Finding the Balance Rate of Uncertain Signed Graphs

2026-05-17

AI Summary

Real-world social, political, and biological networks are often modeled as signed graphs with uncertain edges, yet quantifying their degree of balance remains challenging. This work introduces, for the first time, the notion of a “balance ratio” to measure the balance of such graphs, proves that its exact computation is NP-hard, and proposes the first scalable approximation algorithm. By integrating graph decomposition with Rao–Blackwellized spanning-tree sampling, the algorithm achieves near-linear time complexity. Furthermore, an asymptotically valid confidence interval is constructed using the Delta method. Extensive experiments on multiple real-world datasets demonstrate the method’s efficiency and accuracy, enabling large-scale balance analysis of uncertain signed graphs.

Abstract

Signed graphs are widely used to analyze complex systems such as social, political, and biological networks. The notion of balance, a key concept of signed graphs, reflects the stability of relationships. While it has been extensively studied in deterministic graphs, real-world networks often exhibit uncertainty in their connections, which traditional approaches struggle to address. To bridge this gap, we introduce the concept of balance rate, a metric for quantifying the degree of balance in uncertain signed graphs, and prove that computing it exactly is NP-hard, motivating the need for efficient estimation methods. We propose a novel Rao-Blackwellized spanning-tree estimator that achieves near-linear time complexity per sample by leveraging graph decomposition and structural properties. We also construct asymptotically justified confidence intervals using the Delta method. Experiments on real-world datasets demonstrate the efficiency and effectiveness of our approach, enabling scalable balance analysis in uncertain signed graphs.

Problem

• signed graphs
• balance
• uncertainty
• balance rate
• NP-hard

Innovation

• balance rate
• uncertain signed graphs
• Rao-Blackwellization
• spanning-tree estimator
• Delta method