Degree centrality and root finding in growing random networks

Abstract

We consider growing random networks ${\{{\mathcal{G}}_n\}}_{n\ge 1}$ where, at each time, a new vertex attaches itself to a collection of existing vertices via a fixed number $m\ge 1$ of edges, with probability proportional to a function f (called attachment function) of their degree. It was shown in [BB21] that such network models exhibit two distinct regimes: (i) the persistent regime, corresponding to ${\sum }_{i=1}^{\mathrm{\infty }}f{(i)}^{-2}<\mathrm{\infty }$, where the top K maximal degree vertices fixate over time for any given K, and (ii) the non-persistent regime, with ${\sum }_{i=1}^{\mathrm{\infty }}f{(i)}^{-2}=\mathrm{\infty }$, where the identities of these vertices keep changing infinitely often over time. In this article, we develop root finding algorithms using the empirical degree structure and local network information based on a snapshot of such a network at some large time. In the persistent regime, the algorithm is purely based on degree centrality, that is, for a given error tolerance $\mathit{\epsilon }\in (0,1)$, there exists $K_{\mathit{\epsilon }}\in \mathbb{N}$ such that for any $n\ge 1$, the confidence set for the root in ${\mathcal{G}}_n$, which contains the root with probability at least $1-\mathit{\epsilon }$, consists of the top $K_{\mathit{\epsilon }}$ maximal degree vertices. In particular, the size of the confidence set is stable in the network size. Upper and lower bounds on $K_{\mathit{\epsilon }}$ are explicitly characterized in terms of the error tolerance ε and the attachment function f. In the non-persistent regime, for an appropriate choice of $r_n\to \mathrm{\infty }$ at a rate much smaller than the diameter of the network, the neighborhood of radius $r_n$ around the maximal degree vertex is shown to contain the root with high probability, and a size estimate for this set is obtained. It is shown that, when $f(k)=k^{\mathit{\alpha }},k\ge 1$, for any $\mathit{\alpha }\in (0,1∕2]$, this size grows at a smaller rate than any positive power of the network size.