Emergent behavior in large scale networks

Abstract

We apply mean field asymptotic analysis to explain the emergence of global behavior in large scale networks. The underlying motivating application is epidemics like computer virus spreading, for example, in wide campus local networks. We consider multiple classes of viruses, each type bearing their own statistical characterization - exogenous contamination, contagious propagation, and healing. The network state (distribution of nodes infected by each class in the network) is a jump Markov process, not necessarily reversible, making it a challenge to obtain its invariant distribution. By suitable renormalization, in the limit of a large network (number of nodes,) the macroscopic behavior of the network is described by the solution of a set of deterministic nonlinear differential equations (Riccati type.) We show that, under the heavy traffic assumption, the relevant underlying dynamics induces a coherent nontrivial metastable behavior in a macroscopic space-time scale: a slight imbalance on the effective spreading rate of one class over the others determines a significantly greater steady state predominance of this class over the others, regardless of the initial distribution.