The split-and-drift random graph, a null model for speciation

Abstract

We introduce a new random graph model motivated by biological questions relating to speciation. This random graph is defined as the stationary distribution of a Markov chain on the space of graphs on $\{1, \ldots, n\}$. The dynamics of this Markov chain is governed by two types of events: vertex duplication, where at constant rate a pair of vertices is sampled uniformly and one of these vertices loses its incident edges and is rewired to the other vertex and its neighbors; and edge removal, where each edge disappears at constant rate. Besides the number of vertices $n$, the model has a single parameter $r_n$. Using a coalescent approach, we obtain explicit formulas for the first moments of several graph invariants such as the number of edges or the number of complete subgraphs of order $k$. These are then used to identify five non-trivial regimes depending on the asymptotics of the parameter $r_n$. We derive an explicit expression for the degree distribution, and show that under appropriate rescaling it converges to classical distributions when the number of vertices goes to infinity. Finally, we give asymptotic bounds for the number of connected components, and show that in the sparse regime the number of edges is Poissonian.