On the impossibility of detecting a late change-point in the preferential attachment random graph model

• Main Authors: Kaddouri, Ibrahim, Naulet, Zacharie, Gassiat, Élisabeth
• Format: Journal Article
• Language: English
• Published: 26-07-2024
• Subjects: Mathematics - Probability; Mathematics - Statistics Theory; Statistics - Theory
• DOI: 10.48550/arxiv.2407.18685

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 labeled 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.