Convergence towards an Erd{\H o}s-R\'enyi graph structure in network contraction processes

TL;DR

This paper investigates how networks undergoing contraction processes tend to evolve towards Erdős-Rényi structures, characterized by Poisson degree distributions and loss of correlations, regardless of initial structure.

Contribution

It demonstrates that network contraction leads to convergence towards Erdős-Rényi graphs, extending understanding beyond growth models to contraction dynamics.

Findings

Networks under contraction converge to Erdős-Rényi structures.

Degree distributions become Poisson during contraction.

Degree-degree correlations diminish over time.

Abstract

In a highly influential paper twenty years ago, Barab\'asi and Albert [Science 286, 509 (1999)] showed that networks undergoing generic growth processes with preferential attachment evolve towards scale-free structures. In any finite system, the growth eventually stalls and is likely to be followed by a phase of network contraction due to node failures, attacks or epidemics. Using the master equation formulation and computer simulations we analyze the structural evolution of networks subjected to contraction processes via random, preferential and propagating node deletions. We show that the contracting networks converge towards an Erd{\H o}s-R\'enyi network structure whose mean degree continues to decrease as the contraction proceeds. This is manifested by the convergence of the degree distribution towards a Poisson distribution and the loss of degree-degree correlations.