Emerging properties of the degree distribution in large non-growing networks

Abstract

The degree distribution is a key statistical indicator in network theory, often used to understand how information spreads across connected nodes. In this paper, we focus on non-growing networks formed through a rewiring algorithm and develop kinetic Boltzmann-type models to capture the emergence of degree distributions that characterize both preferential attachment networks and random networks. Under a suitable mean-field scaling, these models reduce to a Fokker-Planck-type partial differential equation with an affine diffusion coefficient, that is consistent with a well-established master equation for discrete rewiring processes. We further analyze the convergence to equilibrium for this class of Fokker-Planck equations, demonstrating how different regimes -- ranging from exponential to algebraic rates -- depend on network parameters. Our results provide a unified framework for modeling degree distributions in non-growing networks and offer insights into the long-time behavior of such systems.