Modular Synchronization in Complex Network with a Gauge Kuramoto Model C. Choi 1 , E. Oh 1,2 , B. Kahng 1 , and D. Kim 1 1 Department of Physics and Astronomy and Center for Theoretical Physics, Seoul National University, Seoul 151-747, Korea bkahng@snu.ac.kr 2 Bioanalysis and Biotransformation Research Center, Korea Institute of Science and Technology, Seoul 136-791, Korea Abstract. We modify the Kuramoto equation(KE) by introducing a gauge term which is a function of link betweenness centrality(BC). The gauge term induces the phase difference from 0 to π between two nodes that belong to different modules. Therefore, a synchronization occurs in each module individually even though the whole network is not synchro- nized globally. By measuring the phase similarity of all pairs of connected nodes, we can detect the modular structure of complex networks. This algorithm requires relatively little computational time O(NL) for net- work with N nodes and L links. Keywords: synchronization, module identification, modular complex network, Kuramoto model. 1 Introduction Every system that has constituents and relationships between themselves can be represented by networks conceptually. Constituents of the system are nodes and their interactions are links which connect a node and another node in networks. Many physical and social systems have been studied via networks. Among many kinds of dynamics on complex networks, synchronization is one of the most popular subjects. Synchronization is a process of adjusting some properties assigned to each node via interactions between the elements of com- plex network. This is so useful that it has been investigated in several kinds of fields - physics, biology, sociology, etc. Module is a set of densely-connected nodes in complex network. Modular complex network has several modules and these modules are connected each other with relatively sparse links. Synchronization in modular complex network shows some new features com- pared to synchronization in complex network having no module structure. Links are dense within modules and sparse between modules. So in synchronization process nodes in same module are synchronized first by intra-module links and the whole network becomes synchronized later by inter-module links or not. We reported on this phenomenon in our previous paper [1]. In this lecture note we follow the outline of the paper briefly and all figures in here are also taken from it. J. Zhou (Ed.): Complex 2009, Part I, LNICST 4, pp. 429–434, 2009. c  ICST Institute for Computer Sciences, Social Informatics and Telecommunications Engineering 2009