1. Modularity Maximization
Lecture 6-2
Modularity Maximization
Ding-Zhu Du
University of Texas at Dallas
First, I want to thank you for you presence. ********In this presentation I will try to introduce The social network which is a theoretical structure to study relationships between individuals, groups, organizations, or even entire societies. 
It is related to a wide range of disciplines. These disciplines include, but are not limited to information science, biology, economics, geography, communication studies, and so on..
The study of social networks begins with the late eighteenth century, two sociologists (Émile [ei'mi:l] Durkheim and Ferdinand ['fɝdənænd] Fer迪南de Tönnies) foreshadowed the idea of social networks in their theories and research of social groups.
Nowadays, we study social networks using network analysis to identify social communities, pick influential person, and design good software.

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
Brief overview of social networks
How to build applications on top of the social network
–  Think about a social network being MS Windows, We can build applications on it.

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

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

14. Outline Modularity Function Greedy Spectral Method Hybrid Method
Brief overview of social networks
How to build applications on top of the social network
–  Think about a social network being MS Windows, We can build applications on it.

15. Increment

16. Greedy Algorithm

17. Outline Modularity Function Greedy Spectral Method and MP
Hybrid Method
Brief overview of social networks
How to build applications on top of the social network
–  Think about a social network being MS Windows, We can build applications on it.

18. Qualified Cut
Community Partition

19. Quadratic Form

20. Spectral Method

21. Linear Program

22. Semi-definite Program
Vector Program
Semi-definite Program

23. Outline Modularity Function Greedy Spectral Method and MP
Hybrid Method
Brief overview of social networks
How to build applications on top of the social network
–  Think about a social network being MS Windows, We can build applications on it.

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
Brief overview of social networks
How to build applications on top of the social network
–  Think about a social network being MS Windows, We can build applications on it.

32. Remark 1 How to evaluate the method for finding a community?
This is an important research problem.
There exist many methods.
Show an example.

33. Clustering

34. Community Detection

35. Remark 2 How to do hierarchy community detection?
This is an important research problem.
There exist many methods.
Show an example.

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 graphs by Santo Fortunato, Physics Reports 486, 75 (2010)

37. THANK YOU!