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中发挥着重要作用。在经典教科书中，讨论了许多统计和概率方面的问题，从那时起，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性的网络的关系结构，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。这对于许多统计任务来说都是正确的，但在涉及到估计、（自举）推理和模型诊断时尤其如此。在本提案中，我们通过采用局部依赖的观点来解决这些问题：在网络中“远离”发生的事情可能被视为独立的，而强依赖仅在局部出现。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，通过社交媒体与有关其工作场所的信息连接的人们的交互建模。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适合与顶点属性一起建模的图形模型，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获得良好的收敛率。然后，我们研究了不同的网络自举方法。第一个bootstrap将基于新的graphon模型，并允许同时对带有顶点属性的网络进行重新采样，第二个bootstrap正在考虑中的是网络的块型bootstrap，它借用了配置模型的思想来重新连接重新采样的子图，利用了局部依赖的思想。我们将研究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。特别地，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图形模型的动态网络的在线监测程序。最后，我们关注的是网络干扰下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，并专注于改变网络结构的干预措施，例如，封锁，我们的目标是避免经常假设的集群干扰或独立集群。在这个项目的所有工作包中，我们重点关注两个方面：第一，发展严谨的理论，第二，为所考虑的具体模型提供可访问的结果和软件。一方面，这使应用研究人员能够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中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.