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Learn about the popular technique of modularity maximization for detecting community structure in networks. Explore the modularity function, greedy algorithms, spectral methods, and hybrid techniques. Discover the resolution limit and the potential of hybrid detection. Evaluate the method for finding communities and learn about hierarchy community detection. Recommended for those interested in network analysis and community detection.
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Lecture 6-2 Modularity Maximization • Ding-Zhu Du • University of Texas at Dallas 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
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.
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
Outline • Modularity Function • Greedy • Spectral Method and MP • Hybrid Method
Qualified Cut Community Partition
Vector Program Semi-definite Program
Outline • Modularity Function • Greedy • Spectral Method and MP • Hybrid Method
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.
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
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
Remark 1 How to evaluate the method for finding a community?
Remark 2 How to do hierarchy community detection?
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)