2015/11/26 by Alessio Brancolini, Benedikt Wirth
Engineering · Mathematics · Physics and Astronomy · #Complex Network Analysis Techniques #Energy (signal processing) #Energy transport #Ramification #Regular polygon #Relaxation (psychology) #Scale (ratio) #Scaling #Slime Mold and Myxomycetes Research #Traffic control and management #Transport network #math.AP #math.CA #math.OC #msc:49Q10 #msc:49Q20 #msc:90B10
paper · pdf · doi:10.1137/15m1050227
published as SIAM J. Math. Anal. 49 (2017), no. 1, 311-359
arxiv created 2015/11/26 · openalex created_date 2016/06/24 · openalex publication_date 2017/01/01 · arxiv updated 2017/11/16 · openalex updated_date 2026/08/06
We consider two variational models for transport networks, an urban planning and a branched transport models, in which the degree of network complexity and ramification is governed by a small parameter ε>0. Smaller ε leads to finer ramification patterns, and we analyze how optimal network patterns in a particular geometry behave as ε→ 0 by proving an energy scaling law. This entails providing constructions of near-optimal networks as well as proving that no other construction can do better. The motivation of this analysis is twofold. On the one hand, it provides a better understanding of the transport network models; for instance, it reveals qualitative differences in the ramification patterns of urban planning and branched transport. On the other hand, the analysis illustrates several variations and refinements of a well-established proof technique based on relaxation and convex duality. Transport networks provide one of the simplest settings in which to explore such variations.