RTD-Lite: Scalable Topological Analysis for Comparing Weighted Graphs in Learning Tasks

2025-03-14

AI Summary

Existing topological alignment methods for large-scale weighted graphs suffer from high computational complexity and incompatibility with end-to-end learning. Method: We propose the first scalable topological distance algorithm with both time and space complexity of O(n²). Our approach avoids persistent homology computation by constructing an auxiliary graph, extracting its minimum spanning tree, hierarchically tracking connected components, and integrating multi-scale differences in connectivity and clustering structure. A differentiable loss function is designed to enable topology-aware embedding learning. Contribution/Results: On synthetic and real-world benchmarks, our method achieves up to数十-fold speedup over state-of-the-art alternatives. When integrated as a regularizer into neural networks, it significantly improves topological fidelity and structural preservation in dimensionality reduction and graph representation learning tasks.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Topological methods for comparing weighted graphs are valuable in various learning tasks but often suffer from computational inefficiency on large datasets. We introduce RTD-Lite, a scalable algorithm that efficiently compares topological features, specifically connectivity or cluster structures at arbitrary scales, of two weighted graphs with one-to-one correspondence between vertices. Using minimal spanning trees in auxiliary graphs, RTD-Lite captures topological discrepancies with $O(n^2)$ time and memory complexity. This efficiency enables its application in tasks like dimensionality reduction and neural network training. Experiments on synthetic and real-world datasets demonstrate that RTD-Lite effectively identifies topological differences while significantly reducing computation time compared to existing methods. Moreover, integrating RTD-Lite into neural network training as a loss function component enhances the preservation of topological structures in learned representations. Our code is publicly available at https://github.com/ArGintum/RTD-Lite

Problem

• Scalable topological analysis for comparing weighted graphs
• Efficiently identifies topological differences in large datasets
• Enhances neural network training by preserving topological structures

Innovation

• RTD-Lite: scalable topological graph comparison algorithm
• Uses minimal spanning trees for efficient discrepancy detection
• Enhances neural network training with topological loss function