Throughput in inhomogeneous planar drainage networks

TL;DR

This paper analyzes traffic flow in inhomogeneous planar drainage networks, showing that, after proper rescaling, the total traffic and tree length converge to the area under a time-inhomogeneous Brownian motion until it hits zero.

Contribution

It introduces a novel probabilistic model for drainage networks on inhomogeneous lattices and establishes convergence results linking traffic flow to Brownian motion.

Findings

Total traffic converges to the area under a time-inhomogeneous Brownian motion.

The maximal path length corresponds to the hitting time of the Brownian motion.

The model captures the effect of inhomogeneity on network throughput.

Abstract

We consider navigation schemes on planar diluted lattices and semi lattices with one discrete and one continuous component. More precisely, nodes that survive inhomogeneous Bernoulli site percolation, or are placed as inhomogeneous Poisson points on shifted copies of $\mathbb{Z}$, forward their individually generated traffic to their respective closest neighbors to the left in the next layer. The resulting drainage network is a tree and we study the amount of traffic that goes through an increasing window at the origin. Our main results show that, properly rescaled, the total traffic, jointly with the total length of the contributing tree part, converges to the area under a time-inhomogeneous Brownian motion until it hits zero. The hitting time corresponds to the limiting maximal path length.