Structure of Optimal Transport Networks Subject to a Global Constraint

TL;DR

This paper analytically characterizes the structure of energy-efficient transport networks with a fixed total volume, showing they are loopless and have limited additional nodes, applicable to various physical systems.

Contribution

It proves that optimal networks minimizing energy dissipation are loopless and have at most N-2 extra nodes, regardless of node placement.

Findings

Optimal networks are loopless.

Number of additional nodes is at most N-2.

Results apply to various physical transport systems.

Abstract

The structure of pipe networks minimizing the total energy dissipation rate is studied analytically. Among all the possible pipe networks that can be built with a given total pipe volume (or pipe lateral surface area), the network which minimizes the dissipation rate is shown to be loopless. Furthermore, such an optimal network is shown to contain at most N-2 nodes in addition to the N sources plus sinks that it connects. These results are valid whether the possible locations for the additional nodes are chosen freely or from a set of nodes (such as points of a grid). Applications of these results to various physical situations and to the efficient computation of optimal pipe networks are also discussed.