Spatial Transportation Networks with Transfer Costs: Asymptotic Optimality of Hub and Spoke Models

TL;DR

This paper analyzes the asymptotic optimality of hub and spoke transportation networks with transfer costs, showing how network length and hop count scale under different constraints, and establishing bounds for optimal network design.

Contribution

It provides the first rigorous analysis of the asymptotic behavior of hub and spoke models with transfer costs, establishing bounds on network length and hop count under various constraints.

Findings

Optimal network length scales as n^{13/10} with 3-hop constraint

Network length grows faster than n^{3/2} with 2-hop constraint

Log-log growth in hops when network length is O(n)

Abstract

Consider networks on $n$ vertices at average density 1 per unit area. We seek a network that minimizes total length subject to some constraint on journey times, averaged over source-destination pairs. Suppose journey times depend on both route-length and number of hops. Then for the constraint corresponding to an average of 3 hops, the length of the optimal network scales as $n^{13/10}$. Alternatively, constraining the average number of hops to be 2 forces the network length to grow slightly faster than order $n^{3/2}$. Finally, if we require the network length to be O(n) then the mean number of hops grows as order $\log \log n$. Each result is an upper bound in the worst case (of vertex positions), and a lower bound under randomness or equidistribution assumptions. The upper bounds arise in simple hub and spoke models, which are therefore optimal in an order of magnitude sense.