Geographic Space as Manifolds

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.