  We investigate the degree sequences of geometric preferential attachment
graphs in general compact metric spaces. We show that, under certain conditions
on the attractiveness function, the behaviour of the degree sequence is similar
to that of the preferential attachment with multiplicative fitness models
investigated by Borgs et al. When the metric space is finite, the degree
distribution at each point of the space converges to a degree distribution
which is an asymptotic power law whose index depends on the chosen point. For
infinite metric spaces, we can show that for vertices in a Borel subset of $S$
of positive measure the degree distribution converges to a distribution whose
tail is close to that of a power law whose index again depends on the set.
