Scale-free networks in complex systems

Abstract

In the past few years, several studies have explored the topology of interactions in different complex systems. Areas of investigation span from biology to engineering, physics and the social sciences. Although having different microscopic dynamics, the results demonstrate that most systems under consideration tend to self-organize into structures that share common features. In particular, the networks of interaction are characterized by a power law distribution, $P(k)\sim k^{-α}$, in the number of connections per node, $k$, over several orders of magnitude. Networks that fulfill this propriety of scale-invariance are referred to as ``scale-free''. In the present work we explore the implication of scale-free topologies in the antiferromagnetic (AF) Ising model and in a stochastic model of opinion formation. In the first case we show that the implicit disorder and frustration lead to a spin-glass phase transition not observed for the AF Ising model on standard lattices. We further illustrate that the opinion formation model produces a coherent, turbulent-like dynamics for a certain range of parameters. The influence, of random or targeted exclusion of nodes is studied.