Distance evolutions in growing preferential attachment graphs

We study the evolution of the graph distance and weighted distance between two fixed vertices in dynamically growing random graph models. More precisely, we consider preferential attachment models with powerlaw exponent τ ϵ (2, 3), sample two vertices ut , vt uniformly at random when the graph has t vertices and study the evolution of the graph distance between these two fixed vertices as the surrounding graph grows. This yields a discrete-time stochastic process in t ' ≥ t , called the distance evolution. We show that there is a tight strip around the function 4 log log(t)-log(log(t'/t)ν1) | log(τ-2)| ν 2 that the distance evolution never leaves with high probability as t tends to infinity. We extend our results to weighted distances, where every edge is equipped with an i.i.d. copy of a nonnegative random variable L.