Urn Modeling of Random Graphs Across Granularity Scales: A Framework for Origin-Destination Human Mobility Networks

TL;DR

This paper introduces a comprehensive framework for modeling human mobility networks using urn models, unifying combinatorial, probabilistic, and continuum approaches, and providing analytical tools for network analysis and urban planning.

Contribution

It develops a unified three-scale framework for origin-destination networks and proves a convergence theorem linking combinatorial and continuum models.

Findings

Simulations match asymptotic predictions closely.

Framework provides formulas for occupancy, vacancy, and coverage.

Universal law applies in large sparse regimes.

Abstract

We model human mobility as a combinatorial allocation process, treating trips as distinguishable balls assigned to location-bins and generating origin-destination (OD) networks. From this analogy, we construct a unified three-scale framework, enumerative, probabilistic, and continuum graphon ensembles, and prove a renormalization theorem showing that, in the large sparse regime, these representations converge to a universal mixed-Poisson law. The framework yields compact formulas for key mobility observables, including destination occupancy, vacancy of unvisited sites, coverage (a stopping-time extension of the coupon collector problem), and overflow beyond finite capacities. Simulations with gravity-like kernels, calibrated on empirical OD data, closely match the asymptotic predictions. By connecting exact combinatorial models with continuum analysis, the results offer a principled toolkit for synthetic network generation, congestion assessment, and the design of sustainable urban mobility policies.