Signed networks consist of both positive and negative relations, and structural balance theory provides an important conceptural framework for understanding their global tension structure. While existing statistical methods mainly focus on assessing empirical evidence of balance in a single observed network, many real-world signed relations evolve over time. This paper develops nonparametric inference for the population degree of structural balance at specified time points in dynamic signed networks, where the target time may or may not coincide with an observed snapshot. We consider a dynamic signed graphon model in which both edge formation and sign generation are governed by smoothly time-varying graphon functions. To exploit temporal smoothness, we construct a kernel-smoothed estimator that borrows information from snapshots near the target time point. Our theoretical analysis establishes a studentized inference procedure and a higher-order distributional approximation based on Edgeworth expansion, showing that temporal smoothing improves inference in sparse networks by reducing variance of observation noise, up to smoothing bias and time-discretization errors. We demonstrate the finite-sample performance and practical usefulness of the proposed method through extensive simulation studies and an application to a dynamic international relation network in political science.

翻译：有符号网络包含正关系和负关系，结构平衡理论为理解其全局张力结构提供了重要概念框架。现有统计方法主要关注评估单个观测网络中平衡性的经验证据，然而许多现实世界中的有符号关系随时间动态演变。本文针对动态有符号网络，在指定时间点（该目标时间可能对应也可能不对应观测快照时刻）上，发展了关于结构平衡总体程度的非参数推断方法。我们考虑一种动态有符号图模型，在该模型中，边形成与符号生成均由随时间平滑变化的图函数控制。为利用时间平滑性，我们构建了核平滑估计量，该估计量从目标时间点附近的快照中借用信息。理论分析建立了学生化推断过程，以及基于Edgeworth展开的高阶分布逼近，结果表明在稀疏网络中，时间平滑通过降低观测噪声方差（伴随平滑偏差与时间离散误差）改进了推断性能。通过广泛的模拟研究及对政治学中动态国际关系网络的实际应用，我们验证了所提方法的有限样本表现与实际效用。