An example of graph limits of growing sequences of random graphs

In this paper, we consider a class of growing random graphs obtained by creating vertices sequentially one by one.At each step, we uniformly choose the neighbors of the newly created vertex; its degree is a random variable with a fixed but arbitrary distribution, depending on the number of existing vertices.Examples from this class turn out to be the Erdős-Rényi random graph, a natural random threshold graph, etc.By working with the notion of graph limits, we define a kernel which, under certain conditions, is the limit of the growing random graph.Moreover, for a subclass of models, the growing graph on any given n vertices has the same distribution as the random graph with n vertices that the kernel defines.The motivation stems from a model of graph growth whose attachment mechanism does not require information about properties of the graph at each iteration.