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Journal of Computational Dynamics (JCD)
 

Modularity revisited: A novel dynamics-based concept for decomposing complex networks

Pages: 191 - 212, Volume 1, Issue 1, June 2014      doi:10.3934/jcd.2014.1.191

 
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Marco Sarich - Freie Universität Berlin, Department of Mathematics and Computer Science, Arnimallee 6, 14195 Berlin, Germany (email)
Natasa Djurdjevac Conrad - Freie Universität Berlin, Department of Mathematics and Computer Science, Arnimallee 6, 14195 Berlin, Germany (email)
Sharon Bruckner - Freie Universität Berlin, Department of Mathematics and Computer Science, Arnimallee 6, 14195 Berlin, Germany (email)
Tim O. F. Conrad - Freie Universität Berlin, Department of Mathematics and Computer Science, Arnimallee 6, 14195 Berlin, Germany (email)
Christof Schütte - Freie Universität Berlin, Department of Mathematics and Computer Science, Arnimallee 6, 14195 Berlin, Germany (email)

Abstract: Finding modules (or clusters) in large, complex networks is a challenging task, in particular if one is not interested in a full decomposition of the whole network into modules. We consider modular networks that also contain nodes that do not belong to one of modules but to several or to none at all. A new method for analyzing such networks is presented. It is based on spectral analysis of random walks on modular networks. In contrast to other spectral clustering approaches, we use different transition rules of the random walk. This leads to much more prominent gaps in the spectrum of the adapted random walk and allows for easy identification of the network's modular structure, and also identifying the nodes belonging to these modules. We also give a characterization of that set of nodes that do not belong to any module, which we call transition region. Finally, by analyzing the transition region, we describe an algorithm that identifies so called hub-nodes inside the transition region that are important connections between modules or between a module and the rest of the network. The resulting algorithms scale linearly with network size (if the network connectivity is sparse) and thus can also be applied to very large networks.

Keywords:  Random walk, metastability, modularity, modules, hubs, complex networks.
Mathematics Subject Classification:  Primary: 60J28, 05C81; Secondary: 05C85.

Received: December 2011;      Revised: July 2012;      Available Online: April 2014.

 References