An Efficient and Exact Algorithm for Locally h-Clique Densest Subgraph Discovery
Xiaojia Xu, Haoyu Liu, Xiaowei Lv, Yongcai Wang, Deying Li
TL;DR
This work tackles locally $h$-clique densest subgraph discovery (L$h$CDS) by introducing IPPV, an exact iterative pipeline that jointly estimates $h$-clique compact numbers and detects top-$k$ non-overlapping locally dense subgraphs. The method combines a convex-programming-based bound derivation, tentative graph decomposition, and both basic and fast flow-based verification to guarantee correctness while remaining scalable. Key innovations include $(k,igpsi_h)$-core based initial bounds, stable $h$-clique groups for tighter pruning, and a reduced-flow verification scheme that preserves exactness. The framework is extended to locally pattern-based densest subgraphs (LhxPDS) via a densest supermodular set decomposition perspective, with extensive experiments on real networks showing IPPV’s efficiency, accuracy, and ability to uncover diverse, near-clique communities across varying $h$.
Abstract
Detecting locally, non-overlapping, near-clique densest subgraphs is a crucial problem for community search in social networks. As a vertex may be involved in multiple overlapped local cliques, detecting locally densest sub-structures considering h-clique density, i.e., locally h-clique densest subgraph (LhCDS) attracts great interests. This paper investigates the LhCDS detection problem and proposes an efficient and exact algorithm to list the top-k non-overlapping, locally h-clique dense, and compact subgraphs. We in particular jointly consider h-clique compact number and LhCDS and design a new "Iterative Propose-Prune-and-Verify" pipeline (IPPV) for top-k LhCDS detection. (1) In the proposal part, we derive initial bounds for h-clique compact numbers; prove the validity, and extend a convex programming method to tighten the bounds for proposing LhCDS candidates without missing any. (2) Then a tentative graph decomposition method is proposed to solve the challenging case where a clique spans multiple subgraphs in graph decomposition. (3) To deal with the verification difficulty, both a basic and a fast verification method are proposed, where the fast method constructs a smaller-scale flow network to improve efficiency while preserving the verification correctness. The verified LhCDSes are returned, while the candidates that remained unsure reenter the IPPV pipeline. (4) We further extend the proposed methods to locally more general pattern densest subgraph detection problems. We prove the exactness and low complexity of the proposed algorithm. Extensive experiments on real datasets show the effectiveness and high efficiency of IPPV.