Exact solution of gyration radius of individual's trajectory for a simplified human mobility model

TL;DR

This paper presents an exact analytical solution for the gyration radius in a simplified human mobility model, explaining empirical observations and revealing a different mechanism for individual displacement patterns.

Contribution

It introduces a simplified model with three activities, deriving an exact gyration radius solution that aligns with empirical data and differs from previous theories.

Findings

Exact gyration radius solution matches empirical data.

Daily movement area is elliptical under model assumptions.

Individuals exhibit characteristic displacements despite population heterogeneity.

Abstract

Gyration radius of individual's trajectory plays a key role in quantifying human mobility patterns. Of particular interests, empirical analyses suggest that the growth of gyration radius is slow versus time except the very early stage and may eventually arrive to a steady value. However, up to now, the underlying mechanism leading to such a possibly steady value has not been well understood. In this Letter, we propose a simplified human mobility model to simulate individual's daily travel with three sequential activities: commuting to workplace, going to do leisure activities and returning home. With the assumption that individual has constant travel speed and inferior limit of time at home and work, we prove that the daily moving area of an individual is an ellipse, and finally get an exact solution of the gyration radius. The analytical solution well captures the empirical observation reported in [M. C. Gonz`alez et al., Nature, 453 (2008) 779]. We also find that, in spite of the heterogeneous displacement distribution in the population level, individuals in our model have characteristic displacements, indicating a completely different mechanism to the one proposed by Song et al. [Nat. Phys. 6 (2010) 818].