Scaling law of individual urban tour behavior

TL;DR

This work identifies a universal scaling law for individual urban tour behavior by showing that tour lengths follow a truncated power-law and introduces the Tour Terminate-Continue (TTC) model, where termination probability scales as $P_{term}(l)=\rho l^{-\gamma}$ and continuation competes with exploration governed by $P_{new}=\frac{\theta}{\theta+\sum m_k}$. The TTC framework reproduces the empirical tour-length distribution and, with a gravity-based extension (d-TTC), the spatial patterns such as the radius of gyration, while also aligning with Heaps' and Zipf's laws for location visits. Analytical results provide closed-form expressions for $P(l)$ and the growth of the number of visited locations $S(n)$, clarifying how $\rho$, $\gamma$, $\theta$, and spatial factors shape mobility. The approach offers a unified, parsimonious mechanism for urban mobility and suggests practical implications for urban planning, logistics, and beyond, with limitations tied to data sources and the need for deeper mechanistic understanding of termination dynamics.

Abstract

Analyzing and modeling the mobility process with tour behavior is fundamental to understanding a wide range of complex systems, including animal foraging, human mobility and freight transportation. However, despite their importance, the distribution of tour length has long been neglected in individual human mobility models. To fill this gap, we analyze Foursquare users' check-in data and find that the distribution of urban tour length follows a truncated power-law distribution. To reproduce the universal scaling law for human mobility in urban areas, we introduce a tour terminate-continue model. Our model reproduces not only the urban tour length distribution but also Heaps' law, Zipf's lawand the distribution of the radius of gyration, providing a new perspective for characterizing individual human mobility.

Figures (11)

• Figure 1: Tour length distributions $P(l)$ of New York (a) and Los Angeles (b), where $P(l)$ denotes the proportion of tours with length $l$ relative to the total number of tours. The main figure shows the distribution of tour lengths in log-log coordinates, where the blue dots represent the empirical data and the black curve represents the fitted values. The diamonds, triangles and squares represent the outputs of the TTC, EPR and SHMU models, respectively. The inset bar chart displays the tour length distribution using linear coordinates. Statistics. Goodness of fit is assessed by the Kolmogorov–Smirnov (KS) distance between the empirical and model distributions; values are given in the legend.
• Figure 2: A schematic diagram of the TTC model. At each step, the individual first departs from the current location and decides whether to terminate the tour and return to the base location (red circle) with a probability $P_{\text{term}}$ related to the current tour length $l$. If the individual chooses to continue the tour, they then decide whether to visit a new location (yellow circle) or a previously visited location (blue circle), according to the competitive mechanism involving parameter $\theta$ and the location visit frequency $m_{k}$.
• Figure 3: Scaling relationships between the probability $P_{\text{term}}$ of terminating the tour and the length $l$ of the current tour in New York (a) and Los Angeles (b), where $P_{\text{term}}(l)$ indicates the probability of terminating a tour with length $l$. Note that the denominator of this probability is the number of tours with length $\geqslant l$. The blue dots represent the empirical data, whereas the black line represents the linear regression fit in the log-log coordinates. The diamonds, triangles and squares represent the outputs of the TTC, EPR and SHMU models, respectively. Statistics. Goodness of fit is quantified by $\mathrm{R^{2}}$ computed in linear space; values are given in the legend.
• Figure 4: Relationship between the number $S$ of locations visited and the number $n$ of trips in New York (a) and Los Angeles (b), where $S(n)$ denotes the total number of distinct locations visited after $n$ trips. Statistics. Statistical measures and notations follow those of Fig. \ref{['fig. 3']}.
• Figure 5: Distributions of the frequency of visits $f_{k}$ to locations in New York (a) and Los Angeles (b), where $k$ is the rank of locations and $f_{k}$ represents the frequency of visiting location $k$. The blue dots represent the empirical data, whereas the black line represents the standard Zipf's law $f_{k}\sim k^{-1}$. The diamonds, triangles, and squares represent the outputs of the TTC, EPR and SHMU models, respectively. Statistics. Statistical measures follow Fig. \ref{['fig. 3']}.