River landscapes and optimal channel networks

Paul Balister, József Balogh, Enrico Bertuzzo, Béla Bollobás, Guido Caldarelli, Amos Maritan, Rossana Mastrandrea, Robert Morris, Andrea Rinaldo

Research output: Contribution to journalArticlepeer-review


We study tree structures termed optimal channel networks (OCNs) that minimize the total gravitational energy loss in the system, an exact property of steady-state landscape configurations that prove dynamically accessible and strikingly similar to natural forms. Here, we show that every OCN is a so-called natural river tree, in the sense that there exists a height function such that the flow directions are always directed along steepest descent. We also study the natural river trees in an arbitrary graph in terms of forbidden substructures, which we call k-path obstacles, and OCNs on a d-dimensional lattice, improving earlier results by determining the minimum energy up to a constant factor for every d ≥ 2. Results extend our capabilities in environmental statistical mechanics.

Original languageEnglish (US)
Pages (from-to)6548-6553
Number of pages6
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number26
StatePublished - Jun 26 2018


  • Graph theory
  • Landscape evolution
  • Slope-area law
  • Spanning trees

ASJC Scopus subject areas

  • General


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