Detecting a late changepoint in the preferential attachment model

Motivated by the problem of detecting a change in the evolution of a network, we consider the preferential attachment random graph model with a time-dependent attachment function. Our goal is to detect whether the attachment mechanism changed over time, based on a single snapshot of the network and without directly observable information about the dynamics. We cast this question as a hypothesis testing problem, where the null hypothesis is a preferential attachment model with a constant affine attachment parameter $δ_0$, and the alternative hypothesis is a preferential attachment model where the affine attachment parameter changes from $δ_0$ to $δ_1$ at an unknown changepoint time $τ_n$. For our analysis we focus on the regime where $δ_0$ and $δ_1$ are fixed, and the changepoint occurs close to the observation time of the network (i.e., $τ_n = n - c n^γ$ with $c>0$ and $γ\in (0, 1)$). This corresponds to the relevant scenario where we aim to detect the changepoint shortly after it has happened. We present two tests based on the number of vertices with minimal degree, and show that these are asymptotically powerful when $\tfrac{1}{2}