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We present analytical results for the structural evolution of random networksundergoing contraction processes via generic node deletion scenarios, namely,random deletion, preferential deletion and propagating deletion. Focusing onconfiguration model networks, which exhibit a given degree distribution$P_0(k)$ and no correlations, we show using a rigorous argument that uponcontraction the degree distributions of these networks converge towards aPoisson 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 thecontracting network at time $t$ with respect to the corresponding Poissondistribution $\pi(k|\langle K \rangle_t)$ with the same mean degree $\langle K\rangle_t$ as a distance measure between $P_t(k)$ and Poisson. The relativeentropy is suitable as a distance measure since it satisfies $S_t \ge 0$ forany 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 entropydecreases monotonically to zero during the contraction process. We thusconclude that the degree distributions of contracting configuration modelnetworks converge towards a Poisson distribution. Since the contractingnetworks remain uncorrelated, this means that their structures converge towardsan Erd{\H o}s-R\'enyi (ER) graph structure, substantiating earlier resultsobtained using direct integration of the master equation and computersimulations [I. Tishby, O. Biham and E. Katzav, {\it Phys. Rev. E} {\bf 100},032314 (2019)]. We demonstrate the convergence for configuration model networkswith degenerate degree distributions (random regular graphs), exponentialdegree distributions and power-law degree distributions (scale-free networks).

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