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

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

Problem Statement
Understanding the statistical regularities of individual daily mobility is a central challenge in urban science and complex systems. A key phenomenon is the "tour," defined as a sequence of trips originating from and returning to a fixed home base. Empirical data reveals a striking scaling law: the probability of an individual returning home after visiting $l$ locations decays as a power law, $P_{ret}(l) \sim l^{-\gamma}$, where $\gamma \in (0, 1)$.

Previous attempts to model this, such as the Tour-Terminate-Continue (TTC) model proposed by Lin et al., successfully reproduce the distribution of tour lengths by treating the power-law decay of $P_{ret}(l)$ as an empirical input. However, this approach leaves the microscopic origin of the exponent $\gamma$ unresolved. The core problem addressed in this work is to provide a mechanistic explanation for why the home-return probability follows a power law and what determines the specific value of the scaling exponent $\gamma$, moving beyond phenomenological description to theoretical derivation.

Methodology
The authors propose a framework grounded in the principle of least effort and cognitive constraints to derive the scaling law from first principles. The methodology proceeds through the following logical steps:

1. Utility Trade-off Model: The decision to terminate a tour (return home) or continue is modeled as a binary choice governed by a trade-off between two utilities:

   • $U_H$: The constant utility of returning home (representing rest and security).
   • $U_L(l)$: The cumulative marginal utility derived from the external tour up to step $l$.
     Using Luce's choice rule, the home-return probability is expressed as $P_{ret}(l) = U_H / (U_H + U_L(l))$.

2. Cognitive Coding and Zipf's Law: To determine the growth pattern of $U_L(l)$, the authors model the individual's retrieval of activity purposes as an optimal coding process. Applying Mandelbrot's information-theoretic framework and Shannon's source coding theorem under the constraint of minimizing average cognitive effort (the principle of least effort), they derive that activity priorities follow a Zipf distribution: $p(r) \sim r^{-\nu}$, where $r$ is the priority rank and $\nu$ is the cognitive heterogeneity exponent.

3. Integration of Spatial Accessibility: The authors argue that spatial accessibility is not an external constraint but is integrated into the cognitive encoding process. Proximate destinations require less cognitive effort, effectively boosting their priority rank. Thus, the sequence of tour steps $l$ aligns with the integrated priority rank $r$ (i.e., $l=r$).

4. Mathematical Derivation: The cumulative utility $U_L(l)$ is calculated as the sum of marginal utilities over the top $l$ ranks. By analyzing the asymptotic behavior of the sum $\sum_{r=1}^{l} r^{-\nu}$, the authors identify that for the regime $0 < \nu < 1$, the cumulative utility grows sublinearly as $U_L(l) \sim l^{1-\nu}$.

5. Validation: The analytical results are validated through agent-based simulations involving $10^8$ agents, comparing simulated $P_{ret}(l)$ and fitted exponents against the theoretical predictions.

Key Results
The analysis yields a direct analytical relationship between the cognitive heterogeneity exponent $\nu$ and the mobility scaling exponent $\gamma$:
$$\gamma = 1 - \nu$$
This derivation demonstrates that:

• The power-law decay of the home-return probability emerges naturally from the sublinear accumulation of tour utility.
• This sublinearity is a direct consequence of the individual's internal Zipf distribution of activity priorities, which itself is the optimal solution for cognitive coding under the principle of least effort.
• The condition $\nu \in (0, 1)$, required for the sublinear growth of utility, naturally ensures that the resulting scaling exponent $\gamma$ falls within the empirically observed range of $(0, 1)$.

Agent-based simulations confirm that the theoretical scaling relation holds, with the fitted exponent $\gamma$ converging to $1-\nu$ as the maximum tour length increases, despite statistical noise at very long tour lengths.

Significance and Claims
The paper claims to provide the first microscopic theoretical underpinning for the empirical TTC model of human mobility. By bridging the gap between individual cognitive constraints (specifically the principle of least effort and optimal coding) and macroscopic mobility patterns, the authors resolve the "black box" of the TTC model.

The significance of this work lies in:

• Mechanistic Explanation: It shifts the understanding of tour behavior from a phenomenological fit to a mechanism driven by cognitive utility trade-offs.
• Unification: It unifies the observation of power-law mobility scaling with the well-established Zipf's law in cognitive science, suggesting that the observed mobility patterns are a direct manifestation of how the brain optimizes the retrieval of activity purposes.
• Theoretical Foundation: It offers a rigorous derivation of the scaling exponent $\gamma$ based on the cognitive parameter $\nu$, providing a foundation for future research linking cognitive science, complex systems, and urban planning.

The authors acknowledge limitations, noting that their current model implicitly internalizes spatial friction through "integrated priority" rather than treating spatial distance as an explicit external constraint, and that it focuses on individual decision-making without incorporating social interactions or collective mobility patterns.