Trip Length Distribution Under Multiplicative Spatial Models of Supply and Demand: Theory and Sensitivity Analysis

We propose new probabilistic models for the spatial distribution of supply and demand and use the models to determine how the trip length distribution is affected by the relative shortage or excess of supply, the spatial clustering of supply and demand, and the degree of attraction or repulsion between supply and demand at different spatial scales. The models have a multiplicative structure and in certain cases possess scale invariance properties. Using detailed population data in metropolitan US regions validates the demand model. The trip length distribution, evaluated under destination choice models of the intervening opportunities type, has quasi-analytic form.We take advantage of this feature to study the sensitivity of the trip length distribution to parameters of the demand, supply and destination choice models. We find that trip length is affected in important but different ways by the spatial density of potential destinations, the dependence among their attractiveness levels, and the correlation between supply and demand at different spatial scales.

💡 Research Summary

The paper introduces a novel probabilistic framework for analyzing trip‑length distributions when both supply (potential destinations) and demand (origins) are spatially heterogeneous. Traditional gravity‑type models assume that trips between two zones are proportional to the product of their populations and decay with distance, which works only at coarse scales. At finer, intra‑urban scales, the spatial arrangement of supply points (e.g., shops, schools, health facilities) and the distribution of demand (population) become decisive.

To capture this, the authors propose multiplicative cascade models for the random measures S(x) and D(x) defined on a unit square. Starting from a uniform density, the region is recursively divided into four sub‑tiles; at each division a random generator vector W is multiplied onto the densities of S and D. In the log‑domain this corresponds to additive Gaussian (log‑normal) fluctuations, possibly combined with a Bernoulli‑type “beta” component that can set the measure to zero on a random subset of tiles. When the generator distribution is identical across scales the cascade is statistically scale‑invariant (multifractal). The log‑normal case is fully described by a variance σ² and a correlation coefficient ρ between ln S and ln D; the beta‑lognormal (β‑LN) case adds parameters P_D and P_S that control the probability that a tile contains any demand or supply at all, allowing for mutual exclusion.

Destination choice follows the classic intervening‑opportunities formulation: let S(d) be the number of supply points within distance d of an origin. The probability that none of them is attractive enough is
 P(L > d) = exp