Dense Subgraph Discovery Meets Strong Triadic Closure

TL;DR

This paper introduces a novel approach to dense subgraph discovery that incorporates strong triadic closure constraints, providing algorithms and heuristics to optimize subgraph density considering strong and weak edges.

Contribution

The paper formulates a new dense subgraph problem with STC constraints, proposes an exact ILP solution and heuristics, and demonstrates their effectiveness on synthetic and real data.

Findings

Exact ILP algorithm finds ground truth in synthetic data.

Heuristics run efficiently on real-world datasets.

The problem generalizes densest subgraph and maximum clique problems.

Abstract

Finding dense subgraphs is a core problem with numerous graph mining applications such as community detection in social networks and anomaly detection. However, in many real-world networks connections are not equal. One way to label edges as either strong or weak is to use strong triadic closure~(STC). Here, if one node connects strongly with two other nodes, then those two nodes should be connected at least with a weak edge. STC-labelings are not unique and finding the maximum number of strong edges is NP-hard. In this paper, we apply STC to dense subgraph discovery. More formally, our score for a given subgraph is the ratio between the sum of the number of strong edges and weak edges, weighted by a user parameter $\lambda$, and the number of nodes of the subgraph. Our goal is to find a subgraph and an STC-labeling maximizing the score. We show that for $\lambda = 1$, our problem is equivalent to finding the densest subgraph, while for $\lambda = 0$, our problem is equivalent to finding the largest clique, making our problem NP-hard. We propose an exact algorithm based on integer linear programming and four practical polynomial-time heuristics. We present an extensive experimental study that shows that our algorithms can find the ground truth in synthetic datasets and run efficiently in real-world datasets.