Finding Adam in random growing trees

TL;DR

This paper develops algorithms to identify the first vertex in large random trees generated by uniform or preferential attachment models, with bounds on the number of vertices needed relative to the desired confidence level.

Contribution

It provides the first algorithms with size bounds independent of tree size and establishes tight bounds for the number of vertices needed based on the attachment model and confidence.

Findings

Optimal K is subpolynomial in 1/ε for uniform attachment

K must be superpolylogarithmic in 1/ε for uniform attachment

K is polynomial in 1/ε for preferential attachment

Abstract

We investigate algorithms to find the first vertex in large trees generated by either the uniform attachment or preferential attachment model. We require the algorithm to output a set of $K$ vertices, such that, with probability at least $1-\epsilon$, the first vertex is in this set. We show that for any $\epsilon$, there exist such algorithms with $K$ independent of the size of the input tree. Moreover, we provide almost tight bounds for the best value of $K$ as a function of $\epsilon$. In the uniform attachment case we show that the optimal $K$ is subpolynomial in $1/\epsilon$, and that it has to be at least superpolylogarithmic. On the other hand, the preferential attachment case is exponentially harder, as we prove that the best $K$ is polynomial in $1/\epsilon$. We conclude the paper with several open problems.