Branching in flow networks with linear congestion

Abstract

In our modern world, we rely on the proper functioning of a variety of networks with complex dynamics. Many of them are prone to congestion due to high loads, which determines their operation and resilience to failures. In this article, we propose a fundamental model of congestion where travel times increase linearly with the load. We show that this model interpolates between shortest path and Ohmic flow dynamics, which both have a broad range of applications. We formulate the model as a quadratic programme and derive a generalization of Ohm's law, where the flow of every link is determined by a potential gradient in a nonlinear way. We provide analytic solutions for fundamental network topologies that elucidate the transition from localized flow to a branched flow. Furthermore, we discuss how to solve the model efficiently for large networks and investigate the resilience to structural damages.