On the impossibility of detecting a late change-point in the preferential attachment random graph model
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866915076569563136 |
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| author | Kaddouri, Ibrahim Naulet, Zacharie Gassiat, Élisabeth |
| author_facet | Kaddouri, Ibrahim Naulet, Zacharie Gassiat, Élisabeth |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18685 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the impossibility of detecting a late change-point in the preferential attachment random graph model Kaddouri, Ibrahim Naulet, Zacharie Gassiat, Élisabeth Statistics Theory Probability 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. |
| title | On the impossibility of detecting a late change-point in the preferential attachment random graph model |
| topic | Statistics Theory Probability |
| url | https://arxiv.org/abs/2407.18685 |