  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.
