A Nonlinear Consensus Algorithm Derived from Statistical Physics

TL;DR

This paper analyzes the ribosome flow model on a ring (RFMR), a mean field approximation of ASEP, showing it converges to equilibrium and functions as a nonlinear consensus algorithm with applications in formation control.

Contribution

It introduces a novel analysis of RFMR using monotone dynamical systems theory, establishing convergence, equilibrium properties, and its role as a nonlinear consensus algorithm.

Findings

RFMR admits a continuum of equilibrium points.

Trajectories of RFMR converge to an equilibrium.

RFMR with homogeneous rates achieves consensus.

Abstract

The asymmetric simple exclusion process (ASEP) is an important model from statistical physics describing particles that hop randomly from one site to the next along an ordered lattice of sites, but only if the next site is empty. ASEP has been used to model and analyze numerous multiagent systems with local interactions ranging from ribosome flow along the mRNA to pedestrian traffic. In ASEP with periodic boundary conditions a particle that hops from the last site returns to the first one. The mean field approximation of this model is referred to as the ribosome flow model on a ring (RFMR). We analyze the RFMR using the theory of monotone dynamical systems. We show that it admits a continuum of equilibrium points and that every trajectory converges to an equilibrium point. Furthermore, we show that it entrains to periodic transition rates between the sites. When all the transition rates are equal all the state variables converge to the same value. Thus, the RFMR with homogeneous transition rates is a nonlinear consensus algorithm. We describe an application of this to a simple formation control problem.