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 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 \(p\). 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 \(t^p\) and \(max \{t2 p - 1, 1\}\), respectively, i.e., they are within at most a polylogarithmic factor from these values with probability at least \(1 - t-A\) for any constant \(A > 0\).