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：网络数据常伴随辅助信息进行采样，或通过长期观测复杂系统收集，从而产生由连续变量索引的多个网络快照。许多统计网络分析方法传统上针对单一网络设计，在此类情境下可应用于聚合网络，但这种方法可能遗漏重要的函数结构。本文提出一种方法，将期望网络显式建模为连续索引（无论是时间还是其他索引变量）的函数。我们通过低维潜在过程对网络期望进行参数化，并用固定有限维函数基表示其分量。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，并将该方法与现有方法进行对比。最后将该方法应用于随时间变化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，并提供可解释且富有意义的结果。