Urban spatial structures from human flow by Hodge–Kodaira decomposition

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 that, when properly applied, can provide an intuitive view of human flow. However, the definition of such a potential in terms of the origin-destination flow matrix and its feasibility remain unresolved. In this case, we use Hodge–Kodaira decomposition, which uniquely decomposes a matrix into a potential-driven (gradient) flow and a curl flow. We depict the potential landscapes in cities resulting from commuting flow and reveal how the landscapes have either changed or remained unchanged by years or methods of transportation. 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 with triangles of places. The potential landscape provides an easy-to-use visualisation tool for showing the attractive places of human flow and will help in a variety of applications such as commerce, urban design, and epidemic spreading.

Identifying sinks and sources of human flows: A new approach to characterizing urban structures

“…Compared to the previous study (Aoki et al, 2022), this study introduces a new statistical center identification method and classifies locations by combination of commuting and shopping flows in the Tokyo metropolitan area.…”

“…Here, we introduce an identification method for city centers based on the human mobility data provided by an OD matrix. First, we mathematically introduce the scalar potential of net movements using the unique decomposition of combinatorial Hodge theory (Jiang et al, 2011; Aoki et al, 2022). Second, we statistically identify the locations that are significantly attractive compared with the counterfactual model.…”

“…Although our method is technically similar to theirs, we use an extension of vector calculus that is applicable to graphs, such that the network structure of the OD flows is preserved. See Aoki et al (2022) for a detailed comparison between our method and that of Mazzoli et al (2019).…”

“…In this study, we propose an alternative center identification method based on human mobility data. To this end, we first define the scalar potential field of human flow by adopting the unique decomposition of a given OD matrix according to combinatorial Hodge theory (Aoki et al, 2022; Jiang et al, 2011) (see the Methods section for details). Using a metaphor for water flowing from a higher to a lower place, the potential landscape provides an intuitive perspective of human flows and reveals the sinks and sources of the flow on a map, as shown in Figure 1(b).…”