A colored network model, corresponding to a colored graph in mathematics, is used for describing the complexity of some inter-connected physical systems. A colored network is consisted of colored nodes and edges. Colored nodes may have identical or nonidentical local dynamics. Colored edges between any pair of nodes denote not only the outer coupling topology but also the inner interactions. In this paper, first, synchronization of edge-colored networks is studied from adaptive control and pinning control approaches. Then, synchronization of general colored networks is considered. To achieve synchronization of a colored network to an arbitrarily given orbit, open-loop control, pinning control and adaptive coupling strength methods are proposed and tested, with some synchronization criteria derived. Finally, numerical examples are given to illustrate theoretical results.
Received 24 July 2012Accepted 20 November 2012Published online 10 December 2012
Lead Paragraph: As is well known, the dynamic complexity of a large-scale network is determined and can be characterized by the interactions among its nodes. Therefore, many network models are presented to describe the complexity, such as weighted networks and directed networks. Most of the existing network models emphasize on outer couplings rather than inner couplings, i.e., assuming that the inner coupling matrix is a constant or even identity matrix. Consider a social relationship network consisting of N individuals, e.g., schoolmates, relatives, and collaborative relationship. For individuals i and j, they may be either schoolmates or relatives but have no collaborative relationship, while for individuals i and k (k≠j), they may only have collaborative relationship but no other relations. That is, the two pairs of nodes have the same outer coupling but different inner couplings, which cannot be well described by the aforementioned network models. In this paper, a colored network corresponding to colored graph in mathematics is introduced to better describe this kind of systems, in which different color edges denote different inner couplings, and different color nodes denote different nodes dynamics. Based on Lyapunov stability theory, sufficient conditions for adaptive synchronization of both edge-colored networks and colored networks are derived through designing proper adaptive controllers and open-loop controllers, respectively. The results will shed some new insights onto both modeling and synchronization of complex networks with intrinsic individual dynamics and interdependent relationships.
This research is jointly supported by the NSFC under Grant Nos. 10805033 and 11072136, the Tianyuan Special Funds of the NSFC under Grant No. 11226242, the Shanghai Univ. Leading Academic Discipline Project (A.13-0101-12-004), and the HK GRF Grant No. CityU 1114/11E. The authors would also like to thank the anonymous referees for their helpful comments and suggestions.
Article outline: I. INTRODUCTION II. COLORED NETWORK MODEL III. SYNCHRONIZATION OF EDGE-COLORED NETWORKS A. Adaptive feedback control B. Pinning control with adaptive coupling IV. SYNCHRONIZATION OF COLORED NETWORKS V. NUMERICAL SIMULATIONS VI. CONCLUSIONS
18.J. Zhou, J. A. Lu, and J. H. Lü, “Adaptive pinning synchronization of a general complex dynamical network,” Proceedings of the 2007 IEEE International Symposium on Circuits and Systems (ISCAS'07), May 27–30, 2007, New Orleans, USA, pp. 2494–2497.
19.T. P. Chen, X. Liu, and W. L. Lu, IEEE Trans. Circ. Syst. I54, 1317 (2007).