On the Concentration of the Maximum Degree in the Duplication-Divergence Models

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 988-1006, March 2024.
Abstract. We present a rigorous and precise analysis of the maximum degree and the average degree in a dynamic duplication-divergence graph model introduced by Solé 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 [math]. This model captures the growth of some real-world processes, e.g., biological or social networks. In this paper, we prove that for some [math], the maximum degree and the average degree of a duplication-divergence graph on [math] vertices are asymptotically concentrated with high probability around [math] and [math], respectively, i.e., they are within at most a polylogarithmic factor from these values with probability at least [math] for any constant [math].