In many real-world networks, data on the edges evolve in continuous time, naturally motivating representations based on point processes. Heterogeneity in edge types further gives rise to multiplex network point processes. In this work, we propose a model for multiplex network data observed in continuous-time. We establish two-to-infinity norm consistency and asymptotic normality for spectral-embedding-based estimation of the model parameters as both network size and time resolution increase. Drawing inspiration from random dot product graph models, each edge intensity is expressed as the inner product of two low-dimensional latent positions: one dynamic and layer-agnostic, the other static and layer-dependent. These latent positions constitute the primary objects of inference, which is conducted via spectral embedding methods. Our theoretical results are established under a histogram estimator of the network intensities and provide justification for applying a doubly unfolded adjacency spectral embedding method for estimation. Simulations and real-data analyses demonstrate the effectiveness of the proposed model and inference procedure.

翻译：在许多现实世界网络中，边上的数据在连续时间内演化，这自然催生了基于点过程的表示方法。边类型的异质性进一步导致了多重网络点过程。在本工作中，我们提出了一个用于连续时间观测的多重网络数据模型。随着网络规模和时间分辨率的增加，我们建立了基于光谱嵌入的模型参数估计的2-∞范数一致性和渐近正态性。受随机点积图模型的启发，每条边的强度被表示为两个低维潜在位置的内积：一个是动态且与层无关的，另一个是静态且依赖于层的。这些潜在位置构成了推断的主要对象，推断通过光谱嵌入方法进行。我们的理论结果是在网络强度的直方图估计量下建立的，并为应用双重展开邻接光谱嵌入方法进行估计提供了依据。模拟和实际数据分析证明了所提模型和推断程序的有效性。