On geometrical scaling of Cayley trees and river networks

Geometrical scaling properties of rooted Cayley trees generated by percolation on the Bethe lattice are analysed. Statistical scaling relations between the characteristic topological length and width dimensions and cluster size as a function of the lattice occupation probability are identified analytically and by simulation. Cayley trees generated with a constant occupation probability exhibit statistical self-affinity at small scales, but approach self-similarity with increasing size, similar to Markov branching models under random topology postulates. The position of the peak of the topological width function is asymptotically invariant with regard to cluster size. However, Cayley trees grown with a constant occupation probability are generally too elongated. A numerical experiment in which the occupation probability on the Bethe lattice decreased systematically on unbranched links as the cluster grew provided average length to width ratios more comparable to river network data. Although the occupation probability uniquely determines the probability of termination, continuation and branching, it is difficult to meaningfully connect it to physical processes involved in river network evolution.