With the emergence of dynamic multiplex networks, corresponding to graphs where multiple types of edges evolve over time, a key inferential task is to determine whether the layers associated with different edge types differ in their connectivity. In this work, we introduce a hypothesis testing framework, under a latent space network model, for assessing whether the layers share a common latent representation. The method we propose extends previous literature related to the problem of pairwise testing for random graphs and enables global testing of differences between layers in multiplex graphs. While we introduce the method as a test for differences between layers, it can easily be adapted to test for differences between time points. We construct a test statistic based on a spectral embedding of an unfolded representation of the graph adjacency matrices and demonstrate its ability to detect differences across layers in the asymptotic regime where the number of nodes in each graph tends to infinity. The finite-sample properties of the test are empirically demonstrated by assessing its performance on both simulated data and a biological dataset describing the neural activity of larval Drosophila.

翻译：随着动态多重网络的出现——对应于多种边类型随时间演化的图结构，一个关键的推断任务是确定与不同边类型相关的层是否在连通性上存在差异。在本工作中，我们引入了一个假设检验框架，基于潜在空间网络模型，用于评估各层是否共享一个共同的潜在表示。我们提出的方法扩展了先前关于随机图成对检验问题的文献，并支持对多重网络中层间差异进行全局检验。尽管我们引入该方法作为层间差异的检验，但它可以轻松调整用于检验时间点间的差异。我们基于图邻接矩阵展开表示的谱嵌入构建了一个检验统计量，并证明了在节点数趋于无穷的渐近条件下其检测层间差异的能力。通过评估该方法在模拟数据和描述果蝇幼虫神经活动的生物数据集上的表现，我们实证展示了该检验在有限样本下的性质。