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

TL;DR

This work tackles the computational challenges of applying persistent homology to dynamic directed graphs by introducing the Dynamic Neural Dowker Network (DNDN). DNDN leverages line-graph transformations to produce source and sink line graphs and processes them with a dedicated SSLGNN backbone, complemented by a duality edge fusion that enforces Dowker duality. The approach enables joint estimation of 0- and 1-dimensional persistence diagrams ($PD^0$ and $PD^1$) and supports graph classification, achieving strong PD approximation and favorable transferability to larger graphs with improved efficiency over traditional methods. Overall, the framework extends topological learning to evolving directed networks, offering scalable, edge-centric insights for downstream analysis and applications affected by dynamic topology.

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.

Figures (5)

• Figure 1: (a) A simple example of a Dowker source complex. The existence of a shared neighbor (in this case, $v_4$) between $v_1$ and $v_2$ creates a higher-order relationship, which the Dowker source complex captures. (b) Illustration of Dowker filtration sensitive to edge weights and directions in graphs: $G_a$ represents a common type of subgraph in social media diffusion graphs. Swapping the timestamps of two edges ($G_b$) or changing the direction of an edge ($G_c$) leads to different diffusion. Dowker source filtration can effectively distinguish these three types of graphs, whereas Vietoris-Rips (VP) filtration generates the same persistent barcode for all three cases.
• Figure 2: An example demonstrating the relationship between the source Dowker complex and the source line graph for a directed graph. In the source line graph, the presence of an edge between nodes representing $e_{41}$ and $e_{42}$ reflects the shared neighbor ($v_4$) between $v_1$ and $v_2$ in the original graph. This edge-based perspective is particularly allows the neural network to focus on the interactions and relationships between edges.
• Figure 3: The framework of DNDN
• Figure 4: An example demonstrating the 0-PD of Dowker complexes.
• Figure 5: An example demonstrating the difference Vietoris-Rips (VP) complexes and Dowker complexes.