Chordless cycle filtrations for dimensionality detection in complex networks via topological data analysis

2025-09-10

AI Summary

This study addresses the problem of accurately estimating the dimension of the latent hyperbolic geometry underlying complex networks—to uncover community structure complexity, improve greedy routing efficiency, and deepen understanding of systemic connectivity. We propose a graph-weighted filtering mechanism based on chordless cycles, integrated with persistent homology and algebraic topological feature extraction, to construct a generalizable, data-driven dimensional estimator. Furthermore, we design a lightweight neural network architecture trained exclusively on synthetic graph datasets, enabling zero-shot transfer to real-world networks without fine-tuning. Experiments demonstrate high-accuracy dimension prediction across diverse synthetic and empirical networks—including social, biological, and Internet topology graphs—significantly enhancing the quality of low-dimensional hyperbolic embeddings and navigability performance. The method exhibits strong robustness and cross-domain effectiveness, validating its applicability to heterogeneous network domains.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Learning with Manifolds
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Social media analysis through the lenses of networks

Abstract

Many complex networks, ranging from social to biological systems, exhibit structural patterns consistent with an underlying hyperbolic geometry. Revealing the dimensionality of this latent space can disentangle the structural complexity of communities, impact efficient network navigation, and fundamentally shape connectivity and system behavior. We introduce a novel topological data analysis weighting scheme for graphs, based on chordless cycles, aimed at estimating the dimensionality of networks in a data-driven way. We further show that the resulting descriptors can effectively estimate network dimensionality using a neural network architecture trained in a synthetic graph database constructed for this purpose, which does not need retraining to transfer effectively to real-world networks. Thus, by combining cycle-aware filtrations, algebraic topology, and machine learning, our approach provides a robust and effective method for uncovering the hidden geometry of complex networks and guiding accurate modeling and low-dimensional embedding.

Problem

• Estimating dimensionality of networks with hyperbolic geometry
• Revealing latent space dimensionality in complex network structures
• Uncovering hidden geometry in social and biological networks

Innovation

• Chordless cycle filtrations for topological graph weighting
• Neural network trained on synthetic graphs for dimensionality estimation
• Combines algebraic topology with machine learning for geometry detection