Robust network formation with biological applications

Abstract

<p>We have provided new results on the structure of optimal transportation networks obtained as minimizers of an energy cost functional posed on a discrete graph. The energy consists of a kinetic (pumping) and a material (metabolic) cost term, constrained by a local mass conservation law. In particular, we have proved that every tree (i.e., graph without loops) represents a local minimizer of the energy with concave metabolic cost. For the linear metabolic cost, we have proved that the set of minimizers contains a loop-free structure. Moreover, we enriched the energy functional such that it accounts also for robustness of the network, measured in terms of the Fiedler number of the graph with edge weights given by their conductivities. We examined fundamental properties of the modified functional, in particular, its convexity and differentiability. We provided analytical insights into the new model by considering two simple examples. Subsequently, we employed the projected subgradient method to find global minimizers of the modified functional numerically. We then presented two numerical examples, illustrating how the optimal graph's structure and energy expenditure depend on the required robustness of the network.</p>