Planted Dense Subgraphs in Dense Random Graphs Can Be Recovered using Graph-based Machine Learning

TL;DR

This paper introduces PYGON, a graph neural network-based algorithm capable of recovering various dense subgraphs in random graphs, regardless of their structure, achieving performance comparable to existing methods for certain subgraph sizes.

Contribution

The paper presents the first learning-based algorithm, PYGON, that can recover dense subgraphs of various structures in dense random graphs, extending beyond planted cliques.

Findings

PYGON can recover cliques of size Θ(√n).

PYGON can recover multiple types of dense subgraphs.

The algorithm works in both directed and undirected graphs.

Abstract

Multiple methods of finding the vertices belonging to a planted dense subgraph in a random dense $G(n, p)$ graph have been proposed, with an emphasis on planted cliques. Such methods can identify the planted subgraph in polynomial time, but are all limited to several subgraph structures. Here, we present PYGON, a graph neural network-based algorithm, which is insensitive to the structure of the planted subgraph. This is the first algorithm that uses advanced learning tools for recovering dense subgraphs. We show that PYGON can recover cliques of sizes $\Theta\left(\sqrt{n}\right)$, where $n$ is the size of the background graph, comparable with the state of the art. We also show that the same algorithm can recover multiple other planted subgraphs of size $\Theta\left(\sqrt{n}\right)$, in both directed and undirected graphs. We suggest a conjecture that no polynomial time PAC-learning algorithm can detect planted dense subgraphs with size smaller than $O\left(\sqrt{n}\right)$, even if in principle one could find dense subgraphs of logarithmic size.