Counting stars is constant-degree optimal for detecting any planted subgraph

We study the computational limits of the following general hypothesis testing problem. Let H=H_{n} be an arbitrary undirected graph. 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 constant degree polynomials for the detection task, when H=H_{n} is any arbitrary graph and for any p=\Omega(1) . Perhaps surprisingly, we prove that an optimal constant degree polynomial is always given by simply 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.