Persistent Local Homology in Graph Learning

Minghua Wang, Yan HU, Ziyun Huang, Di Wang, Jinhui Xu

Abstract: 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.

Submission Length: Regular submission (no more than 12 pages of main content)

Changes Since Last Submission:

Added authors.
Added openreview URL.
Added acknowledgments.
Revised 'local subgraph $g_\varepsilon^s(v)$' to 'punctured local subgraph $g_0^s(v)$' in section 5.1 for consistency.

Code: https://github.com/minghuax/PLH_GL

Assigned Action Editor: ~Rémi_Flamary1

Submission Number: 2043