Scaling properties of topologically random channel networks

The analysis deals with the scaling properties of infinite topologically random channel networks (ITRNs) fast introduced by Shreve (1967, J. Geol., 75: 179–186) to model the branching structure of rivers as a random process. The expected configuration of ITRNs displays scaling behaviour only asymptotically, when the ruler (or ‘yardstick’) length is reduced to a very small extent. The random model can also reproduce scaling behaviour at larger ruler lengths if network magnitude and diameter are functionally related according to a reported deterministic rule. This indicates that subsets of rrRNs can be scaling and, although rrRNs are asymptotically plane-filling due to the law of large numbers, scaling ITRNs can also display fractional dimension.