Persistent Local Homology in Graph Learning

In this study, we introduce Persistent Local Homology (PLH) for graphs, a novel method that synergizes persistent homology with local homology to analyze graph structures. We begin by mathematically formalizing PLH, defining it as the application of persistent homology to annular local subgraphs. This foundation paves the way for the development of a computational pipeline, specifically tailored for PLH, which we explore in various graph learning contexts. Despite its utility, a complexity analysis reveals potential computational bottlenecks in PLH application. To address this, we propose Reduced PLH (rPLH), an efficient variant designed to significantly lower computational complexity. Experimental evaluations with rPLH demonstrate its capability to retain the effectiveness of the original PLH while substantially reducing computational demands. The practical utility of PLH and rPLH is further corroborated through comprehensive experiments on both synthetic and real-world datasets, highlighting their broad applicability and potential in diverse analytical scenarios.