This study details an approach for the analysis of social media collected political data through the lens of Topological Data Analysis, with a specific focus on Persistent Homology and the political processes they represent by proposing a set of mathematical generalizations using Gaussian functions to define and analyze these Persistent Homology categories. Three distinct types of Persistent Homologies were recurrent across datasets that had been plotted through retweeting patterns and analyzed through the k-Nearest-Neighbor filtrations. As these Persistent Homologies continued to appear, they were then categorized and dubbed Nuclear, Bipolar, and Multipolar Constellations. Upon investigating the content of these plotted tweets, specific patterns of interaction and political information dissemination were identified, namely Political Personalism and Political Polarization. Through clustering and application of Gaussian density functions, I have mathematically characterized each category, encapsulating their distinctive topological features. The mathematical generalizations of Bipolar, Nuclear, and Multipolar Constellations developed in this study are designed to inspire other political science digital media researchers to utilize these categories as to identify Persistent Homology in datasets derived from various social media platforms, suggesting the broader hypothesis that such structures are bound to be present on political scraped data regardless of the social media it's derived from. This method aims to offer a new perspective in Network Analysis as it allows for an exploration of the underlying shape of the networks formed by retweeting patterns, enhancing the understanding of digital interactions within the sphere of Computational Social Sciences.

翻译：本研究详细阐述了一种通过拓扑数据分析视角分析社交媒体收集的政治数据的方法，重点聚焦于持久同调及其所代表的政治过程，通过提出一组基于高斯函数的数学泛化，来定义并分析这些持久同调类别。在通过转发模式绘制数据集，并利用k-最近邻过滤进行分析后，三种不同类型的持久同调反复出现。随着这些持久同调的持续显现，它们被分类并命名为单极、双极与多极星座。在探究这些绘制推文的内容时，识别出了特定的互动与政治信息传播模式，即政治个人主义与政治极化。通过聚类分析及高斯密度函数的应用，我以数学方式刻画了每个类别，概括了其独特的拓扑特征。本研究开发的单极、双极与多极星座的数学泛化，旨在激励其他政治科学数字媒体研究人员利用这些类别，以识别来自不同社交媒体平台的数据集中的持久同调，并提出更广泛的假设：无论数据源自何种社交媒体，此类结构必然存在于政治抓取数据中。该方法旨在为网络分析提供新视角，因为它能探索由转发模式所形成的网络底层形状，从而增强计算社会科学领域内对数字互动的理解。