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

TL;DR

This paper investigates the fundamental limits of detecting late change-points in preferential attachment random graph models, showing detection impossibility under certain conditions for unlabeled graphs and possibility when labels are observed.

Contribution

It proves the conjecture that change detection is impossible for unlabeled graphs when the change occurs too late, and demonstrates that labels enable detection if the change is sufficiently early.

Findings

Detection impossible for unlabeled graphs if change occurs within o(n^{1/3})

Detection possible with labeled graphs if and only if the change occurs early enough (Δ_n → ∞)

Highlights a significant difference between labeled and unlabeled graph observations in change-point detection

Abstract

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.