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.

翻译：符号图被广泛用于分析社会、政治和生物网络等复杂系统。平衡作为符号图的关键概念，反映了关系的稳定性。尽管该概念在确定性图中已得到广泛研究，但现实世界中的网络常表现出连接的不确定性，而传统方法难以处理这一问题。为填补这一空白，我们引入了平衡率这一用于量化不确定符号图平衡程度的度量指标，并证明了精确计算该指标是NP难的，这促使我们需要开发高效的估计方法。我们提出了一种新颖的Rao-Blackwellized生成树估计器，通过利用图分解和结构性质，实现了每样本近线性时间复杂度。我们还使用Delta方法构建了渐近有效的置信区间。在真实数据集上的实验证明了我们方法的效率和有效性，从而实现了对不确定符号图的可扩展平衡分析。