Topological Spatial Graph Coarsening

TL;DR

This work introduces Topological Spatial Graph Coarsening, a method to reduce the node count of spatial graphs while preserving their global topological structure. It combines a theta-based edge-collapse coarsening with a triangle-aware graph filtration and persistence-diagram formalism, measuring topological distortion via the Bottleneck distance $d_B$ and balancing it against graph size through a scoring function $S_ heta(G)$. The approach is proven to be equivariant under rotations, translations, and scaling, ensuring consistent behavior under similarity transforms, and is validated on synthetic rings, road networks, and fungi networks, showing substantial size reduction with preserved topological information and classification performance. This topology-aware reduction has practical implications for efficient analysis of spatial networks in transport, biology, and materials science.

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.

Figures (18)

• Figure 1: Examples of pruned graphs for different values of $\theta$. The leftmost graph is the initial graphs.
• Figure 2: A graph (left) and the corresponding persistence diagram (right).
• Figure 3: From left to right. The original and reduced graph. The persistence diagram of the original graph. The persistence diagram of the reduced graph; horizontal dashed line represents the threshold $\theta$ used to compute the reduced graph.
• Figure 4: Illustration of the property of the coarsening procedure when a spatial graph is modified by applying a $(R,A,k)\in Sym(2)$
• Figure 5: Synthetic annulus graph. Left: original graph; right: reduced graph.

Theorems & Definitions (15)

• Definition 1
• Remark 1
• Remark 2
• Definition 2: Triangle-aware graph filtration
• Definition 3: Topology-informed scoring function
• Definition 4
• Definition 5
• Proposition 1
• proof
• Remark 3