Low-degree phase transitions for detecting a planted clique in sublinear time

TL;DR

The paper investigates the limits of sublinear-time detection of a planted clique in a random graph when the clique size is $k=\Theta(n^{1/2+\delta})$. By focusing on non-adaptive query models and restricting computation to $O(\log n)$-degree polynomials, it establishes a sharp phase transition: detection is possible with $|M|=\Theta(n^\gamma)$ only when $\gamma>3(1/2-\delta)$ and impossible when $\gamma<3(1/2-\delta)$. This yields a concrete boundary showing that the best known sublinear runtime $\widetilde{O}(n^{3(1/2-\delta)})$ for detection cannot be improved within the non-adaptive low-degree framework. The upper bound constructs a concrete non-adaptive mask and a low-degree test that strongly separates the planted and null graphs, matching the lower-bound threshold up to constants. Overall, the work advances understanding of computational limits in sublinear regimes and highlights the role of query design and low-degree methods in planted-clique detection.

Abstract

We consider the problem of detecting a planted clique of size $k$ in a random graph on $n$ vertices. When the size of the clique exceeds $Θ(\sqrt{n})$, polynomial-time algorithms for detection proliferate. We study faster -- namely, sublinear time -- algorithms in the high-signal regime when $k = Θ(n^{1/2 + δ})$, for some $δ> 0$. To this end, we consider algorithms that non-adaptively query a subset $M$ of entries of the adjacency matrix and then compute a low-degree polynomial function of the revealed entries. We prove a computational phase transition for this class of non-adaptive low-degree algorithms: under the scaling $\lvert M \rvert = Θ(n^γ)$, the clique can be detected when $γ> 3(1/2 - δ)$ but not when $γ< 3(1/2 - δ)$. As a result, the best known runtime for detecting a planted clique, $\widetilde{O}(n^{3(1/2-δ)})$, cannot be improved without looking beyond the non-adaptive low-degree class. Our proof of the lower bound -- based on bounding the conditional low-degree likelihood ratio -- reveals further structure in non-adaptive detection of a planted clique. Using (a bound on) the conditional low-degree likelihood ratio as a potential function, we show that for every non-adaptive query pattern, there is a highly structured query pattern of the same size that is at least as effective.

Figures (2)

• Figure 1: Phase diagram for detecting a clique of size $k = \Theta(n^{1/2+\delta})$ using $|M| = \Theta(n^\gamma)$ non-adaptive queries to the adjacency matrix. If arbitrary computation is allowed on the query results, detection is impossible in the red region and possible otherwise racz2019finding. If a low-degree test must be applied to the query results, our upper bound Theorem \ref{['thm:main']}(b) achieves detection in the green (easy) region; in this region, there is also an algorithm for clique detection whose runtime is dominated by the query complexity $\Theta(n^\gamma)$mardia2020finding. Below the line $\delta = 0$, it is known that low-degree polynomials cannot detect the clique, even if the entire input is revealed barak2019nearlyhopkins2018statistical. Our lower bound Theorem \ref{['thm:main']}(a) fills in the rest of the hard (yellow) region.
• Figure 2: Illustration of the $\mathsf{Donate}$ process (Algorithm \ref{['alg:donation']}) using notation from the proof of Lemma \ref{['lem: donation']}. Panel (a) shows the neighborhoods of $u$ (denoted as $N_u \cup \mathsf{Both}$) and $v$ (denoted as $N_v \cup \mathsf{Both}$) before $\mathsf{Donate}(v \rightarrow u)$. Panel (b) shows the result: Each neighbor of $v$ not originally connected to $u$ (i.e. in $N_v$) is disconnected from $v$ and connected to $u$ instead.

Theorems & Definitions (22)

• Definition 2.1: Mask and mask degree
• Definition 2.2: Erdős--Rényi distribution $G(n, 1/2)$ and masked Erdős--Rényi distribution $G(n, M)$
• Definition 2.3: Clique indicator distributions: ${\sf Clique}(n, k)$ and ${\sf Clique}(n, k, S)$
• Definition 2.4: Planted Clique distributions: $G(n, 1/2, k)$ and ${\sf G}(n,k,M)$
• Definition 2.5: Strong/weak separation
• Theorem 1
• Definition 3.1: Low-degree likelihood ratio upper bound: ${\sf LDUB}(n,M)$
• Definition 3.2: Conditional low-degree likelihood ratio upper bound: ${\sf Cond}(n,M,S)$
• Lemma 3.4: Donation cannot hurt low-degree algorithms
• proof