Imbalance of positive and negative links induces regularity Neeraj Kumar Kamal a , Sudeshna Sinha a,b,⇑ a The Institute of Mathematical Sciences, CIT Campus, Chennai 600 113, India b Indian Institute of Science Education and Research (IISER) Mohali, Transit Campus: MGSIPAP Complex, Sector 26, Chandigarh 160 019, India article info Article history: Received 20 July 2010 Accepted 4 December 2010 Available online 8 January 2011 abstract We investigate the effect of the interplay of positive and negative links, on the dynamical regularity of a random weighted network, with neuronal dynamics at the nodes. We inves- tigate how the mean J and the variance of the weights of links, influence the spatiotempo- ral regularity of this dynamical network. We find that when the connections are predominantly positive (i.e. the links are mostly excitatory, with J > 0) the spatiotemporal fixed point is stable. A similar trend is observed when the connections are predominantly negative (i.e. the links are mostly inhibitory, with J < 0). However, when the positive and negative feedback is quite balanced (namely, when the mean of the connection weights is close to zero) one observes spatiotemporal chaos. That is, the balance of excitatory and inhibitory connections preserves the chaotic nature of the uncoupled case. To be brought to an inactive state one needs one type of connection (either excitatory or inhibitory) to dominate. Further we observe that larger network size leads to greater spatiotemporal reg- ularity. We rationalize our observations through mean field analysis of the network dynamics. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction An important prototype of a complex system is a large interactive network of nonlinear dynamical elements. The basic components of such complex networks are: (i) local nodal dynamics, modelled by nonlinear maps or differen- tial equations capable of yielding a rich variety of temporal patterns, and (ii) transmission of information among these local dynamical units by coupling connections of varying strengths and underlying topologies, described by a con- nectivity matrix. Examples of complex dynamical networks include food webs, biological neural networks, electrical power grids, social and economic relations, coauthorship and citation networks of scientists, cellular and metabolic networks, etc. [1–3]. This ubiquity of various real and artificial net- works has stimulated the recent surge of research in this field [1–10]. Now many social, technological, biological and econom- ical systems are best described by weighted networks [3,4], whose properties and dynamics depend not only on the underlying topological structure, but also on the connec- tion weights among their nodes. However, most existing research work on complex dynamical network models have concentrated on network structures, with connection weights among their nodes being either 1 or 0 [2]. Further, the few existing studies of weighted random dynamical networks, consider the weights of the connec- tivity matrix to be drawn from a zero mean gaussian or uniform distributions [2]. So the interesting scenario of distributions skewed towards positive or negative weights has not been adequately addressed. Nor has the dynamical significance of the imbalance of positive–negative links been sufficiently studied. The focus of our work here is this unexplored problem. In particular, here we consider random weighted net- works of chaotic maps, with the nodal dynamics modelling neuronal activity. Namely, we have a network where the 0960-0779/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2010.12.002 ⇑ Corresponding author at: The Institute of Mathematical Sciences, CIT Campus, Chennai 600 113, India. E-mail addresses: neeraj@imsc.res.in (N.K. Kamal), sudeshna@imsc. res.in (S. Sinha). Chaos, Solitons & Fractals 44 (2011) 71–78 Contents lists available at ScienceDirect Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos