Limits of randomly grown graph sequences

Motivated in part by various sequences of graphs growing under random rules (such as Internet models), Borgs, Chayes, Lovász, Sós, Szegedy and Vesztergombi introduced convergent sequences of dense graphs and their limits. In this paper we use this framework to study one of the motivating classes of examples, namely randomly growing graphs. We prove the (almost sure) convergence of several such randomly growing graph sequences, and determine their limit. The analysis is not always straightforward: in some cases the cut-distance from a limit object can be directly estimated, while in other cases densities of subgraphs can be shown to converge.