Geographic Space as Manifolds

TL;DR

This paper introduces the Geographic Manifold, a low-dimensional spatial representation capturing geographic constraints, which simplifies complex spatial analytics and is grounded in the intrinsic properties of geographic phenomena.

Contribution

It formally proves the existence of the Geographic Manifold and demonstrates its effectiveness in simplifying spatial analysis tasks.

Findings

Effective low-dimensional representation of geographic space.

Simplification of location choice and propagation analyses.

Grounded in intrinsic Euclidean properties of geographic phenomena.

Abstract

The communications and interrelations between different locations on the Earth's surface have far-reaching implications for both social and natural systems. Effective spatial analytics ideally require a spatial representation, where geographic principles are succinctly expressed within a defined metric space. However, common spatial representations, including map-based or network-based approaches, fall short by incompletely or inaccurately defining this metric space. Here we show, by introducing an inverse friction factor that captures the spatial constraints in spatial networks, that a homogeneous, low-dimensional spatial representation - termed the Geographic Manifold - can be achieved. We illustrate the effectiveness of the Geographic Manifold in two classic scenarios of spatial analytics - location choice and propagation, where the otherwise complicated analyses are reduced to straightforward regular partitioning and concentric diffusing, respectively on the manifold with a high degree of accuracy. We further empirically explain and formally prove the general existence of the Geographic Manifold, which is grounded in the intrinsic Euclidean low-dimensional statistical physics properties of geographic phenomena. This work represents a step towards formalizing Tobler's famous First Law of Geography from a geometric approach, where a regularized geospace thereby yielded is expected to contribute in learning abstract spatial structure representations for understanding and optimization purposes.