On the Order of Random Channel Networks

Abstract

The order of a stream with no tributaries is defined to be 1. In general, when two streams of orders $\alpha $ and $\beta $ flow together, the larger stream thus produced has order $\max \{ \alpha ,\beta \}$ or $\alpha + 1$, according as $\alpha \ne \beta $ or $\alpha = \beta $. The order $\Omega $ of a river network $\mathcal{N}$ is the order of the highest ordered stream in $\mathcal{N}$. Our object is to investigate the distribution of $\Omega $ for random networks with n sources. It follows from our results that the distribution of $\Omega $ is very highly concentrated about $1 + \frac{1}{2}\log_2 n$.