%T River Networks and Optimal Channel Networks
%L eprints3738
%X River networks represent a perfect example of a physical
phenomenon that can be described by means of graph theory.
Water collected by rainfall flows from one point to another
one (downstream) in the river basin creates a spanning (water
flows uniformly on the terrain and therefore from every point
of the basin we have water flow) tree (water cannot flow uphill).
Rivers on Earth and even those that might have been
present on Mars all display similar statistical properties
thereby calling for a model based on basic properties. 
A class of models named Optimal Channel Networks (OCN) derive 
the final configuration by minimising a given cost function.
The physical inspiration for the minimization problem traces
back to the ideas of Nobel laureate Prigogine on a general
theory of irreversible processes in open dissipative systems.
Actually, theoretical results from OCN allowed to provide an
explanation to universal allometric behaviour in a variety
of different physical situations from species distribution to food webs optimisation alternative to the traditional
approach. In the specific case of river networks, the OCN
model postulates that the total gravitational energy loss in the
system is minimised. Empirical and theoretical works focus
generally on two dimensional case, while recently (inspired by
vascular systems) also the three dimensional case has been
analysed.
Here we devise some new analytical results that illustrate
the role and the properties of the structure that minimises
the cost function proposed in the ABM and we also provide
some insight about the structure of the absolute minimum by varying some of the parameters of the model. In what follows we will give a theoretical characterization of river networks and provide a simple rule to distinguish spanning trees from natural river trees. Furthermore, we extend the study of OCNs embedded on a lattice finding a lower and upper bound for the energy of an OCN in any dimension D.
%I IMT Institute for Advanced Studies Lucca
%A Paul Balister
%A J?szef Balogh
%A B?la Bollob?s
%A Guido Caldarelli
%A Rossana Mastrandrea
%A Rob Morris