  We study logical limit laws for preferential attachment random graphs. In
this random graph model, vertices and edges are introduced recursively: at time
$1$, we start with vertices $0,1$ and $m$ edges between them. At step $n+1$ the
vertex $n+1$ is introduced together with $m$ edges joining the new vertex with
$m$ vertices chosen from $1,\ldots,n$ independently with probabilities
proportional to their degrees plus a positive parameter $\delta$. We prove that
this random graph obeys the convergence law for first-order sentences with at
most $m-2$ variables.
