September 2008 EPL, 83 (2008) 68003 www.epljournal.org doi: 10.1209/0295-5075/83/68003 Modular synchronization in complex networks with a gauge Kuramoto model E. Oh 1,2 , C. Choi 2 , B. Kahng 2(a) and D. Kim 2 1 Bioanalysis and Biotransformation Research Center, Korea Institute of Science and Technology Seoul 136-791, Korea 2 Department of Physics and Astronomy and Center for Theoretical Physics, Seoul National University Seoul 151-747, Korea received 12 June 2008; accepted in final form 6 August 2008 published online 12 September 2008 PACS 89.75.-k – Complex systems PACS 89.65.-s – Social and economic systems Abstract – We modify the Kuramoto model for synchronization on complex networks by introducing a gauge term that depends on the edge betweenness centrality (BC). The gauge term introduces additional phase difference between two vertices from 0 to π as the BC on the edge between them increases from the minimum to the maximum in the network. When the network has a modular structure, the model generates the phase synchronization within each module, however, not over the entire system. Based on this feature, we can distinguish modules in complex networks, with relatively little computational time of O(NL), where N and L are the number of vertices and edges in the system, respectively. We also examine the synchronization of the modified Kuramoto model and compare it with that of the original Kuramoto model in several complex networks. Copyright c  EPLA, 2008 Complex networks have drawn considerable attention from diverse disciplines such as sociology, information science, physics, biology and so on [1]. Many complex networks in real world contain modules within them, which form in a self-organized way to achieve the effi- ciency functionally or regionally. Such modular systems can exhibit collective synchronized patterns within each module, not forming the global synchronization [2] as can be found in the cortex of neural network [3] or differ- ent synchronization transition behaviors depending on the patterns of inter-modular connections [4]. In this letter, we study the modular synchronization pattern generated from a modified Kuramoto equation (KE), which we call the gauge KE, dφ i (t) dt =Ω i - J N  j=1 a ij sin(φ i (t) - φ j (t) - ηg(b ij )). (1) Here, φ i is the phase of vertex i,Ω i is the natural frequency of vertex i selected from the Gaussian distri- bution e -Ω 2 /2 / √ 2π, J is the overall coupling constant and a ij is the (i, j )-th component of the adjacency matrix, which is one when the vertices i and j are connected, and (a) E-mail: bkahng@snu.ac.kr zero otherwise. η is a control parameter. The extra phase term g(b ij ), we call the gauge term below, is defined as g(b ij )= b ij - b min b max - b min π, (2) where b min and b max are the minimum and the maximum edge betweenness centrality (BC) [5] or load [6], respec- tively, in the system. Here, the edge BC or load is the amount of effective traffic passing through a given edge when every pair of vertices sends and receives a unit packet that travels along the shortest path between them. Then the gauge term g(b ij ) is in the range from 0 to π depending on the BC of edge. When η = 0, the gauge KE recovers the standard KE [7] which becomes fully synchronized when J is sufficiently large. The KE with the extra phase of the form sin(φ i - φ j - c)(c = constant) was studied first in [8]. The effect of the extra phase is to destroy the synchronization. Intuitively, one expect that the BCs on intra-module links are smaller than those on inter-module. Thus, each module can be synchronized, while the entire system is not. Moreover, the gauge term induces an effec- tive coupling that can be negative at the edges connecting different modules. Due to this negative coupling, the aver- age phase of each module may have velocity different from 68003-p1