Looking for vertex number one

Abstract

Given an instance of the preferential attachment graph $G_{n}=([n],E_{n})$, we would like to find vertex 1, using only “local” information about the graph; that is, by exploring the neighborhoods of small sets of vertices. Borgs et al. gave an algorithm which runs in time $O(\log^{4}n)$, which is local in the sense that at each step, it needs only to search the neighborhood of a set of vertices of size $O(\log^{4}n)$. We give an algorithm to find vertex 1, which w.h.p. runs in time $O(\omega\log n)$ and which is local in the strongest sense of operating only on neighborhoods of single vertices. Here $\omega=\omega(n)$ is any function that goes to infinity with $n$.