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

TL;DR

This paper analytically demonstrates that the degree distribution of contracting random networks converges to a Poisson distribution, indicating a transition towards Erdős-Rényi graph structure, using relative entropy as a measure.

Contribution

It provides a rigorous analytical proof that various types of contracting configuration model networks evolve towards a Poisson degree distribution, confirming earlier simulation results.

Findings

Degree distributions converge to Poisson during contraction.

Relative entropy decreases monotonically to zero.

Networks tend towards Erdős-Rényi structure.

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 $S_t=S[P_t(k) || \pi(k|\langle K \rangle_t)]$ of the degree distribution $P_t(k)$ of the contracting network at time $t$ with respect to the corresponding Poisson distribution $\pi(k|\langle K \rangle_t)$ with the same mean degree $\langle K \rangle_t$ as a distance measure between $P_t(k)$ and Poisson. The relative entropy is suitable as a distance measure since it satisfies $S_t \ge 0$ for any degree distribution $P_t(k)$, while equality is obtained only for $P_t(k) = \pi(k|\langle K \rangle_t)$. 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 Erd{\H o}s-R\'enyi (ER) graph structure, substantiating earlier results obtained using direct integration of the master equation and computer simulations [I. Tishby, O. Biham and E. Katzav, {\it Phys. Rev. E} {\bf 100}, 032314 (2019)]. 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).