Looking for vertex number one

TL;DR

This paper presents a highly local and efficient algorithm for identifying a specific vertex in a preferential attachment graph using only small neighborhood explorations, improving upon previous methods.

Contribution

It introduces a new algorithm that finds vertex 1 in preferential attachment graphs with minimal local exploration, operating only on neighborhoods of individual vertices.

Findings

Runs in time O(ω log n) with high probability

Operates solely on neighborhoods of single vertices

Improves locality over previous algorithms

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 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 $n$.