On the number of isolated vertices in a growing random graph

Abstract

This paper studies the properties of the number of isolated vertices in a random graph where vertices arrive one-by-one at times $1,2,\ldots$\,. They are connected by edges to the previous vertices independently with the same probability. Assuming that the probability of an edge tends to zero, we establish the asymptotics of large, normal, and moderate deviations for the stochastic process of the number of the isolated vertices considered at times inversely proportional to that probability. In addition, we identify the most likely trajectory for that stochastic process to follow conditioned on the event that at a large time the graph is found with a large number of isolated vertices.