Sublinear preferential attachment combined with the growing number of choices

We prove almost sure convergence of the maximum degree in an evolving graph model combining a growing number of local choices with sublinear preferential attachment. At each step in the growth of the graph, a new vertex is introduced. Then we draw a random number of edges from it to existing vertices, chosen independently by the following rule. For each edge, we consider a sample of the growing size of vertices chosen with probabilities proportional to the sublinear function of their degrees. Then new vertex attaches to the vertex with the highest degree from the sample. Depending on the growth rate of the sample and the sublinear function, the maximum degree could be of the sublinear order, of the linear order or having almost all edges drawing to it. The prove using various stochastic approximation processes and a large deviation approach.