Author's personal copy Available online at www.sciencedirect.com Journal of the Franklin Institute 351 (2014) 358–372 Synchronization in complex networks under structural evolution A. Anzo, J.G. Barajas-Ramírez n IPICyT, División de Matemáticas Aplicadas, Camino a la Presa San José 2055, Col. Lomas 4a Secc. C.P.78216, San Luis Potosí, SLP, Mexico Received 1 June 2013; received in revised form 17 August 2013; accepted 1 September 2013 Available online 17 September 2013 Abstract We investigate the effects of structural evolution on the stability of synchronized behavior in complex networks. By structural evolution we mean processes that change the topology of the network. In particular, we consider structural evolution as two simultaneous processes: on one hand, the topology changes according to an arbitrary switching law among a set of admissible patterns of connection; on the other hand, the strength of connection evolves according to an adaptive law. Our results show that by constraining the admissible patterns of connection, and using an adaptive law based on the difference between the nodes, we can guarantee the stability of the synchronized solution of the network despite structural changes. Additionally, we extend our results by considering alternative structural evolution processes, namely, a node-based adaptive strategy and a resetting switching law. We illustrate our results with numerical simulation. & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. 1. Introduction The study of complex networks has attracted considerable attention from different areas of science and technology. This is mainly due to its potential applicability to the analysis, modeling, and control of real-world networks like the Internet, power grids, transportation networks, among many others (see [1–4] and reference therein). One of the most significant aspects of network science is the structural analysis of networks, in particular the interplay between its structural features and dynamical properties. In this sense, one of the main topics of research is the effect of network topology on the synchronization of dynamical networks [5–7]. A basic hypothesis in the www.elsevier.com/locate/jfranklin 0016-0032/$32.00 & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2013.09.001 n Corresponding author. Tel.: þ52 444 834 200. E-mail addresses: andres.anzo@ipicyt.edu.mx (A. Anzo), jgbarajas@ipicyt.edu.mx, gbrslp@hotmail.com (J.G. Barajas-Ramírez).