  We consider the preferential attachment model with location-based choice
introduced by Haslegrave, Jordan and Yarrow as a model in which condensation
phenomena can occur [Haslegrave et al. 2020]. In this model every vertex
carries an independent and uniformly drawn location. Starting from an initial
tree the model evolves in discrete time. At every time step, a new vertex is
added to the tree by selecting $r$ candidate vertices from the graph with
replacement according to a sampling probability proportional to these vertices'
degrees. The new vertex then connects to one of the candidates according to a
given probability associated to the ranking of their locations. In this paper,
we introduce a function that describes the phase transition when condensation
can occur. Considering the noncondensation phase, we use stochastic
approximation methods to investigate bounds for the (asymptotic) proportion of
vertices inside a given interval of a given maximum degree. We use these bounds
to observe a power law for the asymptotic degree distribution described by the
aforementioned function. Hence, this function fully characterises the
properties we are interested in. The power law exponent takes the critical
value one at the phase transition between the condensation - noncondensation
phase.
