Max Q s.t. condition (1) • This may give an improvement. • Is it possible to do? • (1) can be written as linear constraints • Q can be written as a quadratic function • Thus, Max Q s.t. (1) can be formulated as a quadratic programming, which can be transformed into a semi-definite programming

27. Linear Constraints

28. Linear Constraints

29. Modularity Density Modularity Density function (Li et al. 2008)

30. Opt D s.t. condition (1) • This may give an improvement. • Is it possible to do? • (1) can be written as linear constraints • Q can be written as a fractional function • Thus, Max D s.t. (1) can be formulated as a Geometric Programming.

31. Outline • Community Structure • Connection-Based Detection • Influence-Based Detection • Remarks

32. Remark 1 How to evaluate the method for finding a community?

33. Clustering

34. Community Detection

35. Remark 2 How to do hierarchy community detection?

36. Survey • Introductory review: Communities in networks by M. A. Porter, J.-P. Onnela, and P. J. Mucha, Notices of the American Mathematical Society 56, 1082 (2009) • Comprehensive review: Community detection in graphsby Santo Fortunato, Physics Reports 486, 75 (2010)

37. THANK YOU!