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

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 analyse 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 modelling degree distributions in non-growing networks and offer insights into the long-time behaviour of such systems.