Unveiling Densest Multilayer Subgraphs via Greedy Peeling

The densest subgraphs in multilayer (ML) graphs unveil intricate relationships that are missed by simple graph representations, offering profound insights and applications across diverse domains. In this paper, we present a layer-oriented view of existing density measures for ML graphs and highlight their problems in identifying the densest subgraphs under the layer-oriented densities, including inefficiency, poor approximation ratios, and the lack of a unified algorithmic framework. In light of this, we introduce a new family of vertex-oriented density measures called generalized density. The two parameters $q$ and $p$ allow the generalized density to flexibly adjust its focus in the density evaluation. We investigate the problem of finding the ML subgraph that maximizes the generalized density and show that the problem can be solved using a unified greedy vertex peeling framework with strong approximation guarantees for half of the $(q, p)$ parameter space. Specifically, for four regimes of $(q, p)$, we design tailored vertex-peeling strategies that lead to approximation algorithms with provable approximation ratios and precise time complexity bounds. We also develop a highly efficient implementation that reduces the execution time of greedy peeling to near-linear time for two of the four explored regimes of $(q, p)$. Extensive experiments on ten real-world ML graphs reveal that our generalized density and greedy peeling algorithms can effectively uncover different types of dense ML subgraphs in large-scale ML graphs.