Fully Dynamic Algorithm for Top-$k$ Densest Subgraphs

TL;DR

This paper introduces a fully dynamic, efficient algorithm for maintaining top-$k$ densest subgraphs in large, evolving graphs, significantly outperforming existing methods in speed and density quality.

Contribution

The paper presents a novel fully-dynamic algorithm that updates top-$k$ densest subgraphs efficiently using local updates, unlike previous methods that require global re-computation.

Findings

The algorithm often finds denser subgraphs than competitors.

It achieves up to five orders of magnitude speedup.

Theoretical analysis supports its efficiency and effectiveness.

Abstract

Given a large graph, the densest-subgraph problem asks to find a subgraph with maximum average degree. When considering the top-$k$ version of this problem, a na\"ive solution is to iteratively find the densest subgraph and remove it in each iteration. However, such a solution is impractical due to high processing cost. The problem is further complicated when dealing with dynamic graphs, since adding or removing an edge requires re-running the algorithm. In this paper, we study the top-$k$ densest-subgraph problem in the sliding-window model and propose an efficient fully-dynamic algorithm. The input of our algorithm consists of an edge stream, and the goal is to find the node-disjoint subgraphs that maximize the sum of their densities. In contrast to existing state-of-the-art solutions that require iterating over the entire graph upon any update, our algorithm profits from the observation that updates only affect a limited region of the graph. Therefore, the top-$k$ densest subgraphs are maintained by only applying local updates. We provide a theoretical analysis of the proposed algorithm and show empirically that the algorithm often generates denser subgraphs than state-of-the-art competitors. Experiments show an improvement in efficiency of up to five orders of magnitude compared to state-of-the-art solutions.