We consider the problem of late change-point detection under the preferential attachment random graph model with time dependent attachment function. This can be formulated as a hypothesis testing problem where the null hypothesis corresponds to a preferential attachment model with a constant affine attachment parameter $\delta_0$ and the alternative corresponds to a preferential attachment model where the affine attachment parameter changes from $\delta_0$ to $\delta_1$ at a time $\tau_n = n - \Delta_n$ where $0\leq \Delta_n \leq n$ and $n$ is the size of the graph. It was conjectured in Bet et al. that when observing only the unlabeled graph, detection of the change is not possible for $\Delta_n = o(n^{1/2})$. In this work, we make a step towards proving the conjecture by proving the impossibility of detecting the change when $\Delta_n = o(n^{1/3})$. We also study change-point detection in the case where the labelled graph is observed and show that change-point detection is possible if and only if $\Delta_n \to \infty$, thereby exhibiting a strong difference between the two settings.

翻译：我们研究了时间依赖依附函数下偏好依附随机图模型中的晚期变点检测问题。该问题可表述为一个假设检验问题：零假设对应具有恒定仿射依附参数 $\delta_0$ 的偏好依附模型，而备择假设对应仿射依附参数在时刻 $\tau_n = n - \Delta_n$ 从 $\delta_0$ 变为 $\delta_1$ 的偏好依附模型，其中 $0\leq \Delta_n \leq n$，$n$ 为图的规模。Bet 等人曾猜想，当仅观测未标记图时，对于 $\Delta_n = o(n^{1/2})$ 的情况，变点检测是不可能的。本工作中，我们通过证明 $\Delta_n = o(n^{1/3})$ 时变点检测的不可能性，向该猜想的证明迈进一步。我们还研究了标记图可观测情况下的变点检测问题，证明变点检测当且仅当 $\Delta_n \to \infty$ 时可行，从而揭示了两种设定间的显著差异。