Many real-world networks evolve dynamically over time and present different types of connections between nodes, often called layers. In this work, we propose a latent position model for these objects, called the dynamic multiplex random dot product graph (DMPRDPG), which uses an inner product between layer-specific and time-specific latent representations of the nodes to obtain edge probabilities. We further introduce a computationally efficient spectral embedding method for estimation of DMPRDPG parameters, called doubly unfolded adjacency spectral embedding (DUASE). The DUASE estimates are proved to be both consistent and asymptotically normally distributed. A key strength of our method is the encoding of time-specific node representations and layer-specific effects in separate latent spaces, which allows the model to capture complex behaviors while maintaining relatively low dimensionality. The embedding method we propose can also be efficiently used for subsequent inference tasks. In particular, we highlight the use of the ISOMAP algorithm in conjunction with DUASE as a way to efficiently capture trends and global changepoints within a network, and the use of DUASE for graph clustering. Applications on real-world networks describing geopolitical interactions between countries and financial news reporting demonstrate practical uses of our method.

翻译：许多现实世界的网络随时间动态演化，并在节点间呈现出不同类型的连接，这些连接通常被称为层。在本研究中，我们为这类对象提出了一种潜在位置模型，称为动态多重随机点积图（DMPRDPG），该模型利用节点在层特定和时间特定的潜在表示之间的内积来获得边概率。我们进一步引入了一种计算高效的谱嵌入方法，用于估计DMPRDPG参数，称为双重展开邻接谱嵌入（DUASE）。我们证明了DUASE估计量既具有一致性，又具有渐近正态分布性。我们方法的一个关键优势在于将时间特定的节点表示和层特定的效应编码在独立的潜在空间中，这使得模型能够在保持相对较低维度的同时捕捉复杂行为。我们提出的嵌入方法也能高效地用于后续的推理任务。特别地，我们强调了将ISOMAP算法与DUASE结合使用，作为有效捕捉网络内趋势和全局变化点的一种方式，以及将DUASE用于图聚类。在描述国家间地缘政治互动和金融新闻报道的真实世界网络上的应用，展示了我们方法的实际用途。