Nonlinear preferential rewiring in fixed-size networks as a diffusion process

TL;DR

This paper introduces a fixed-size network model with nonlinear preferential rewiring, revealing a phase transition from homogeneous to heterogeneous structures and describing the evolution with a nonlinear diffusion equation.

Contribution

It presents a novel network model with nonlinear preferential detachment and reattachment, analyzing phase transitions and diffusion dynamics in degree distributions.

Findings

Degree distributions undergo a second order phase transition at alpha = beta.

Stationary states follow power-law distributions with exponents related to alpha and beta.

The temporal evolution is governed by a nonlinear diffusion equation.

Abstract

We present an evolving network model in which the total numbers of nodes and edges are conserved, but in which edges are continuously rewired according to nonlinear preferential detachment and reattachment. Assuming power-law kernels with exponents alpha and beta, the stationary states the degree distributions evolve towards exhibit a second order phase transition - from relatively homogeneous to highly heterogeneous (with the emergence of starlike structures) at alpha = beta. Temporal evolution of the distribution in this critical regime is shown to follow a nonlinear diffusion equation, arriving at either pure or mixed power-laws, of exponents -alpha and 1-alpha.