Topological Spatial Graph Coarsening

TL;DR

This paper introduces a novel, parameter-free method for reducing spatial graphs by collapsing short edges while preserving topological features, using a new filtration based on persistent diagrams, and proves its invariance under geometric transformations.

Contribution

It presents a topological spatial graph coarsening framework that balances graph reduction with topological preservation, utilizing a new triangle-aware filtration and proving geometric invariance.

Findings

Significantly reduces graph size while maintaining topological features

Effective on both synthetic and real spatial graphs

Parameter-free and invariant under geometric transformations

Abstract

Spatial graphs are particular graphs for which the nodes are localized in space (e.g., public transport network, molecules, branching biological structures). In this work, we consider the problem of spatial graph reduction, that aims to find a smaller spatial graph (i.e., with less nodes) with the same overall structure as the initial one. In this context, performing the graph reduction while preserving the main topological features of the initial graph is particularly relevant, due to the additional spatial information. Thus, we propose a topological spatial graph coarsening approach based on a new framework that finds a trade-off between the graph reduction and the preservation of the topological characteristics. The coarsening is realized by collapsing short edges. In order to capture the topological information required to calibrate the reduction level, we adapt the construction of classical topological descriptors made for point clouds (the so-called persistent diagrams) to spatial graphs. This construction relies on the introduction of a new filtration called triangle-aware graph filtration. Our coarsening approach is parameter-free and we prove that it is equivariant under rotations, translations and scaling of the initial spatial graph. We evaluate the performances of our method on synthetic and real spatial graphs, and show that it significantly reduces the graph sizes while preserving the relevant topological information.