  We consider the connected component of the partial duplication model for a
random graph, a model which was introduced by Bhan, Galas and Dewey as a model
for gene expression networks. The most rigorous results are due to Hermann and
Pfaffelhuber, who show a phase transition between a subcritical case where in
the limit almost all vertices are isolated and a supercritical case where the
proportion of the vertices which are connected is bounded away from zero.
  We study the connected component in the subcritical case, and show that, when
the duplication parameter $p<e^{-1}$, the degree distribution of the connected
component has a limit, which we can describe in terms of the stationary
distribution of a certain Markov chain and which follows an approximately power
law tail, with the power law index predicted by Ispolatov, Krapivsky and
Yuryev. Our methods involve analysing the quasi-stationary distribution of a
certain continuous time Markov chain associated with the evolution of the
graph.
