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A Symmetric Kernel Matrix for Community Detection in Directed Complex Networks

Firouzi, Hamid Reza | 2015

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  1. Type of Document: M.Sc. Thesis
  2. Language: English
  3. Document No: 47706 (52)
  4. University: Sharif University of Technology, International Campus, Kish Island
  5. Department: Science and Engineering
  6. Advisor(s): Roosta Azad, Reza; Zarei, Mina
  7. Abstract:
  8. Complex directed networks are widely used in representing, modeling and summarizing various homo and heterogeneous real networks such as protein-protein interactions, social and twitting networks, brain connectomes, chipsets mapping and marketing. While working on undirected networks in the hope for detecting communities or any other static and dynamic structures, has been on agenda of many physician and computer scientists, dealing with directed ones seems more challengeable for quick advances. Availability of theoretical basis especially symmetric adjacency matrix recommends modeling with undirected graphs but we believe that directed graphs are more general and practical to model real networks. A+A^T and A.A^T have been employed in a recent paper to symmetrize assymetric adjacency matrix A^ of a directed graph. Another well-defined asymmetric representing matrix for directed graphs is ‘all- pairs shortest paths’. We propose a kernel matrix by symmetrization of S providing a similarity measure based on shortest cycles each pair of nodes co-occur on. Thanks to the kernel matrix we have proposed an exact formula for our directed modularity which we show can be optimized by k-way spectral clustering through a trace optimization problem. The resolution limit which observed as a drawback for general modularity is absent here. The freedom of using wide variety of clustering methods at the optimization step, opens our hands to adjust the features of derived communities (i.e. overlapping or size inequality) as well as precision versus speed. The proposed modularity has been evaluated through one spectral optimization process on a number of synthesized directed complex networks as well as on a real biological network. The results are remarkably satisfying
  9. Keywords:
  10. Community Detection ; Modularity ; Spectral Clustering ; Kernel Matrix ; Directed Complex Networks ; Shortest Cycles

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