Statistical Law of Stream Numbers

TLDR

Horton's law of stream numbers is hypothesized to arise from the statistics of many randomly merging stream channels, analogous to how the ideal gas law emerges from random molecular collisions, and networks with identical first‑order Strahler streams exhibit comparable topological complexity, suggesting a topologically random population of channel networks. In a topologically random population, the most probable networks largely follow Horton's law but display systematic deviations: the most probable network order yields a geometric mean bifurcation ratio near 4, and when both order and first‑order streams are fixed, the second‑order bifurcation ratio remains close to 4, causing bifurcation ratios to vary with order and the Horton diagram to be concave upward, straight, or concave downward depending on whether the geometric mean ratio is below, equal to, or above 4. Statistical comparison with 172 published stream‑number datasets confirms that natural channel networks lacking geologic controls are topologically random, supporting the view that Horton's law of stream numbers largely results from random network development governed by chance.

Abstract

The statistical nature and remarkable generality of Horton's law of stream numbers suggest the speculation that the law of stream numbers arises from the statistics of a large number of randomly merging stream channels in somewhat the same fashion that the law of perfect gases arises from the statistics of a large number of randomly colliding gas molecules. The fact that networks with the same number of first-order Strahler streams are comparable in topological complexity suggests equating "randomly merging stream channels" with a topologically random population of channel networks, defined as a population within which all topologically distinct networks with given number of first-order streams are equally likely. In a topologically random population the most probable networks approximately obey Horton's law but exhibit certain systematic deviations. For networks with given number of first-order streams, the most probable network order is that which makes the geometric mean bifurcation ratio closest to 4. For networks with both order and number of first-order streams specified, the most probable networks have the property that the bifurcation ratio of the second-order streams is always close to 4 and, hence, that the bifurcation ratios respectively decrease, remain unchanged, or increase with order and the corresponding curves on the Horton diagram are respectively concave upward, straight, or concave downward according as the geometric mean bifurcation ratio is less than, equal to, or greater than 4. Statistical comparison of these properties with 172 published sets of stream numbers strongly supports the conclusion that, as speculated, populations of natural channel networks developed in the absence of geologic controls are topologically random and, hence, that the law of stream numbers is indeed largely a consequence of random development of channel networks according to the laws of chance.