Practical and Performant Enhancements for Maximization of Algebraic Connectivity

2025-11-11

AI Summary

Existing graph sparsification methods (e.g., MAC) for large-scale, long-horizon graph state estimation suffer from poor scalability and real-time capability due to reliance on manually predefined edge sets, high computational overhead, and difficulty in online deployment. This work proposes an efficient graph sparsification framework grounded in algebraic connectivity maximization. We design a dedicated optimization solver incorporating an adaptive step-size strategy and an automatic connectivity preservation mechanism—eliminating the need for manual edge selection throughout the process. Experiments demonstrate that our method achieves an average 2× speedup in runtime, significantly improves convergence rate and solution quality, and maintains strong graph connectivity while enabling online, real-time state estimation. The framework establishes a scalable, robust, and practical paradigm for long-horizon graph state estimation.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Intelligent Robots: State Estimation
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Long-term state estimation over graphs remains challenging as current graph estimation methods scale poorly on large, long-term graphs. To address this, our work advances a current state-of-the-art graph sparsification algorithm, maximizing algebraic connectivity (MAC). MAC is a sparsification method that preserves estimation performance by maximizing the algebraic connectivity, a spectral graph property that is directly connected to the estimation error. Unfortunately, MAC remains computationally prohibitive for online use and requires users to manually pre-specify a connectivity-preserving edge set. Our contributions close these gaps along three complementary fronts: we develop a specialized solver for algebraic connectivity that yields an average 2x runtime speedup; we investigate advanced step size strategies for MAC's optimization procedure to enhance both convergence speed and solution quality; and we propose automatic schemes that guarantee graph connectivity without requiring manual specification of edges. Together, these contributions make MAC more scalable, reliable, and suitable for real-time estimation applications.

Problem

• Enhancing graph sparsification scalability for large long-term state estimation
• Accelerating algebraic connectivity optimization with specialized solvers and strategies
• Automating connectivity preservation to eliminate manual edge specification requirements

Innovation

• Specialized solver for algebraic connectivity speedup
• Advanced step size strategies to enhance convergence
• Automatic schemes guaranteeing graph connectivity without manual edges