STATISTICAL LAW OF STREAM NUMBERS FOR SUBNETWORKS IN INFINITE TOPOLOGICALLY RANDOM CHANNEL NETWORKS

Abstract

Werner proved by the combinatorial theory that the relation of stream numbers and orders of infinite topologically random channel networks obeys statistically Horton's law with the bifurcation ratio 4 (Werner, 1972). A problem, however, still remains how to adapt the conclusion to natural channel networks, the orders of which are given. It is the purpose of this paper to lead an equation to describe the statistical relationship of stream numbers and orders of subnetworks in infinite topologically random channel networks, the orders of which are given, introducing parameters different from the bifurca-tion ratio. The combinatorial analysis shows that the mean mel of the number of the ‘excess’ (cf. Smart, 1967) streams of order l merging directly into a stream of order m is 2m-l-1 in infinite topologically random networks. Therefore, mem-1=1=e1 and _??_ 2 =K. Using the values of rl and K, the following equation is finally deduced. _??_ where mμl is the mean of the number of the streams of order Z merging into the streams of various orders higher than l in a subnetwork of order m, which is a portion of infinite topologically random channel networks. The above equation is convenient to describe the statistical law of stream numbers for finite channel networks and can explain the fact that Norton's diagrams of many of natural channel networks are concave upward. And the probability S (l) of drawing a stream of order l at random from the streams com-prising an infinite topologically random channel network is given as follows : _??_ This quite agrees with the equation by Shreve (1967, p. 182).