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

TL;DR

This paper rigorously analyzes the maximum and average degrees in a duplication-divergence graph model, showing they concentrate around specific power-law functions of the number of vertices, reflecting real-world network growth.

Contribution

It provides the first precise probabilistic concentration results for degrees in the duplication-divergence model, a key model for biological and social networks.

Findings

Maximum degree concentrates around t^p

Average degree concentrates around max{t^{2p-1}, 1}

Results hold with high probability for large t

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 Sol\'e, 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 $A > 0$.