AI 中文总结
研究针对网络集合统计分析难题，引入功能拓扑数据分析框架 funTDA，整合功能与拓扑数据分析工具，通过模拟研究和实际数据应用，展示其区分不同网络配置能力，并用主成分分析和假设检验评估拓扑差异。
AI 中文摘要
对网络集合进行统计分析在基因调控、社会和金融网络等众多应用领域愈发重要。由于网络的顶点和边不在欧几里得空间，直接应用传统统计方法并不简单。核心挑战在于定义不同规模和结构类型网络间有意义的相似性或距离度量。为此引入功能拓扑数据分析框架（funTDA），它整合功能数据分析和拓扑数据分析工具。通过模拟研究及两个实际数据应用（简·奥斯汀和查尔斯·狄更斯小说中词共现网络、17 个感染 H3N2 流感个体的基因调控网络），展示了 funTDA 区分不同网络配置的能力，还通过主成分分析和假设检验评估了网络拓扑差异。
英文摘要
Statistical analysis of collections of networks, where each network is treated as the primary unit of observation, is of growing importance across a wide range of application domains, including gene regulatory, social, and financial networks. As networks consist of vertices and edges that do not naturally reside in Euclidean space, the direct application of conventional statistical methodologies, such as the computation of means and covariances, principal component analysis, and hypothesis testing, to samples of networks is not straightforward. A central challenge lies in defining meaningful measures of similarity or distance between networks of potentially varying sizes and structural types (e.g., directed, undirected, weighted or unweighted), particularly when no predefined node correspondence exists. To address these challenges, we introduce a framework termed functional topological data analysis (funTDA), which integrates tools from functional data analysis and topological data analysis to facilitate exploratory data analysis and inference on samples of networks. The proposed framework enables the computation of summary statistics, including means and variances, and supports the application of principal component analysis and hypothesis testing to topological features extracted from network data. Through simulation studies involving networks with varying connectivity structures, we demonstrate the ability of funTDA to distinguish between distinct network configurations. The methodology is illustrated through two real-data applications: networks constructed from pairwise word co-occurrences in novels by Jane Austen and Charles Dickens, and gene regulatory networks derived from gene expression measurements for seventeen individuals exposed to H3N2 influenza. In both applications, differences in network topology are assessed using principal component analysis and hypothesis testing.
