Provably Finding a Hidden Dense Submatrix among Many Planted Dense Submatrices via Convex Programming

TL;DR

This work addresses the densest m×n submatrix problem by extending convex nuclear-norm relaxations to settings with many planted dense blocks modeled by a heterogeneous stochastic block model. The authors develop a dual-certificate framework to prove sufficient conditions for exact recovery in both random and adversarial data, and they demonstrate phase-transition behavior via synthetic experiments and real networks. A practical ADMM-based solver solves the convex program efficiently, and empirical results on graphs including maximum cliques in benchmark networks and ASOIAF character networks corroborate the theory. The results unify and strengthen recovery guarantees for planted cliques, bicliques, and dense subgraphs, with implications for robust module detection in complex networks.

Abstract

We consider the densest submatrix problem, which seeks the submatrix of fixed size of a given binary matrix that contains the most nonzero entries. This problem is a natural generalization of fundamental problems in combinatorial optimization, e.g., the densest subgraph, maximum clique, and maximum edge biclique problems, and has wide application the study of complex networks. Much recent research has focused on the development of sufficient conditions for exact solution of the densest submatrix problem via convex relaxation. The vast majority of these sufficient conditions establish identification of the densest submatrix within a graph containing exactly one large dense submatrix hidden by noise. The assumptions of these underlying models are not observed in real-world networks, where the data may correspond to a matrix containing many dense submatrices of varying sizes. We extend and generalize these results to the more realistic setting where the input matrix may contain \emph{many} large dense subgraphs. Specifically, we establish sufficient conditions under which we can expect to solve the densest submatrix problem in polynomial time for random input matrices sampled from a generalization of the stochastic block model. Moreover, we also provide sufficient conditions for perfect recovery under a deterministic adversarial. Numerical experiments involving randomly generated problem instances and real-world collaboration and communication networks are used empirically to verify the theoretical phase-transitions to perfect recovery given by these sufficient conditions.

Figures (8)

• Figure 1: Examples of matrices sampled from planted submatrix and balanced planted submatrix models with heterogeneous probabilities $\{p_{rs}\}$.
• Figure 2: Examples of easy and hard to solve instances of the densest submatrix problem sampled in Experiment 1. Here, orange squares indicate the nodes $U_1$ in the graph $G$ with adjacency matrix $\boldsymbol{A}$ that induce the planted subgraph.
• Figure 3: Examples of easy, hard, and impossible graphs for recovery of the planted submatrix sampled in Experiment 2. Here, orange squares indicate the nodes $U_1$ in the graph $G$ with adjacency matrix $\boldsymbol{A}$ that induce the planted subgraph.
• Figure 4: Recovery counts for Experiments 1 and 2. Darker squares indicate more recoveries out of 10 trials for each $(q,m)$-pair.
• Figure 5: Adjacency matrices of the JAZZ, KARATE, DOLPHINS, and LESMIS networks and recovered maximum cliques.

Theorems & Definitions (19)

• Definition 1.1: Densest Submatrix Problem
• Remark 1: Interpretation of “with high probability”
• Definition 1.2: Planted Submatrix Model
• Definition 1.3: Balanced Planted Submatrix Model
• Theorem 1.1: Perfect Recovery in the Balanced Planted Submatrix Model
• Theorem 2.1
• Theorem 2.2
• Theorem 3.1
• Lemma 3.1
• Lemma 3.2