A geometric preferential attachment model with fitness

We study a random graph $G_n$, which combines aspects of geometric random graphs and preferential attachment. The resulting random graphs have power-law degree sequences with finite mean and possibly infinite variance. In particular, the power-law exponent can be any value larger than 2. The vertices of $G_n$ are $n$ sequentially generated vertices chosen at random in the unit sphere in $\mathbb R^3$. A newly added vertex has $m$ edges attached to it and the endpoints of these edges are connected to old vertices or to the added vertex itself. The vertices are chosen with probability proportional to their current degree plus some initial attractiveness and multiplied by a function, depending on the geometry.