Instability in a Network Coevolving with a Particle System

TL;DR

This paper investigates how a coupled system of particles and a network can become unstable, leading to hub formation or fat-tailed degree distributions, with a phase transition at a critical particle density.

Contribution

It introduces a novel coupled dynamics model showing how particle flux influences network rewiring, leading to phase transitions and dynamic scaling behaviors.

Findings

Network forms star-shaped structure at low density

High density leads to fat-tailed degree distribution

Identifies a critical particle density for phase transition

Abstract

We study a coupled dynamics of a network and a particle system. Particles of density $\rho$ diffuse freely along edges, each of which is rewired at a rate given by a decreasing function of particle flux. We find that the coupled dynamics leads to an instability toward the formation of hubs and that there is a dynamic phase transition at a threshold particle density $\rho_c$. In the low density phase, the network evolves into a star-shaped one with the maximum degree growing linearly in time. In the high density phase, the network exhibits a fat-tailed degree distribution and an interesting dynamic scaling behavior. We present an analytic theory explaining mechanism for the instability and a scaling theory for the dynamic scaling behavior.