On dynamic random graphs with degree homogenization via anti-preferential attachment probabilities

We analyze a dynamic random undirected graph in which newly added vertices are connected to those already present in the graph either using, with probability $p$, an anti-preferential attachment mechanism or, with probability $1-p$, a preferential attachment mechanism. We derive the asymptotic degree distribution in the general case and study the asymptotic behaviour of the expected degree process in the general and that of the degree process in the pure anti-preferential attachment case. Degree homogenization mainly affects convergence rates for the former case and also the limiting degree distribution in the latter. Lastly, we perform a simulative study of a variation of the introduced model allowing for anti-preferential attachment probabilities given in terms of the current maximum degree of the graph.