Degree centrality and root finding in growing random networks

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 an attachment function $f$ of their degree. It was shown in \cite{BBpersistence} that such network models exhibit two regimes: (i) the persistent regime, corresponding to $\sum_{i=1}^{\infty}f(i)^{-2} < \infty$, where the top $K$ maximal degree vertices fixate over time for any given $K$, and (ii) the non-persistent regime, with $\sum_{i=1}^{\infty}f(i)^{-2} = \infty$, where the identities of these vertices keep changing infinitely often over time. 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 $\varepsilon \in (0,1)$, there exists $K_{\varepsilon}$ 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 $1 - \varepsilon$, consists of the top $K_{\varepsilon}$ maximal degree vertices. In particular, the size of the confidence set is stable in the network size. Upper and lower bounds on $K_{\varepsilon}$ are explicitly characterized in terms of the error tolerance $\varepsilon$ and the attachment function $f$. In the non-persistent regime, for an appropriate choice of $r_n \rightarrow \infty$ 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 $f(k) = k^{\alpha}, k \ge 1,$ for any $\alpha \in (0,1/2]$, this size grows at a smaller rate than any positive power of the network size.