[STAT/AP] Bas Lodewijks: Super-linear preferential attachment with fitness

13 maart 2023 15:45 t/m 17:45 - Locatie: EEMCS Lecture Hall G | Zet in mijn agenda

We consider a model of randomly growing trees called super-linear preferential attachment with fitness. In this model, we start with a root labelled 1 with fitness (or weight) W_1, and at each step n at least 2 a new vertex n with weight W_n (an i.i.d. copy of W_1) is introduced and connected to one vertex already present in the tree. Conditionally on the tree created so far (including the vertex-fitnesses W_1, ..., W_(n-1), vertex n connects to vertex v with a probability proportional to f(deg_n-1(v), W_v), where deg_n-1(v) is the degree of vertex v in the tree of size n-1 created so far.
We focus on the case where the function f grows super-linear in its first argument. In particular, we shall discuss the two examples f(j,W)= j^p W and f(j,W)=j^p+W (multiplicative fitness and additive fitness, respectively), where p>1 is a constant called the super-linear exponent. We will identify a phase transition in the structure of the tree, in particular whether the limiting infinite tree contains a unique vertex with infinite degree or a unique infinite path almost surely. Moreover, we quantify the phase-transition in terms of the tail-behaviour of the fitness distribution and the super-linear exponent p.