Potential Landscapes Reveal Spatiotemporal Structure in Urban Mobility: Hodge Decomposition and Principal Component Analysis of Tokyo Before and During COVID-19

TL;DR

This paper tackles the challenge of interpreting high-dimensional origin–destination mobility data by proposing a two-step framework that first uses combinatorial Hodge theory to convert OD flows into a time series of potential landscapes (preserving imbalanced flows) and then applies PCA to express these landscapes as a small set of static spatial patterns with time-varying scores. The approach yields interpretable spatiotemporal insights for Tokyo before and during COVID-19, showing pronounced weekday–holiday differences and pandemic-related reductions in peak mobility. The first three principal components capture the majority of variance (≈96%), providing a compact trajectory of mobility dynamics and linking spatial patterns to commuting and industrial activity. The method offers a scalable, interpretable tool for urban planning and public health interventions that can be extended to other urban settings with similar OD data.

Abstract

Understanding human mobility is vital to solving societal challenges, such as epidemic control and urban transportation optimization. Recent advancements in data collection now enable the exploration of dynamic mobility patterns in human flow. However, the vast volume and complexity of mobility data make it difficult to interpret spatiotemporal patterns directly, necessitating effective information reduction. The core challenge is to balance data simplification with information preservation: methods must retain location-specific information about human flows from origins to destinations while reducing the data to a comprehensible level. This study proposes a two-step dimensionality reduction framework: First, combinatorial Hodge theory is applied to the given origin--destination (OD) matrices with timestamps to construct a set of potential landscapes of human flow, preserving imbalanced trip information between locations. Second, principal component analysis (PCA) expresses the time series of potential landscapes as a linear combination of a few static spatial components, with their coefficients representing temporal variations. The framework systematically decouples the spatial and temporal components of the given data. By implementing this two-step reduction method, we reveal large weight variations during a pandemic, characterized by an overall decline in mobility and stark contrasts between weekdays and holidays. These findings demonstrate the effectiveness of our framework in uncovering complex mobility patterns and its potential to inform urban planning and public health interventions.

Figures (12)

• Figure 1: Methodology for examining spatiotemporal mobility patterns. (Left) Raw, hourly OD data derived from mobility datasets serve as the foundation of the analysis, capturing detailed flow patterns between spatial grids. (Middle) Hourly OD data transformed into a series of hourly potential landscapes. (Right) Principal component axes representing the contribution of each spatial grid to the corresponding eigenvector $\mathbf{w}_{k}$. Hourly dynamics are defined by the trajectory of score changes over time.
• Figure 2: Illustration of potential extraction by combinatorial Hodge decomposition. Input edge flow $Y$ in the left panel can be decomposed into three orthogonal components: gradient, harmonic, and curl flows. The gradient flow is the focal component of this reduction method, which can be described by differences in node potentials, denoted by red numbers. The other elements are cyclic, and incoming and outgoing fluxes are balanced, while the gradient component represents the imbalanced flow that induces the temporal changes of population mass.
• Figure 3: Threshold distances $\theta$ across four scenarios: (a) 2019 Weekday; (b) 2019 Holiday; (c) 2021 Weekday; and (d) 2021 Holiday.
• Figure 4: Temporal evolution of the potential landscape $-V(=s)$ for weekdays in 2019 in the Tokyo metropolitan area. Grid colors represent the potential value in each spatial grid. Black contour lines delineate the borders of Tokyo's special wards, marking the primary urban center. Green contour lines denote the Yamanote railway loop, which encloses the urban core of Tokyo. Subfigures how different hours: (a) 00:00; (b) 03:00; (c) 06:00; (d) 09:00; (e) 12:00; (f) 15:00; (g) 18:00; (h) 21:00.
• Figure 5: Temporal evolution of the PC1, PC2, and PC3 scores, and their corresponding eigenvectors on weekdays in 2019. (a) Time series of the principal component scores over a day. (b) and (c) Relations between principal components in PC2 vs. PC1 and PC3 vs. PC1 spaces, respectively. The corresponding eigenvectors are shown in (d) $\mathbf{w}_{1}$, (e) $\mathbf{w}_{2}$, and (f) $\mathbf{w}_{3}$. Numbers on the maps (d)--(f) indicate key locations: 1-Yokohama, 2-Chiba, 3-Kawasaki, 4-Funabashi, 5-Ikebukuro, 6-Shinjuku, and 7-Tokyo Station. Contour lines mirror those in Fig. \ref{['fig_potential_dynamics']}.