Analysis of the convergence of the degree distribution of contracting random networks towards a Poisson distribution using the relative entropy.

Abstract

We present analytical results for the structural evolution of random networks undergoing contraction processes via generic node deletion scenarios, namely, random deletion, preferential deletion, and propagating deletion. Focusing on configuration model networks, which exhibit a given degree distribution P_{0}(k) and no correlations, we show using a rigorous argument that upon contraction the degree distributions of these networks converge towards a Poisson distribution. To this end, we use the relative entropy S_{t}=S[P_{t}(k)||π(k|〈K〉_{t})] of the degree distribution P_{t}(k) of the contracting network at time t with respect to the corresponding Poisson distribution π(k|〈K〉_{t}) with the same mean degree 〈K〉_{t} as a distance measure between P_{t}(k) and Poisson. The relative entropy is suitable as a distance measure since it satisfies S_{t}≥0 for any degree distribution P_{t}(k), while equality is obtained only for P_{t}(k)=π(k|〈K〉_{t}). We derive an equation for the time derivative dS_{t}/dt during network contraction and show that the relative entropy decreases monotonically to zero during the contraction process. We thus conclude that the degree distributions of contracting configuration model networks converge towards a Poisson distribution. Since the contracting networks remain uncorrelated, this means that their structures converge towards an Erdős-Rényi (ER) graph structure, substantiating earlier results obtained using direct integration of the master equation and computer simulations [Tishby etal., Phys. Rev. E 100, 032314 (2019)2470-004510.1103/PhysRevE.100.032314]. We demonstrate the convergence for configuration model networks with degenerate degree distributions (random regular graphs), exponential degree distributions, and power-law degree distributions (scale-free networks).