Spatial Topology and its Structural Analysis based on the Concept of Simplicial Complex

TL;DR

This paper presents a novel model using simplicial complexes to analyze spatial relationships in geographic systems, capturing local and global structural properties through centrality measures.

Contribution

The paper introduces a new approach to spatial relationship modeling using simplicial complexes, enabling detailed structural analysis of urban systems.

Findings

Effective representation of spatial relationships as simplicial complexes.

Quantitative measures of structural properties using centrality metrics.

Application demonstrated through urban system case study.

Abstract

This paper introduces a model that identifies spatial relationships for a structural analysis based on the concept of simplicial complex. The spatial relationships are identified through overlapping two map layers, namely a primary layer and a contextual layer. The identified spatial relationships are represented as a simplical complex, in which simplices and vertices respectively represent two layers of objects. The model relies on the simplical complex for structural representation and analysis. To quantify structural properties of individual primary objects (or equivalently simplices), and the simplicial complex as a whole, we define a set of centrality measures by considering multidimensional chains of connectivity, i.e. the number of contextual objects shared by a pair of primary objects. With the model, the interaction and relationships with a geographic system are modeled from both local and global perspectives. The structural properties and modeling capabilities are illustrated with a simple example and a case study applied to the structural analysis of an urban system.