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

The connected component of the partial duplication graph