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 $A > 0$.

翻译：摘要：本文对Solé、Pastor-Satorras等人提出的动态复制-发散图模型中的最大度与平均度进行了严格精确的分析。该模型通过复制-发散机制实现图增长：即迭代创建某节点的副本，并以概率$p$随机调整新节点的邻域。此模型能够刻画生物网络或社交网络等实际过程的演化特征。我们证明：对于某些$0 < p < 1$，在顶点数为$t$的复制-发散图中，最大度与平均度分别以高概率渐近集中于$t^p$和$\max\{t^{2p - 1}, 1\}$附近。具体而言，对任意常数$A > 0$，这些统计量以至少$1 - t^{-A}$的概率落在上述值的多项式对数因子范围内。