  We study an evolving spatial network in which sequentially arriving vertices
are joined to existing vertices at random according to a rule that combines
preference according to degree with preference according to spatial proximity.
We investigate phase transitions in graph structure as the relative weighting
of these two components of the attachment rule is varied.
  Previous work of one of the authors showed that when the geometric component
is weak, the limiting degree sequence of the resulting graph coincides with
that of the standard Barab\'asi--Albert preferential attachment model. We show
that at the other extreme, in the case of a sufficiently strong geometric
component, the limiting degree sequence coincides with that of a purely
geometric model, the on-line nearest-neighbour graph, which is of interest in
its own right and for which we prove some extensions of known results. We also
show the presence of an intermediate regime, in which the behaviour differs
significantly from both the on-line nearest-neighbour graph and the
Barab\'asi--Albert model; in this regime, we obtain a stretched exponential
upper bound on the degree sequence.
  Our results lend some mathematical support to simulation studies of Manna and
Sen, while proving that the power law to stretched exponential phase transition
occurs at a different point from the one conjectured by those authors.
