Synchronization in random networks with given expected degree sequences Paolo Checco a, * , Mario Biey a , Ljupco Kocarev b a Dipartimento di Elettronica, Politecnico di Torino, Torino, Italy b Institute for Nonlinear Science, University of California, San Diego, La Jolla, CA, USA Accepted 22 May 2006 Abstract Synchronization in random networks with given expected degree sequences is studied. We also investigate in details the synchronization in networks whose topology is described by classical random graphs, power-law random graphs and hybrid graphs when N !1. In particular, we show that random graphs almost surely synchronize. We also show that adding small number of global edges to a local graph makes the corresponding hybrid graph to synchronize. Ó 2006 Elsevier Ltd. All rights reserved. 1. Introduction The study of complex systems pervades all of science, from cell biology to ecology, from computer science to mete- orology. A paradigm of a complex system is a network [1], where complexity may come from different sources: topo- logical structure, network evolution, connection and node diversity, and/or dynamical evolution. Examples of networks include food webs [2,3], electrical power grids, cellular and metabolic networks, the World-Wide Web [4], the Internet backbone [5], neural networks, and co-authorship and citation networks of scientists. These networks consist of nodes which are interconnected by a mesh of links. The macroscopic behavior of a network is determined by both the dynam- ical rules governing the nodes and the flow occurring along the links. Real networks of interacting dynamical systems – be they neurons, power stations or lasers – are complex. Many real-world networks are small-world [6] and/or scale-free networks [7]. The presence of a power-law connectivity dis- tribution, for example, makes the Internet a scale-free network. The research on complex networks has been focused so far on the their topological structure [8]. However, most networks offer support for various dynamical processes. In this paper we propose to study one aspect of dynamical processes in non-trivial complex network topologies, namely their synchronization behavior. The general question of network synchronizability, for many aspects, is still an open and outstanding research prob- lem [9,10]. There are, in general, two classes of results which give criteria under which a network of oscillators synchro- nizes. The first class of results uses Lyapunov’s direct method by constructing a Lyapunov function which decreases along trajectories and gives analytical criteria for local or global synchronization. For example, in [11], the authors gave 0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.05.063 * Corresponding author. E-mail addresses: paolo.checco@polito.it (P. Checco), mario.biey@polito.it (M. Biey), lkocarev@ucsd.edu (L. Kocarev). Chaos, Solitons and Fractals 35 (2008) 562–577 www.elsevier.com/locate/chaos