The expected degree distribution in transient duplication divergence models

TL;DR

This paper analyzes the degree distribution in duplication-divergence models, establishing a central limit theorem for large degrees and demonstrating asymptotic similarity across various model formulations.

Contribution

It introduces a comprehensive analysis of degree distributions in generalized duplication-divergence models, including a central limit theorem for large degrees.

Findings

Central limit theorem for the logarithm of degree distribution

Asymptotic similarity of different model formulations

Analysis of vertices with large degrees

Abstract

We study the degree distribution of a randomly chosen vertex in a duplication--divergence graph, under a variety of different generalizations of the basic model of Bhan, Galas and Dewey (2002) and V\'azquez, Flammini, Maritan and Vespignani (2003). We pay particular attention to what happens when a non-trivial proportion of the vertices have large degrees, establishing a central limit theorem for the logarithm of the degree distribution. Our approach, as in Jordan (2018) and Hermann and Pfaffelhuber (2021), relies heavily on the analysis of related birth--catastrophe processes, and couplings are used to show that a number of different formulations of the process have asymptotically similar expected degree distributions.