Network data are often sampled with auxiliary information or collected through the observation of a complex system over time, leading to multiple network snapshots indexed by a continuous variable. Many methods in statistical network analysis are traditionally designed for a single network, and can be applied to an aggregated network in this setting, but that approach can miss important functional structure. Here we develop an approach to estimating the expected network explicitly as a function of a continuous index, be it time or another indexing variable. We parameterize the network expectation through low dimensional latent processes, whose components we represent with a fixed, finite-dimensional functional basis. We derive a gradient descent estimation algorithm, establish theoretical guarantees for recovery of the low dimensional structure, compare our method to competitors, and apply it to a data set of international political interactions over time, showing our proposed method to adapt well to data, outperform competitors, and provide interpretable and meaningful results.

翻译：网络数据常伴随辅助信息采样，或通过复杂系统随时间观测而收集，从而产生由连续变量索引的多个网络快照。传统统计网络分析中的许多方法通常针对单一网络设计，在此场景下可应用于聚合网络，但该方法可能遗漏重要的函数型结构。本文提出一种方法，将期望网络显式估计为连续索引变量（时间或其他索引变量）的函数。我们通过低维潜在过程对网络期望进行参数化，其分量采用固定有限维函数基表示。我们推导了梯度下降估计算法，建立了低维结构恢复的理论保证，将本方法与现有方法进行比较，并将其应用于随时间演变的国际政治互动数据集，结果表明所提方法能良好适应数据、优于现有方法，并提供可解释且有意义的结果。