Topological phase transition in the Scheidegger model of river networks.

Transport networks are found at the heart of myriad natural systems, yet are poorly understood, except for the case of river networks. The Scheidegger model, in which rivers are convergent random walks, has been studied only in the case of flat topography, ignoring the variety of curved geometries found in nature. Embedding this model on a cone, we find a convergent and a divergent phase, corresponding to few, long basins and many, short basins, respectively, separated by a singularity, indicating a phase transition. Quantifying basin shape using Hacks law l ~ a(h) gives distinct values for h, providing a method of testing our hypotheses. The generality of our model suggests implications for vascular morphology, in particular, differing number and shapes of arterial and venous trees.