arXiv subjects

Danillo Barros de Souza

Publications and source records attributed to Danillo Barros de Souza.

3 recordsLinked to original sources

arXiv2026

Efficient Decomposition of Forman-Ricci Curvature on Vietoris-Rips Complexes and Data Applications

Discrete Forman-Ricci curvature (FRC) is an efficient tool that characterizes essential geometrical features and associated transitions of real-world networks, extending seamlessly to higher-dimensional computations in simplicial complexes. In this article, we provide two major advancements: First, we give a decomposition for FRC that inhently allows a local computations of FRC. Second, we construct a set-theoretical proof enabling an efficient algorithms for the local computation of FRC in Vietoris-Rips (VR) complexes. Our findings open new avenues for geometric computations in VR complexes and highlight an essential yet under-explored aspect of data analysis and visualisation: the geometry underpinning statistical patterns.

math.GT↗

arXiv2025

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

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.

math.CO↗

arXiv2024

Efficient set-theoretic algorithms for computing high-order Forman-Ricci curvature on abstract simplicial complexes

Forman-Ricci curvature (FRC) is a potent and powerful tool for analysing empirical networks, as the distribution of the curvature values can identify structural information that is not readily detected by other geometrical methods. Crucially, FRC captures higher-order structural information of clique complexes of a graph or Vietoris-Rips complexes, which is not readily accessible to alternative methods. However, existing FRC platforms are prohibitively computationally expensive. Therefore, herein we develop an efficient set-theoretic formulation for computing such high-order FRC in simplicial complexes. Significantly, our set theory representation reveals previous computational bottlenecks and also accelerates the computation of FRC. Finally, We provide a pseudo-code, a software implementation coined FastForman, as well as a benchmark comparison with alternative implementations. We envisage that FastForman will be used in Topological and Geometrical Data analysis for high-dimensional complex data sets. Moreover, our development paves the way for future generalisations towards efficient computations of FRC on cell complexes.

physics.data-an↗