Universal expansion of human mobility across urban scales

TL;DR

This work treats individual mobility as a networked process by decomposing daily trajectories into spatially coherent modules. It uncovers a universal expansion law: the module radius scales sublinearly with distance from home as $r_c \sim d_c^{\kappa}$ with $\kappa$ around $0.6$, a pattern that holds across the US and two African countries. Moreover, modules align with a nested urban hierarchy defined via the H3 indexing system, satisfying $L_c \sim \log(d_c)$ and $\log(R) \sim L$, implying $\log(R) \sim \log(d_c)$. The findings are robust across data sources, module definitions, and demographic factors, and they bridge home-centric mobility theories with hierarchical urban structure, with implications for epidemic modeling, mobility equity, and urban resilience.

Abstract

Human mobility is a fundamental process underpinning socioeconomic life and urban structure. Classic theories, such as egocentric activity spaces and central place theory, provide crucial insights into specific facets of movement, like home-centricity and hierarchical spatial organization. However, identifying universal characteristics or an underlying principle that quantitatively links these disparate perspectives has remained a challenge. Here, we reveal such a connection by analyzing the spatial structure of individual daily mobility trajectories using network-based modules. We discover a universal scaling law: the spatial extent (radius) of these mobility modules expands sublinearly with increasing distance from home, a pattern consistent across three orders of magnitude. Furthermore, we demonstrate that these modules precisely map onto the nested hierarchy of urban systems, corresponding to local, city-level, and regional scales as distance from home increases. These findings deepen our understanding of human mobility dynamics and demonstrate the profound connection between classical urban theory, human geography, and mobility studies.

Figures (6)

• Figure 1: The spatial expansion of modules. (a) The spatial distribution of modules in anonymized cell phone users' trajectories. By representing trajectories as networks and applying the Louvain method, (b) the example trajectory network is divided into five modules, each encompassing spatially and topologically proximate locations. (c) Modules located far from home are in larger spatial coverage. By analyzing all U.S., Senegal, and Ivory Coast data, (d-i) module radius $r_c$ increases with distance from home $d_c$ in a power-law manner, $r_c \sim d_c^{\kappa}$. Specifically, for U.S. data in the West, Northeast, Midwest, and South regions, the value of $\kappa$ is approximately $0.61 \pm 0.03$. For Senegal and Ivory Coast data, $\kappa$ is approximately $0.58$.
• Figure 1: The spatial expansion of modules regarding convex hull area size. Module convex hull area size, $A_c$, increases sub-linearly with its distance from home $d_c$. The exponent is around 0.55 for the U.S. data, 0.52 for the Senegal data, and 0.44 for the Ivory Coast data.
• Figure 2: Spatial hierarchical levels and expansion law. With the destination Boston, Massachusetts as the example, (a) the urban environment is structured into four hierarchical levels $L$, with higher-level units covering lower-level units. We add the convex hull of aggregated H3 cells at each level to better illustrate the geographical extent. (b) Users’ module networks at the destination with varying distances from home (e.g., 10km, 100km, 1000km). We assign modules a specific level $L_c$ if that hierarchical level unit encompasses at least 80% of the module. (c) The distribution of spatial levels $L_c$ for module networks in (b). Modules that are at greater distances from home tend to travel across higher hierarchical levels. By analyzing all U.S., Senegal, and Ivory Coast data, (d) the spatial levels of modules at varying distances from home follow $L_c \sim \log(d_c)$. (e) The spatial size of hierarchical levels follows $\log(R) \sim L$, leading to $\log(R) \sim L=L_c \sim \log(d_c)$.
• Figure 2: Part-1-Module radius versus distance from home, for populations in different states. By categorizing users based on the states of their home locations, the spatial expansion of the module remains consistent.
• Figure 3: Part-2-Module radius versus distance from home, for populations in different states. By categorizing users based on the states of their home locations, the spatial expansion of the module remains consistent.