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. lidong.wu@utdallas.edu

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

Model-Based Detection Modularity Maximization Is the most popular one

Outline Modularity Function Greedy Spectral Method and MP Hybrid Method 1. Brief overview of social networks 2. How to build applications on top of the social network –  Think about a social network being MS Windows, We can build applications on it.

Modularity Function (Newman 2006)

Modularity Function (Newman 2006)

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.

Modularity Function

Modularity Function (Newman 2006)

Modularity Function (digraph)

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

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

Outline Modularity Function Greedy Spectral Method Hybrid Method 1. Brief overview of social networks 2. How to build applications on top of the social network –  Think about a social network being MS Windows, We can build applications on it.

Increment

Greedy Algorithm

Outline Modularity Function Greedy Spectral Method and MP Hybrid Method 1. Brief overview of social networks 2. How to build applications on top of the social network –  Think about a social network being MS Windows, We can build applications on it.

Qualified Cut Community Partition

Quadratic Form

Spectral Method

Linear Program

Semi-definite Program Vector Program Semi-definite Program

Outline Modularity Function Greedy Spectral Method and MP Hybrid Method 1. Brief overview of social networks 2. How to build applications on top of the social network –  Think about a social network being MS Windows, We can build applications on it.

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.

Hybrid Detection: a Possible Research Direction

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

Linear Constraints

Linear Constraints

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

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.

Outline Community Structure Connection-Based Detection Influence-Based Detection Remarks 1. Brief overview of social networks 2. How to build applications on top of the social network –  Think about a social network being MS Windows, We can build applications on it.

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

Clustering

Community Detection

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

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)

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