Urban spatial structures from human flow by Hodge-Kodaira decomposition

TL;DR

This paper introduces a novel application of Hodge-Kodaira decomposition to urban human flow data, revealing potential landscapes that help visualize and understand city dynamics and spatial structures.

Contribution

It applies Hodge-Kodaira decomposition to human flow matrices, providing a new potential-based perspective and visualization of urban spatial structures based on commuting flows.

Findings

Gradient component dominates in some metropolitan areas, indicating potential-driven flow.

Curl component is significant in areas with circular or triangular flow patterns.

Potential landscapes effectively visualize attractive places and flow dynamics in cities.

Abstract

Human flow in cities indicates social activity and can reveal urban spatial structures based on human behaviours for relevant applications. Scalar potential is a mathematical concept, and if successfully introduced, it can provide an intuitive perspective of human flow. However, the definition of such a potential to the origin-destination flow matrix and determination of its plausibility remain unsolved. Here, we apply Hodge-Kodaira decomposition, in which a matrix is uniquely decomposed into a potential-driven (gradient) flow and a curl flow. We depict the potential landscapes in cities due to commuting flow and reveal how the landscapes have been changed or unchanged by years or transport methods. We then determine how well the commuting flow is described by the potential, by evaluating the percentage of the gradient component for metropolitan areas in the USA and show that the gradient component is almost 100\% in several areas; in other areas, however, the curl component is dominant, indicating the importance of circular flow along triangles of places. The potential landscape provides an easy-to-use visualisation tool to show the attractive places of human flow and will aid in various applications in commerce, urban design, and epidemic spreading.