Counting Stars is Constant-Degree Optimal For Detecting Any Planted Subgraph
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Yu, Xifan Zadik, Ilias Zhang, Peiyuan |
| author_facet | Yu, Xifan Zadik, Ilias Zhang, Peiyuan |
| contents | We study the computational limits of the following general hypothesis testing problem. Let H=H_n be an \emph{arbitrary} undirected graph on n vertices. We study the detection task between a ``null'' Erdős-Rényi random graph G(n,p) and a ``planted'' random graph which is the union of G(n,p) together with a random copy of H=H_n. Our notion of planted model is a generalization of a plethora of recently studied models initiated with the study of the planted clique model (Jerrum 1992), which corresponds to the special case where H is a k-clique and p=1/2.
Over the last decade, several papers have studied the power of low-degree polynomials for limited choices of H's in the above task. In this work, we adopt a unifying perspective and characterize the power of \emph{constant degree} polynomials for the detection task, when \emph{H=H_n is any arbitrary graph} and for \emph{any p=Ω(1).} Perhaps surprisingly, we prove that the optimal constant degree polynomial is always given by simply \emph{counting stars} in the input random graph. As a direct corollary, we conclude that the class of constant-degree polynomials is only able to ``sense'' the degree distribution of the planted graph H, and no other graph theoretic property of it. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_17766 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Counting Stars is Constant-Degree Optimal For Detecting Any Planted Subgraph Yu, Xifan Zadik, Ilias Zhang, Peiyuan Statistics Theory Computational Complexity Data Structures and Algorithms We study the computational limits of the following general hypothesis testing problem. Let H=H_n be an \emph{arbitrary} undirected graph on n vertices. We study the detection task between a ``null'' Erdős-Rényi random graph G(n,p) and a ``planted'' random graph which is the union of G(n,p) together with a random copy of H=H_n. Our notion of planted model is a generalization of a plethora of recently studied models initiated with the study of the planted clique model (Jerrum 1992), which corresponds to the special case where H is a k-clique and p=1/2. Over the last decade, several papers have studied the power of low-degree polynomials for limited choices of H's in the above task. In this work, we adopt a unifying perspective and characterize the power of \emph{constant degree} polynomials for the detection task, when \emph{H=H_n is any arbitrary graph} and for \emph{any p=Ω(1).} Perhaps surprisingly, we prove that the optimal constant degree polynomial is always given by simply \emph{counting stars} in the input random graph. As a direct corollary, we conclude that the class of constant-degree polynomials is only able to ``sense'' the degree distribution of the planted graph H, and no other graph theoretic property of it. |
| title | Counting Stars is Constant-Degree Optimal For Detecting Any Planted Subgraph |
| topic | Statistics Theory Computational Complexity Data Structures and Algorithms |
| url | https://arxiv.org/abs/2403.17766 |