This study presents a novel representation learning model tailored for dynamic networks, which describes the continuously evolving relationships among individuals within a population. The problem is encapsulated in the dimension reduction topic of functional data analysis. With dynamic networks represented as matrix-valued functions, our objective is to map this functional data into a set of vector-valued functions in a lower-dimensional learning space. This space, defined as a metric functional space, allows for the calculation of norms and inner products. By constructing this learning space, we address (i) attribute learning, (ii) community detection, and (iii) link prediction and recovery of individual nodes in the dynamic network. Our model also accommodates asymmetric low-dimensional representations, enabling the separate study of nodes' regulatory and receiving roles. Crucially, the learning method accounts for the time-dependency of networks, ensuring that representations are continuous over time. The functional learning space we define naturally spans the time frame of the dynamic networks, facilitating both the inference of network links at specific time points and the reconstruction of the entire network structure without direct observation. We validated our approach through simulation studies and real-world applications. In simulations, we compared our methods link prediction performance to existing approaches under various data corruption scenarios. For real-world applications, we examined a dynamic social network replicated across six ant populations, demonstrating that our low-dimensional learning space effectively captures interactions, roles of individual ants, and the social evolution of the network. Our findings align with existing knowledge of ant colony behavior.

翻译：本研究提出了一种专为动态网络设计的表示学习模型，该模型描述了群体内个体间持续演化的关系。该问题可纳入函数型数据分析的降维研究范畴。通过将动态网络表示为矩阵值函数，我们的目标是将此类函数型数据映射到低维学习空间中的一组向量值函数。该学习空间被定义为度量函数空间，支持范数和内积的计算。通过构建此学习空间，我们解决了动态网络中的（i）属性学习，（ii）社区发现，以及（iii）链接预测与个体节点恢复问题。我们的模型还支持非对称的低维表示，从而能够分别研究节点的调控角色与接收角色。关键在于，该学习方法考虑了网络的时间依赖性，确保表示随时间连续变化。我们定义的函数学习空间自然地跨越动态网络的时间框架，既便于推断特定时间点的网络链接，也支持在无直接观测的情况下重建整个网络结构。我们通过仿真研究和实际应用验证了所提方法。在仿真中，我们在多种数据损坏场景下，将本方法的链接预测性能与现有方法进行了比较。在实际应用方面，我们分析了在六个蚂蚁群体中复现的动态社交网络，结果表明我们的低维学习空间能有效捕捉个体蚂蚁的交互行为、角色分工以及网络的社会演化规律。我们的发现与现有关于蚁群行为的知识相一致。