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arXiv · 2407.18685

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

Ibrahim Kaddouri·Zacharie Naulet·Élisabeth Gassiat
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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 $δ_0$ and the alternative corresponds to a preferential attachment model where the affine attachment parameter changes from $δ_0$ to $δ_1$ at a time $τ_n = n - Δ_n$ where $0\leq Δ_n \leq n$ 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 $Δ_n = o(n^{1/2})$. In this work, we make a step towards proving the conjecture by proving the impossibility of detecting the change when $Δ_n = o(n^{1/3})$. 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 $Δ_n \to \infty$, thereby exhibiting a strong difference between the two settings.

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Ibrahim Kaddouri, Zacharie Naulet, Élisabeth Gassiat. 2024-12-23. On the impossibility of detecting a late change-point in the preferential attachment random graph model. https://arxiv.org/abs/2407.18685

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