A Flexible Latent Space Model for Multilayer Networks

has a distinct edge type defined through a type of relation-Entities often interact with each other through multiple types of relations, which can be represented as multilayer networks. Multilayer networks among the same set of nodes usually share common structures, while each layer can also possess its distinct node connecting behaviors. This paper proposes a flexible latent space model for multilayer networks for the purpose of capturing such characteristics. Specifically, the proposed model embeds each node with a latent vector shared among layers and a layer-specific effect for each layer; both elements together with a layerspecific connectivity matrix determine edge formations. To fit the model, we develop a projected gradient descent algorithm for efficient parameter estimation. We also establish theoretical properties of the maximum likelihood estimators and show that the upper bound of the common latent structure's estimation error is inversely proportional to the number of layers under mild conditions. The superior performance of the proposed model is demonstrated through simulation studies and applications to two real-world data examples.