Synchronization in dynamical networks: Evolution along commutative graphs S. Boccaletti, 1 D.-U. Hwang, 1 M. Chavez, 2 A. Amann, 3 J. Kurths, 4 and L. M. Pecora 5 1 CNR- Istituto dei Sistemi Complessi, Largo E. Fermi, 6-50125 Florence, Italy 2 Laboratoire de Neurosciences Cognitives et Imagerie Cérébrale (LENA) CNRS UPR-640, Hôpital de la Salpêtrière, 47 Boulevard de l’Hôpital, 75651 Paris CEDEX 13, France 3 Tyndall National Institute, Lee Maltings, Cork, Ireland 4 Institut für Physik, Universität Potsdam, Am Neuen Palais, PF 601553, D-14415 Potsdam, Germany 5 Code 6362, Naval Research Laboratory, Washington, D.C. 20375, USA Received 5 December 2006; revised manuscript received 18 April 2006; published 5 July 2006 Starting from an initial wiring of connections, we show that the synchronizability of a network can be significantly improved by evolving the graph along a time dependent connectivity matrix. We consider the case of connectivity matrices that commute at all times, and compare several approaches to engineer the corre- sponding commutative graphs. In particular, we show that synchronization in a dynamical network can be achieved even in the case in which each individual commutative graphs does not give rise to synchronized behavior. DOI: 10.1103/PhysRevE.74.016102 PACS numbers: 89.75.Hc, 05.45.Xt, 87.18.Sn Complex networks, i.e., collections of dynamical nodes connected by a wiring of edges exhibiting complex topologi- cal properties, are the prominent candidates to describe the occurrence of collective dynamics in many areas of science 1. Of particular interest is the existence of synchronized states in such networks. These states indeed are at the basis for the emergence of coherent global behaviors in both nor- mal and abnormal brain functions 2, and play a crucial role in determining the food web dynamics in ecological systems 3. So far, synchronized behaviors 4 have been mostly stud- ied in the limit of static networks e.g., networks whose wir- ing of connections is fixed with the emphasis focusing on how the complexity in the overall topology influences the propensity of the coupled units to synchronize 5,6. In par- ticular, it has been established that proper weighting proce- dures in static complex networks are able to greatly enhance the appearance of synchronized behavior 7. The very opposite limit of blinking networks 8 has also been considered, where the wiring of connections is rapidly i.e., with a characteristic time scale much shorter than that of the networked system’s dynamics switching among dif- ferent configurations. Under these conditions, it has been found that synchronous motion can be established for suffi- ciently rapid switching times even in the case in which each visited wiring configuration would prevent synchronization under static conditions. None of these two limits, however, seems an adequate description of many relevant phenomena occurring in natural systems. For instance, properly modeling processes such as mutation in biological systems 9, synaptic plasticity in neu- ronal networks 10, or adaptation in social or financial mar- ket dynamics 11 would require accounting for time varying networks whose evolution takes place over characteristic time scales that are commensurate with those of the nodes’ dynamics. In this paper, we assess the conditions for the appearance of synchronized states in dynamical networks, without mak- ing any explicit hypothesis on the time scale responsible for the variation of the coupling wiring. We consider a network of N coupled identical systems, whose evolution is described by x ˙ i = fx i  -   j=1 N G ij thx j , i =1,..., N . 1 Here x  R m is the m-dimensional vector describing the state of the ith node, fx : R m → R m governs the local dynam- ics of the nodes, hx : R m → R m is a vectorial output func- tion,  is the coupling strength, dots stand for temporal de- rivatives, and G ij t  R are the time varying elements of a zero row sum  j G ij t =0 " i and " t N  N symmetric connectivity matrix Gt with strictly positive diagonal terms G ii t  0 " i and " t and negative off diagonal terms G ij t  0 " i  j and " t, specifying the evolution in strength and topology of the underlying connection wiring. Being symmetric Gt admits at all times a set  i tv i t of real eigenvalues of associated orthonormal eigenvectors, such that Gtv i t =  i tv i t and v j T · v i =  ij . It is worth noticing that the zero row sum condition im- posed on Gt can be actually encompassed by a diffusion process when, for instance, the topology of the connectivity matrix reduces to that of a unidimensional chain. In general, a possible way of physically realizing this property is a dif- fusion process of the output function onto the first neighbor- hood of a given node defined as the set of vertices that are adjacent to the node in the complex topology imposed by the connectivity wiring. Such a condition and Geršgorin’s circle theorem 12 ensure that i the spectrum is entirely semipositive, i.e.,  i t  0 " i and " t; ii  1 t 0 with associated eigenvector v 1 t = 1  N 1,1,...,1 T that entirely defines a synchronization manifold x i t = x s t , " i, whose stability will be the object of our study; and iii all the other eigenvalues  i ti =2,..., N,  i t  0 for con- nected graphs have associated eigenvectors v i t spanning the transverse manifold of x s tin the m  N-dimensional phase space of Eq. 1. Let x i t = x i t - x s t = x i,1 t ,..., x i,m t be the de- PHYSICAL REVIEW E 74, 016102 2006 1539-3755/2006/741/0161025 ©2006 The American Physical Society 016102-1