Urn modeling of random graphs across granularity scales: A framework for origin-destination human mobility networks

We model human mobility as a combinatorial allocation process, treating trips as distinguish- able balls assigned to location-bins and generating origin–destination (OD) networks. From this analogy, we construct a unified three-scale framework, enumerative, probabilistic, and contin- uum 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 un- visited sites, coverage (a stopping-time extension of the coupon collector problem), and overflow beyond finite capacities. A key contribution is the explicit and testable treatment of cross-scale consistency in OD modeling. Numerical simulations with gravity-like kernels, calibrated on empir- ical 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.