Finding the Balance Rate of Uncertain Signed Graphs

TL;DR

This paper introduces a new metric called balance rate for uncertain signed graphs, proves its computational hardness, and proposes an efficient estimator with confidence intervals, enabling scalable analysis of complex networks.

Contribution

It defines the balance rate for uncertain signed graphs, proves NP-hardness of exact computation, and develops a Rao-Blackwellized estimator with confidence intervals for practical analysis.

Findings

Estimator achieves near-linear time complexity per sample.

Experiments demonstrate the method's efficiency and effectiveness.

The approach enables scalable balance analysis in real-world datasets.

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.