ON THE CONCENTRATION OF THE MAXIMUM DEGREE IN THE DUPLICATION-DIVERGENCE MODELS.

We present a rigorous and precise analysis of the maximum degree and the average degree in a dynamic duplication-divergence graph model introduced by Sol\'e et al. [Adv. Complex Syst., 5 (2002), pp. 43-54] in which the graph grows according to a duplication-divergence mechanism, i.e., by iteratively creating a copy of some node and then randomly alternating the neighborhood of a new node with probability p. This model captures the growth of some real-world processes, e.g., biological or social networks. In this paper, we prove that for some 0 < p < 1, the maximum degree and the average degree of a duplication-divergence graph on t vertices are asymptotically concentrated with high probability around tp and max\{ t2p-1, 1}, respectively, i.e., they are within at most a polylogarithmic factor from these values with probability at least 1-t-A for any constant A >0. [ABSTRACT FROM AUTHOR]