Random self‐similar river networks and derivations of generalized Horton Laws in terms of statistical simple scaling

Recent data analyses of a dozen or so large networks support the presence of simple scaling in the probability distributions of many topologic, geometric, and hydraulic-geometric variables. We have labeled them “generalized Horton laws” in contrast to traditional Horton laws which are stated only in terms of means. These data sets also show that Horton's bifurcation ratios are significantly larger than 4, which can not be predicted by the well-known random model. Motivated by the need to understand these data analyses of large river networks within a common theoretical framework, a new class of models with statistically self-similar network topology is introduced. The network construction process can be viewed as a statistical generalization of the type of a recursive process which generates many fractal geometrical objects. Unlike the well-known random model, our random channel networks do not belong to the class of binary Galton-Watson stochastic branching processes. The generalized Horton laws are predicted in the limit of large Horton-Strahler order as a scale parameter for a subclass of random self-similar network models. A derivation of Horton's law of stream numbers as a consequence of the simple scaling of distributions of basin magnitude gives it a new theoretical significance. A further subclass also exhibits Tokunga self-similarity, which is an average property of the side tributary structure in river networks. Comparisons with empirical observations for three large basins analyzed here show that the generalized Horton laws are well predicted by this new class of random self-similar network models, and analytically predicted relationships between topologic and geometric Horton ratios are also seen in data.