Exploration of Network Scaling: Variations on Optimal Channel Networks

Metabolic allometry, a common pattern in nature, is a close-to-3/4-power scaling law between metabolic rate and body mass in organisms, across and within species. An analogous relationship between metabolic rate and water volume in river networks has also been observed. Optimal Channel Networks (OCNs), at local optima, accurately model many scaling properties of river systems, including metabolic allometry. OCNs are embedded in two-dimensional space; this work extends the model to three dimensions. In this paper we compare characteristics of 3d OCNs with 2d OCNs and with organic metabolic networks, studying the scaling behaviors of area, length, volume, and energy. In addition, we take a preliminary look at comparing Steiner trees with OCNs. We find that the three-dimensional OCN has predictable characteristics analogous to those of the two-dimensional version, as well as scaling properties similar to metabolic networks in biological organisms.