Dynamic random graphs with vertex removal

TL;DR

This paper introduces a dynamic random graph model with vertex addition and removal, analyzing its limiting behavior, component structure, degree distribution, and phase transition between subcritical and supercritical regimes.

Contribution

It provides the first detailed analysis of a vertex-removing dynamic random graph, including convergence, component structure, degree bounds, and phase transition characterization.

Findings

Converges to a local limit with explicit description

Identifies subcritical and supercritical regimes for giant component

Provides bounds for the critical parameter and degree distribution

Abstract

We introduce and analyse a Dynamic Random Graph with Vertex Removal (DRGVR) defined as follows. At every step, with probability $p > 1/2$ a new vertex is introduced, and with probability $1-p$ a vertex, chosen uniformly at random among the present ones (if any), is removed from the graph together with all edges adjacent to it. In the former case, the new vertex connects by an edge to every other vertex with probability inversely proportional to the number of vertices already present. We prove that the DRGVR converges to a local limit and determine this limit. Moreover, we analyse its component structure and distinguish a subcritical and a supercritical regime with respect to the existence of a giant component. As a byproduct of this analysis, we obtain upper and lower bounds for the critical parameter. Furthermore, we provide precise expression of the maximum degree (as well as in- and out-degree for a natural orientation of the DRGVR). Several concentration and stability results complete the study.