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

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 $n-\mathrm{\Omega }(\sqrt{n})$. They further make the striking conjecture that detection becomes impossible for any test if the change occurs at time $n-o(\sqrt{n})\mathrm{.}$ Kaddouri et al.~\cite{kaddouri2024impossibility} make a step forward by proving the detection is impossible if the change occurs at time $n-o(n^{1\mathrm{/}3})\mathrm{.}$ In this paper, we resolve the conjecture affirmatively, proving that detection is indeed impossible if the change occurs at time $n-o(\sqrt{n})\mathrm{.}$ 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.