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Counting Stars is Constant-Degree Optimal For Detecting Any Planted Subgraph

Xifan Yu, Ilias Zadik, Peiyuan Zhang

TL;DR

The power of constant degree polynomials for the detection task is characterized and it is proved that an optimal constant degree polynomial is always given by simply counting stars in the input random graph.

Abstract

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.

Counting Stars is Constant-Degree Optimal For Detecting Any Planted Subgraph

TL;DR

The power of constant degree polynomials for the detection task is characterized and it is proved that an optimal constant degree polynomial is always given by simply counting stars in the input random graph.

Abstract

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.
Paper Structure (20 sections, 20 theorems, 127 equations, 8 figures)

This paper contains 20 sections, 20 theorems, 127 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Main Result: Optimality of Star Counts
  4. A simple criterion using only the degree profile of $H$
  5. Characterization of the Optimal Test
  6. Applications
  7. Tightness of Theorem \ref{['thm:main']}
  8. Getting Started for Proof Sections
  9. Proof of Main Theorem
  10. Proof Strategy
  11. Key Lemmas
  12. Auxillary Lemmas
  13. Proof of Theorem \ref{['thm:main']}: Putting it all together
  14. Proof of Key Lemmas
  15. Proof of Main Corollary
  16. ...and 5 more sections

Key Result

Theorem 2.4

Suppose $H=H_n$ is an arbitrary subgraph, $D = O(1)$ and $p = \Omega(1)$. Then, the following holds for testing $\mathbb{P}$ and $\mathbb{Q}$ in the planted subgraph detection task corresponding to planting a copy of $H$ per Definition dfn:planted.

Figures (8)

  • Figure 1: A generic star graph on the left, and the star graph $K_{1,5}$ on the right.
  • Figure 2: Phase-transition diagram characterized by Theorem \ref{['thm:degree-characterization']}. The x-axis is the maximum degree $\Delta$ and the y-axis is the number of edges $m$ of subgraph $H$. CD is an abbreviation of constant degree.
  • Figure 3: Phase-transition diagram for $\mathrm{PDS}(k,p,q)$, with $k = n^{\beta}$, $p = 1 - n^{-\gamma}$ and $q = n^{\alpha}$.
  • Figure 4: Phase-transition diagram of planted bipartite clique, where $H = K_{a,b}$ with $a \geq b$. We use parameter configuration $p = 1 - n^{-\gamma}$, $a = n^\alpha$ and $b = n^{\beta}$.
  • Figure 5: An example of $4$ pairs of copies of $\mathbf{S}_1$ and $\mathbf{S}_2$ numbered by $(1), (2), (3), (4)$, where the labellings of copies of $\mathbf{S}_1$ are marked with blue letters, and the labellings of copies of $\mathbf{S}_2$ are marked with red letters.
  • ...and 3 more figures

Theorems & Definitions (58)

  • Definition 1.1: Planted subgraph detection task
  • Definition 2.1: Strong separation
  • Definition 2.2: (Low-degree) advantage
  • Definition 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Corollary 2.8
  • Definition 3.1
  • ...and 48 more