Dynamic Neural Dowker Network: Approximating Persistent Homology in Dynamic Directed Graphs

TL;DR

This paper presents the Dynamic Neural Dowker Network, a novel framework that efficiently approximates persistent homology in dynamic directed graphs, capturing high-order topological features and improving dynamic graph analysis.

Contribution

The paper introduces DNDN, combining line graph transformations, SSLGNN, and a duality edge fusion mechanism to approximate dynamic Dowker filtration for the first time.

Findings

DNDN accurately approximates dynamic Dowker filtration.

DNDN outperforms existing methods in dynamic graph classification.

The approach effectively captures high-order topological features.

Abstract

Persistent homology, a fundamental technique within Topological Data Analysis (TDA), captures structural and shape characteristics of graphs, yet encounters computational difficulties when applied to dynamic directed graphs. This paper introduces the Dynamic Neural Dowker Network (DNDN), a novel framework specifically designed to approximate the results of dynamic Dowker filtration, aiming to capture the high-order topological features of dynamic directed graphs. Our approach creatively uses line graph transformations to produce both source and sink line graphs, highlighting the shared neighbor structures that Dowker complexes focus on. The DNDN incorporates a Source-Sink Line Graph Neural Network (SSLGNN) layer to effectively capture the neighborhood relationships among dynamic edges. Additionally, we introduce an innovative duality edge fusion mechanism, ensuring that the results for both the sink and source line graphs adhere to the duality principle intrinsic to Dowker complexes. Our approach is validated through comprehensive experiments on real-world datasets, demonstrating DNDN's capability not only to effectively approximate dynamic Dowker filtration results but also to perform exceptionally in dynamic graph classification tasks.