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.