A Proof of The Changepoint Detection Threshold Conjecture in Preferential Attachment Models

TL;DR

This paper proves the conjecture that detecting a changepoint in preferential attachment networks is impossible if it occurs too close to the end, specifically within $n - o(\sqrt{n})$, and confirms the optimality of existing estimators.

Contribution

It resolves the longstanding conjecture by proving detection impossibility at the $n - o(\sqrt{n})$ threshold and establishes the optimality of current changepoint estimators.

Findings

Detection is impossible if the changepoint occurs at time $n - o(\sqrt{n})$.

Estimating the changepoint with error smaller than $o(\sqrt{n})$ is impossible.

Confirms the order-optimality of existing changepoint estimators.

Abstract

We investigate the problem of detecting and estimating a changepoint in the attachment function of a network evolving according to a preferential attachment model on $n$ vertices, using only a single final snapshot of the network. Bet et al.~\cite{bet2023detecting} show that a simple test based on thresholding the number of vertices with minimum degrees can detect the changepoint when the change occurs at time $n-\Omega(\sqrt{n})$. They further make the striking conjecture that detection becomes impossible for any test if the change occurs at time $n-o(\sqrt{n}).$ Kaddouri et al.~\cite{kaddouri2024impossibility} make a step forward by proving the detection is impossible if the change occurs at time $n-o(n^{1/3}).$ In this paper, we resolve the conjecture affirmatively, proving that detection is indeed impossible if the change occurs at time $n-o(\sqrt{n}).$ Furthermore, we establish that estimating the changepoint with an error smaller than $o(\sqrt{n})$ is also impossible, thereby confirming that the estimator proposed in Bhamidi et al.~\cite{bhamidi2018change} is order-optimal.