On the concentration of the maximum degree in the duplication-divergence models

08/08/2023

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by   Alan Frieze, et al.

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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, Pastor-Satorras et al. 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{t^2 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.