Guaranteed Recovery of Planted Cliques and Dense Subgraphs by Convex Relaxation

TL;DR

This paper presents a convex relaxation approach to exactly recover dense subgraphs, including planted cliques, in noisy graphs, demonstrating theoretical guarantees and numerical effectiveness.

Contribution

It introduces a convex relaxation method for dense subgraph recovery with proven guarantees even under adversarial and random noise conditions.

Findings

Exact recovery of planted cliques under certain conditions

Convex relaxation performs well in noisy and adversarial settings

Numerical simulations confirm the method's effectiveness

Abstract

We consider the problem of identifying the densest k-node subgraph in a given graph. We write this problem as an instance of rank-constrained cardinality minimization and then relax using the nuclear and 11 norms. Although the original combinatorial problem is NP-hard, we show that the densest k-subgraph can be recovered from the solution of our convex relaxation for certain program inputs. In particular, we establish exact recovery in the case that the input graph contains a single planted clique plus noise in the form of corrupted adjacency relationships. We consider two constructions for this noise. In the first, noise is introduced by an adversary deterministically deleting edges within the planted clique and placing diversionary edges. In the second, these edge corruptions are performed at random. Analogous recovery guarantees for identifying the densest subgraph of fixed size in a bipartite graph are also established, and results of numerical simulations for randomly generated graphs are included to demonstrate the efficacy of our algorithm.