Finding the Balance Rate 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.
Cite
@article{arxiv.2605.17492,
title = {Finding the Balance Rate of Uncertain Signed Graphs},
author = {Zeyu Wang and Kudria Sergei and Jingbang Chen and Jiawei Chen and Xinyu Wang and Xiaodong Luo and Can Wang},
journal= {arXiv preprint arXiv:2605.17492},
year = {2026}
}