Multilayer (or multiple) networks are widely used to represent diverse patterns of relationships among objects in increasingly complex real-world systems. Identifying a common invariant subspace across network layers has become an active area of research, as such a subspace can filter out layer-specific noise, facilitate cross-network comparisons, reduce dimensionality, and extract shared structural features of scientific interest. One statistical approach to detecting a common subspace is hypothesis testing, which evaluates whether the observed networks share a common latent structure. In this paper, we propose an empirical likelihood (EL) based test for this purpose. The null hypothesis states that all network layers share the same invariant subspace, whereas under the alternative hypothesis at least two layers differ in their subspaces. We study the asymptotic behavior of the proposed test via Monte Carlo approximation and assess its finite-sample performance through extensive simulations. The simulation results demonstrate that the proposed method achieves satisfactory size and power, and its practical utility is further illustrated with a real-data application.

翻译：多层（或多重）网络被广泛用于表示日益复杂的现实世界系统中对象间多样化的关系模式。识别跨网络层的公共不变子空间已成为一个活跃的研究领域，因为此类子空间能够滤除层特异性噪声、促进跨网络比较、降低维度并提取具有科学意义的共享结构特征。检测公共子空间的一种统计方法是假设检验，该方法评估观测网络是否共享共同的潜在结构。本文为此提出一种基于经验似然（EL）的检验方法。原假设设定所有网络层共享相同的不变子空间，而在备择假设下至少有两个层的子空间存在差异。我们通过蒙特卡洛近似研究了所提检验的渐近性质，并通过大量仿真评估了其有限样本性能。仿真结果表明，所提方法取得了令人满意的检验水平与功效，其实际效用通过真实数据应用得到了进一步展示。