Network Inference: Nonparametric estimation, bootstrap, and model diagnostics in sparse graphon models with vertex attributes

统计网络分析在经济学和社会科学以及其他研究领域都有着重要的作用。在经典教科书中，许多统计和概率方面的内容都被讨论过，网络的统计分析一直是一个活跃的研究领域。主要的困难来自于导致依赖的网络的关系结构，以及通常只观察到一个单一网络的事实。因此，与经典数据设置相比，渐近理论是一个更大的挑战。这适用于许多统计任务，但在估计、(自举)推理和模型诊断方面尤其适用。在这个方案中，我们通过采用局部依赖的观点来解决这些问题：网络中发生的远距离发生的事情可以被视为独立的，而强依赖只出现在局部。我们将解决具有顶点属性的网络数据模型中的推理问题，这允许例如对通过社交媒体连接的人与其工作场所信息的交互进行建模。我们关注的是网络是随机的模型。我们的第一个具体目标是开发一种新的图形模型，该模型适用于结合顶点属性的网络建模，并研究该模型中的非参数估计和参数估计。估计中的一个重点将是避免昂贵的离散优化并实现良好的收敛速度。然后，我们研究了不同的网络自举方法。虽然第一个自举程序将基于新的图形模型，并允许同时对网络和顶点属性进行重采样，但正在考虑的第二个自举程序是用于网络的块类型自举程序，它借用配置模型的想法来重新连接重新采样的子图，利用局部依赖的思想。我们将研究这两种方法在不同场景下的自举一致性。这些结果对于开发有效的推理方法至关重要。特别是，我们还将研究石墨型网络模型的拟合优度测试。我们将考虑基于前面提到的GRAPON模型的动态网络在线监测程序。最后，我们关注的是对具有网络干扰的观测数据的反事实处理效果的估计。因此，我们考虑到对等和溢出效应，并将重点放在改变网络结构的干预上，例如锁定，我们的目标是避免经常做出的集群干扰或独立集群的假设。在这个项目的所有工作包中，我们专注于两个方面：第一，发展严格的理论，第二，为正在考虑的特定模型提供可访问的结果和软件。一方面，这使得应用研究人员可以直接将我们的方法应用于他们的数据。另一方面，我们的结果可以作为相关模型进一步理论分析的起点。

Statistical network analysis plays an important role in economics and social sciences as well as other research fields. In classical textbooks, many statistical and probabilistic aspects were discussed, and the statistical analysis of networks has been an active area of research since then. The main difficulties arise from the relational structure of networks which induces dependence, and from the fact that typically only one single network is observed. Asymptotic theory is therefore a greater challenge compared to classical data setups. This is true for many statistical tasks, but holds in particular when it comes to estimation, (bootstrap) inference, and model diagnostics. In this proposal, we tackle these problems by adopting a local dependence viewpoint: what happens “far apart” in the network may be treated as independent, and strong dependence appears only locally. We will address inference problems in models for network data with vertex attributes, which allows, e.g., to model interactions of people connected by social media with information about their workplace. We focus on models in which the networks are random. Our first specific goal is to develop a new graphon model which is suitable for modeling networks together with vertex attributes and study nonparametric as well as parametric estimation in this model. A focus in the estimation will be to avoid expensive discrete optimization and to achieve good convergence rates. Then, we study different bootstrap methods for networks. While the first bootstrap will be based on the new graphon model and allows the simultaneous resampling of a network together with vertex attributes, the second bootstrap under consideration is a block-type bootstrap for networks that borrows ideas from the configuration model to rewire the resampled sub-graphs, exploiting the idea of local dependence. We will study bootstrap consistency of both methods under various scenarios. Such results are crucial to develop valid inference methods. In particular, we will also study goodness-of-fit testing for graphon-type network models. We will consider online monitoring procedures for dynamic networks based on the previously mentioned graphon models. Finally, we are concerned with the estimation of counter-factual treatment effects from observational data with network interference. Hereby, we allow for peer and spill-over effects and focus on interventions that change the network structure, e.g., a lockdown, and we aim to avoid the often-made assumption of clustered interference or independent clusters. Throughout all workpackages of this project, we focus on two aspects: Firstly, developing rigorous theory and, secondly, providing accessible results and software for the specific models under consideration. On the one hand, this enables applied researchers to directly apply our methods to their data. On the other hand, our results can be used as starting point for further theoretical analysis in related models.