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

2025-04-30

AI Summary

Computing Forman-Ricci curvature (FRC) locally and efficiently in Vietoris–Rips (VR) complexes remains challenging due to its conventional dependence on global combinatorial structure. Method: We introduce, for the first time, a decomposable explicit formula for FRC and establish a rigorous set-theoretic proof framework, enabling fully localized computation of FRC on VR complexes with linear time complexity. Our approach integrates discrete differential geometry, combinatorial topology, and set-theoretic analysis to overcome the global dependency bottleneck. Contribution/Results: Experiments demonstrate substantial gains in efficiency for extracting geometric features from high-dimensional networks, uncovering latent geometric structures underlying statistical patterns in data. Moreover, our method identifies critical geometric phase transitions—previously overlooked by conventional approaches—in multiple real-world classification tasks. This work establishes a novel geometric foundation for graph neural networks and topological machine learning.

Technology Category

Machine Learning: Learning with ManifoldsKnowledge Representation and Reasoning: Geometric, Spatial, and Temporal ReasoningComputer Vision: Other Foundations of Computer Vision

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methodsSocial Networks and Social Media: Computational social science

Abstract

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, enabling local computations of FRC. Second, we construct a set-theoretical proof enabling an efficient algorithm for the local computation of FRC in Vietoris-Rips (VR) complexes.Strikingly, this approach reveals critical information and geometric insights often overlooked by conventional classification techniques. Our findings open new avenues for geometric computations in VR complexes and highlight an essential yet under-explored aspect of data classification: the geometry underpinning statistical patterns.

Problem

• Decompose Forman-Ricci curvature for local computations in networks
• Develop efficient algorithm for curvature in Vietoris-Rips complexes
• Reveal geometric insights overlooked by traditional classification methods

Innovation

• Decomposes Forman-Ricci curvature for local computations
• Efficient algorithm for curvature in Vietoris-Rips complexes
• Reveals geometric insights overlooked by traditional methods