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

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 P0(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 St=S[Pt(k)||π(k|(K)t)] of the degree distribution Pt(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 Pt(k) and Poisson. The relative entropy is suitable as a distance measure since it satisfies St≥0 for any degree distribution Pt(k), while equality is obtained only for Pt(k)=π(k|(K)t). We derive an equation for the time derivative dSt/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 Erdos-Rényi (ER) graph structure, substantiating earlier results obtained using direct integration of the master equation and computer simulations [Tishby et al., 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).