Braess's Paradox and Power-Law Nonlinearities in Five-Arc and Six-Arc Two-Terminal Networks

Abstract

We extend a general network theorem of Calvert and Keady (CK) relating to the minimum number of arcs needed to guarantee the occurrence of the Braess Paradox. We rephrase the CK theorems and express our proof in the terminology of traffic networks.

CK described their theorem in relation to a two-terminal network of liquid in pipes. “Approximately stated, it is: if every relationship between flow and head difference is not a power law, with the same (power) 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.” In relation to the original Braess situation of traffic network flows, the relationship is between flow and link-cost on a congested link.

Our extended theorem shows that 5 pipes (roads, links, arcs) arranged in a Wheatstone Bridge (WB) network (as in the original Braess network) are necessary and sufficient to produce a Braess paradox (BP) in a two-terminal network (not limited to liquid in pipes) if at least one of the five has a different conductivity law (not power s).