Balister, Paul and Balogh, Jószef and Bollobás, Béla and Caldarelli, Guido and Mastrandrea, Rossana and Morris, Rob
*River Networks and Optimal Channel Networks.*
Working Paper
(Unpublished)

## Abstract

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.

Item Type: | Working Paper (Working Paper) |
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Subjects: | Q Science > QA Mathematics Q Science > QC Physics |

Research Area: | Economics and Institutional Change |

Depositing User: | Rossana Mastrandrea |

Date Deposited: | 04 Aug 2017 08:43 |

Last Modified: | 04 Aug 2017 08:43 |

URI: | http://eprints.imtlucca.it/id/eprint/3738 |

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