Multilayer networks are a network data structure in which elements in a population of interest have multiple modes of interaction or relation, represented by multiple networks called layers. We propose a novel class of models for cross-layer dependence in multilayer networks, aiming to learn how interactions in one or more layers may influence interactions in other layers of the multilayer network, by developing a class of network separable models which separate the network formation process from the layer formation process. In our framework, we are able to extend existing single layer network models to a multilayer network model with cross-layer dependence. We establish non-asymptotic bounds on the error of estimators and demonstrate rates of convergence for both maximum likelihood estimators and maximum pseudolikelihood estimators in scenarios of increasing parameter dimension. We additionally establish non-asymptotic error bounds on the multivariate normal approximation and elaborate a method for model selection which controls the false discovery rate. We conduct simulation studies which demonstrate that our framework and method work well in realistic settings which might be encountered in applications. Lastly, we illustrate the utility of our method through an application to the Lazega lawyers network.

翻译：多层网络是一种网络数据结构，其中目标群体中的元素具有多种交互或关系模式，这些模式由称为层的多个网络表示。我们提出了一类用于多层网络中跨层依赖的新型模型，旨在通过开发一类网络可分离模型（将网络形成过程与层形成过程分离），学习一个或多个层中的交互如何影响多层网络中其他层的交互。在我们的框架中，我们能够将现有的单层网络模型扩展到具有跨层依赖的多层网络模型。我们建立了估计器误差的非渐近界，并展示了在参数维度增加的情况下，最大似然估计量和最大伪似然估计量的收敛速率。此外，我们建立了多元正态近似的非渐近误差界，并阐述了一种控制错误发现率的模型选择方法。我们进行的模拟研究证明，我们的框架和方法在实际应用中可能遇到的情境下表现良好。最后，我们通过将方法应用于Lazega律师网络，展示了其实用性。