Jaccard-constrained dense subgraph discovery

TL;DR

This paper introduces a novel approach for discovering dense, temporally consistent subgraphs in evolving networks by maximizing a combined density and Jaccard similarity measure, with proven NP-hardness and efficient algorithms.

Contribution

It formulates the problem of Jaccard-constrained dense subgraph discovery in temporal networks, proves its NP-hardness, and proposes efficient iterative and greedy algorithms for practical solution.

Findings

Algorithms are efficient and scalable.

Methods successfully find ground truth in synthetic data.

Real-world datasets yield interpretable dense subgraphs.

Abstract

Finding dense subgraphs is a core problem in graph mining with many applications in diverse domains. At the same time many real-world networks vary over time, that is, the dataset can be represented as a sequence of graph snapshots. Hence, it is natural to consider the question of finding dense subgraphs in a temporal network that are allowed to vary over time to a certain degree. In this paper, we search for dense subgraphs that have large pairwise Jaccard similarity coefficients. More formally, given a set of graph snapshots and a weight $\lambda$, we find a collection of dense subgraphs such that the sum of densities of the induced subgraphs plus the sum of Jaccard indices, weighted by $\lambda$, is maximized. We prove that this problem is NP-hard. To discover dense subgraphs with good objective value, we present an iterative algorithm which runs in $\mathcal{O}(n^2k^2 + m \log n + k^3 n)$ time per single iteration, and a greedy algorithm which runs in $\mathcal{O}(n^2k^2 + m \log n + k^3 n)$ time, where $k$ is the length of the graph sequence and $n$ and $m$ denote number of nodes and total number of edges respectively. We show experimentally that our algorithms are efficient, they can find ground truth in synthetic datasets and provide interpretable results from real-world datasets. Finally, we present a case study that shows the usefulness of our problem.