Mechanism underlying the scaling law of home-return probability in human mobility

Individual daily mobility exhibits a striking scaling law: the probability of returning home after a tour of $l$ locations decays as $P_{\mathrm{r}\mathrm{e}\mathrm{t}}(l)\sim l^{-\gamma }$. While the tour-terminate-continue (TTC) model reproduces this behavior, it relies on this power law as an empirical input, leaving the microscopic origin of $\gamma$ unresolved. Here we show that this scaling emerges from a utility trade-off governed by cognitive constraints. By invoking the principle of least effort, we demonstrate that individual activity priorities follow Zipf’s law, $p(r)\sim r^{-\nu }$, which directly dictates the sublinear accumulation of tour utility, $U_L(l)\sim l^{1-\nu }$. Luce’s choice rule then yields $P_{\mathrm{r}\mathrm{e}\mathrm{t}}(l)\sim l^{-(1-\nu )}$, giving the exact exponent $\gamma =1-\nu$. Agent-based simulations confirm this analytical relation. Our framework bridges the gap between individual cognitive constraints and the scaling law of tour behavior, providing a microscopic theoretical underpinning for human mobility.