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 dataset 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：网络数据通常通过辅助信息采样或随时间收集，导致多个以连续变量为索引的网络快照。统计网络分析中的许多方法传统上设计用于单个网络，可以在此设置中应用于聚合网络，但这种方法可能会忽略重要的功能结构。在这里，我们开发了一种估计预期网络的方法，将其明确地表示为连续指标的函数，无论是时间还是其他索引变量。我们通过低维潜在过程参数化网络预期，将其组件用具有固定的有限维函数基础表示。我们推导了一个梯度下降估计算法，建立了低维结构恢复的理论保证，比较了我们的方法与竞争对手，将其应用于随时间变化的国际政治互动数据集，显示我们提出的方法能够良好地适应数据，优于竞争对手，并提供可解释和有意义的结果。