  Duplication graphs are graphs that grow by duplication of existing vertices,
and are important models of biological networks, including protein-protein
interaction networks and gene regulatory networks. Three models of graph growth
are studied: pure duplication growth, and two two-parameter models in which
duplication forms one element of the growth dynamics. A power-law degree
distribution is found to emerge in all three models. However, the parameter
space of the latter two models is characterized by a range of parameter values
for which duplication is the predominant mechanism of graph growth. For
parameter values that lie in this ``duplication-dominated'' regime, it is shown
that the degree distribution either approaches zero asymptotically, or
approaches a non-zero power-law degree distribution very slowly. In either
case, the approach to the true asymptotic degree distribution is characterized
by a dependence of the scaling exponent on properties of the initial degree
distribution. It is therefore conjectured that duplication-dominated,
scale-free networks may contain identifiable remnants of their early structure.
This feature is inherited from the idealized model of pure duplication growth,
for which the exact finite-size degree distribution is found and its asymptotic
properties studied.
