We consider growing random networks {Gn}n≥1 where, at each time, a new vertex attaches itself to collection of existing vertices via fixed number m≥1 edges, with probability proportional function f (called attachment function) their degree. It was shown in [BB21] that such network models exhibit two distinct regimes: (i) the persistent regime, corresponding ∑i=1∞f(i)−2<∞, where top K maximal de...