Persistent Homology generalizations for Social Media Network Analysis

Abstract: 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.