Infinite Topologically Random Channel Networks

Abstract

Individual channel networks ordinarily are portions of far larger, essentially infinite, networks. The overall network is by definition at infinite topologically random network if the populations of subnetworks within it are topologically random. From such a network, the probability of randomly drawing a link, a subnetwork, or a basin with Strahler order $$\omega is 1/2^{\omega}$$; and that of randomly drawing a stream of order $$\omega is 3/4^{\omega}$$. The probability of drawing a link of magnitude $$\mu$$, that is, one having $$\mu$$ sources ultimately tributary to it, is equal to the probability of a first passage through the origin at step $$2\mu - 1$$ in a symmetric random walk, a fact which suggests a useful mathematical analogy between random walks and infinite topologically random networks. Assuming uniform link length equal to the constant of channel maintenance, which in turn is the reciprocal of drainage density, the probability distributions for links and streams of various orders may be interpreted as crude geomorphological "laws" analogous to Horton's laws of drainage composition. These distributions predict geometric-series "laws" in which, using Strahler orders, the bifurcation ratio is 1/4, the link-number ratio is 1/2, the length ratio is 2, the cumulative-length ratio is 4, and the basin-area ratio is 4, all in good agreement with the observed ratios. They also predict values of 4/3 and 2/3, respectively, for the dimensionless ratios of total number of Strahler streams to network magnitude and of Strahler stream frequency to the square of the drainage density, in agreement with the values of 1.34 and 0.694 found empirically.