Multilayer networks have become increasingly ubiquitous across diverse scientific fields, ranging from social sciences and biology to economics and international relations. Despite their broad applications, the inferential theory for multilayer networks remains underdeveloped. In this paper, we propose a flexible latent space model for multilayer directed networks with various edge types, where each node is assigned with two latent positions capturing sending and receiving behaviors, and each layer has a connection matrix governing the layer-specific structure. Through nonlinear link functions, the proposed model represents the structure of a multilayer network as a tensor, which admits a Tucker low-rank decomposition. This formulation poses significant challenges on the estimation and statistical inference for the latent positions and connection matrices, where existing techniques are inapplicable. To tackle this issue, a novel unfolding and fusion method is developed to facilitate estimation. We establish both consistency and asymptotic normality for the estimated latent positions and connection matrices, which paves the way for statistical inference tasks in multilayer network applications, such as constructing confidence regions for the latent positions and testing whether two network layers share the same structure. We validate the proposed method through extensive simulation studies and demonstrate its practical utility on real-world data.

翻译：多层网络在社会科学、生物学、经济学与国际关系等众多科学领域中日益普遍。尽管其应用广泛，多层网络的推断理论仍发展不足。本文针对具有多种边类型的多层有向网络提出一种灵活的潜在空间模型，其中每个节点被赋予两个潜在位置以捕捉其发送与接收行为，且每层具有一个控制该层特定结构的连接矩阵。通过非线性链接函数，所提模型将多层网络的结构表示为一个张量，该张量允许进行Tucker低秩分解。这一形式化表达为潜在位置与连接矩阵的估计与统计推断带来了显著挑战，现有技术难以适用。为解决此问题，本文开发了一种新颖的展开与融合方法以促进估计。我们为估计的潜在位置与连接矩阵建立了一致性与渐近正态性，从而为多层网络应用中的统计推断任务（如构建潜在位置的置信区域、检验两个网络层是否共享相同结构）奠定了基础。我们通过大量模拟研究验证了所提方法的有效性，并在真实数据上展示了其实用价值。