Degree centrality and root finding in growing random networks

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 degree fixate over time for any given K, and (ii) non-persistent ∑i=1∞f(i)−2=∞, identities these keep changing infinitely often time. In this article, we develop root finding algorithms using empirical structure local information based on snapshot some large algorithm is purely centrality, is, error tolerance ε∈(0,1), there exists Kε∈N n≥1, confidence set Gn, which contains least 1−ε, consists Kε vertices. particular, size stable size. Upper lower bounds are explicitly characterized terms ε f. an appropriate choice rn→∞ rate much smaller than diameter network, neighborhood radius rn around contain high probability, estimate obtained. that, when f(k)=kα,k≥1, α∈(0,1∕2], grows positive power