Braess's paradox and power-law nonlinearities in networks

Abstract

We study flows in physical networks with a potential function defined over the nodes and a flow defined over the arcs. The networks have the further property that the flow on an arc a is a given increasing function of the difference in potential between its initial and terminal node. An example is the equilibrium flow in water-supply pipe networks where the potential is the head and the Hazen-Williams rule gives the flow as a numerical factor k a times the head difference to a power s > 0 (and s ≅ 0.54). In the pipe-network problem with Hazen-Williams nonlinearities, having the same s > 0 on each arc, given the consumptions and supplies, the power usage is a decreasing function of the conductivity factors k a . There is also a converse to this. Approximately stated, it is: if every relationship between flow and head difference is not a power law, with the same s on each arc, given at least 6 pipes, one can arrange (lengths of) them so that Braess's paradox occurs, i.e. one can increase the conductivity of an individual pipe yet require more power to maintain the same consumptions.