Alternative set-theoretical algorithms for efficient computations of cliques in Vietoris-Rips complexes

TL;DR

The paper tackles the NP-hard problem of identifying cliques in dense networks and Vietoris-Rips complexes by diagnosing bottlenecks in classic algorithms and introducing two set-theoretical, dynamic-neighborhood strategies. The proposed methods—Evolving Neighbourhood Boundary and Evolving Multi-Layer Node-Neighbourhood—rely on discretized set operations and real-time neighborhood maintenance to efficiently extend cliques to higher dimensions. Extensive benchmarks against NetworkX and Gudhi across complete graphs, point-cloud-derived VR complexes, and unweighted real networks demonstrate superior memory efficiency and faster runtimes, enabling scalable computation of higher-order structures. These results have practical implications for topological data analysis tasks and related geometric-inference applications in neuroscience, biology, and network science, where efficient higher-order clique computation is essential.

Abstract

Identifying cliques in dense networks remains a formidable challenge, even with significant advances in computational power and methodologies. To tackle this, numerous algorithms have been developed to optimize time and memory usage, implemented across diverse programming languages. Yet, the inherent NP-completeness of the problem continues to hinder performance on large-scale networks, often resulting in memory leaks and slow computations. In the present study, we critically evaluate classic algorithms to pinpoint computational bottlenecks and introduce novel set-theoretical approaches tailored for network clique computation. Our proposed algorithms are rigorously implemented and benchmarked against existing Python-based solutions, demonstrating superior performance. These findings underscore the potential of set-theoretical techniques to drive substantial performance gains in network analysis.

Figures (6)

• Figure 1: Example of cliques of dimensions $d \in \{0,1,2,3\}$, and the construction of a Vietories-Rips complex for different distance parameters $\varepsilon$. Higher dimensional cliques appear as the cutoff distance increases. For instance, tetrahedron (blue faces), pentahedron (orange faces) and hexahedron (green faces).
• Figure 2: Finding cliques in a simple network as the neighbourhood is updated. The idea of the algorithms relies on the extension of the iterated edges to higher-order simplices. In this case, the ordering of edges is $(1,2),(2,3),(1,5),(3,5),(1,3),(3,4),(4,5)$. One new edge can give rise to several cliques, for instance, edge $(1,3)$ at iteration 8, which provides clique expansions at iterations $9$, $10$ (two triangles - not visible) and $11$ (one tetrahedron).
• Figure 3: Benchmark comparison between algorithms for finding cliques in VR-complexes from complete networks ($d_{\max}$=4).
• Figure 4: Benchmark comparison between algorithms for finding cliques in VR-complexes from point cloud data ((ism_test_horse.pcd)).
• Figure 5: Benchmark comparison between algorithms for finding cliques in VR-complexes from point cloud data ( ism_test_cat.pcd).