Predicting Braess' Paradox in Supply and Transport Networks

TL;DR

This paper introduces a novel method to predict and understand Braess' paradox in supply and transport networks by mapping the problem to electrostatic dipole currents, enabling targeted network improvements.

Contribution

It presents a dual electrostatic current mapping and an approximate rerouting alignment criterion to predict Braessian edges and improve network robustness.

Findings

Exact mapping to electrostatic dipole currents

Efficient prediction of Braessian edges

Mitigation of network overload by weakening specific edges

Abstract

Reliable functioning of supply and transport networks fundamentally support many non-equilibrium dynamical systems, from biological organisms and ecosystems to human-made water, gas, heat, electricity and traffic networks. Strengthening an edge of such a network lowers its resistance opposing a flow and intuitively improves the robustness of the system's function. If, in contrast, it deteriorate operation by overloading other edges, the counterintuitive phenomenon of \emph{Braess' paradox} emerges. How to predict which edges enhancements may trigger Braess' paradox remains unknown to date. To approximately locate and intuitively understand such Braessian edges, we here present a differential perspective on how enhancing any edge impacts network-wide flow patterns. First, we exactly map the prediction problem to a dual problem of electrostatic dipole currents on networks such that simultaneously finding \textit{all} Braessian edges is equivalent to finding the currents in the resistor network resulting from a constant current across one edge. Second, we propose a simple approximate criterion -- rerouting alignment -- to efficiently predict Braessian edges, thereby providing an intuitive topological understanding of the phenomenon. Finally, we show how to intentionally weaken Braessian edges to mitigate network overload, with beneficial consequences for network functionality.