Summary
The planted clique problem asks how to find a hidden group of friends (a clique) inside a large network where connections are mostly random. It is believed that the smallest hidden group that can be found efficiently has size proportional to the square root of the network size. The authors introduce new clever algorithms that use color-coding and fast matrix operations to detect and recover these hidden groups faster, even when the group is smaller than previously possible. Their work improves the balance between how quickly the algorithms run and how small the hidden clique can be. These results build on and extend previous methods to better spot hidden structures in complex data.
What this means in practice
- •For data scientists: Detect smaller tightly connected groups hidden in large random networks more efficiently, improving community detection in social and biological data.
- •For network security teams: Identify covert clusters or anomalous tightly connected nodes within network traffic graphs faster, aiding threat detection in cybersecurity.
Tested on simulated data.
Abstract
In the planted clique problem, one observes either an Erdős--Rényi graph on $n$ vertices or such a graph with a clique added to $k = k(n)$ vertices, and seeks to detect or recover the clique. It is widely believed that $k = Θ(\sqrt{n})$ is the smallest clique size for which polynomial-time algorithms exist for these tasks. We develop new algorithms in this regime using color-coding to estimate signed subgraph counts, further accelerated with fast matrix multiplication. We first show that, for each $t \geq 1$, for $c(t)$ a constant associated to the order of growth of the number of connected graphs of treewidth at most $t$, cliques of size $k = λ\sqrt{n}$ planted in a random location with $λ> 1 / \sqrt{c(t)}$ can be detected and recovered in time $n^{t + 1 + o(1)}$. For instance, since $c(1) = e$, this recovers by counting signed trees the performance of the $\widetilde{O}(n^2)$-time message-passing algorithm of Deshpande--Montanari (2015) that succeeds when $λ> 1 / \sqrt{e} \approx 0.6066$. For $t \geq 3$, the exact value of $c(t)$ is not known, but lower bounds on it give a hierarchy of slower polynomial-time algorithms that succeed for smaller $λ$. We further show that the above algorithm for $t = 2$ can be implemented in time $n^{ω+ o(1)}$ for $ω$ the constant of square matrix multiplication and succeeds when $λ> 0.3320$; under the folklore conjecture that $ω= 2$, this runs in the nearly-linear time of the algorithm of Deshpande--Montanari while finding smaller cliques. Second, we show that the above algorithm for $t = 1$ can be combined with the boosting scheme of Alon--Krivelevich--Sudakov (1998) using rectangular matrix multiplication, giving improved runtimes for smaller $λ$. Taken together, our results achieve the best known tradeoff between runtime and signal strength $λ$.