Emergence of Lévy flights in Distributed Consensus Systems

In this paper, we consider a multi-agent model which generates a collective super-diffusion behavior. Although such complex behaviors are ubiquitous in many natural and human-made systems, discovering mechanisms of their emergence is mostly an open research area. Our model is motivated to solve consensus problem under limitations on information exchange including link failures and additive noise. We use orthogonal decomposition approach to analyze the system and establish several equivalent necessary and sufficient conditions for Mean Square (MS) stability of part of this system. We show that the emergence of the super-diffusion behavior is introduced by the loss of MS stability and prove it to be Lévy flights for a special system. This work is the first, to the best of our knowledge, to establish the intimate relationship between propagation of uncertainties in networks, the MS stability robustness and the emergence of Lévy flights, which may have far reaching consequence on the understanding and engineering of complex systems.