Layered Hodge Decomposition for Urban Transit Networks

Modeling the amount of passenger flow along any given line segment of an urban transit network is a challenging task due to the complexity of the system. In this paper, we embark on a characterization of these flows on the basis of a combination of (1) a layered decomposition of the origin-destination matrix, and (2) the Hodge decomposition, a discrete algebraic topology technique that partitions flows into gradient, solenoidal, and harmonic components. We apply our method to data from the London Underground. We find that the layered decomposition estimates the contribution of each origin-destination pair to the flow on each network link, and that the solenoidal and harmonic flows can be described by simple equations, thereby reducing much of the solution to determining gradient flows. Our exploratory analysis suggests it may be feasible to develop solution methods for the transit flow problem with a complexity equivalent to the solution of a hydraulic or electric circuit.