Distances and large deviations in the spatial preferential attachment model

This paper considers two asymptotic properties of a spatial preferential-attachment model introduced by E. Jacob and P. Morters (In Algorithms and Models for the Web Graph (2013) 14-25 Springer). First, in a regime of strong linear reinforcement, we show that typical distances are at most of doubly-logarithmic order. Second, we derive a large deviation principle for the empirical neighbourhood structure and express the rate function as solution to an entropy minimisation problem in the space of stationary marked point processes.