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

10 We pursue the analysis of the maximum degree in a dynamic duplication-divergence graph model 11 defined by Solé et al. in which a new node arriving at time t first randomly selects an existing 12 node and connects to its neighbors with probability p, and then connects to the other nodes 13 with probability r/t. This model is often said to capture the growth of some real-world processes 14 e.g. biological or social networks. However, there are only a handful of rigorous results concerning 15 this model. Here we study the distribution of the maximum degree of a vertex in graphs generated 16 by this model. 17 In this paper we solve an open problem by proving that for 2 < p < 1 with high probability the 18 maximum degree is asymptotically concentrated around t, i.e. it deviates from this value by at 19 most a polylogarithmic factor. Our findings are a step towards a better understanding of the overall 20 structure of graphs generated by this model, especially the degree distribution, compression, and 21 symmetry, which are important open problems in this area. 22 2012 ACM Subject Classification Mathematics of computing → Random graphs; Theory of com23 putation → Random network models 24

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