Visually Comparing Graph Vertex Ordering Algorithms through Geometrical and Topological Approaches

Abstract

Graph vertex ordering is widely employed in spatial data analysis, especially in urban analytics, where street graphs serve as spatial discretization for modeling and simulation. It is also crucial for visualization, as many methods require vertices to be arranged in a well-defined order to reveal non-trivial patterns. The goal of vertex ordering methods is to preserve neighborhood relations, but the structural complexity of real-world graphs often introduces distortions. Comparing different ordering methods is therefore essential to identify the most suitable one for each application. Existing metrics for assessing spatial vertex ordering typically focus on global quality, which hinders the identification of localized distortions. Visual evaluation is particularly valuable, as it allows analysts to compare methods within a single visualization, assess distortions, identify anomalous regions, and, in urban contexts, explain spatial inconsistencies. This work presents a visualization-assisted tool for assessing vertex ordering techniques, with a focus on urban analytics. We evaluate geometric and topological ordering approaches using urban street graphs. The visual tool integrates existing and newly proposed metrics, validated through experiments on data from multiple cities. Results demonstrate that the proposed methodology effectively supports users in selecting suitable vertex ordering techniques, tuning hyperparameters, and identifying regions with high ordering distortions.