Collective effects of link failures in linear flow networks

The reliable operation of supply networks is crucial for the proper functioning of many systems, ranging from biological organisms such as the human blood transport system or plant leaves to man-made systems such as power grids or gas pipelines. Whereas the failure of single transportation links has been analysed thoroughly, the understanding of multiple failures is becoming increasingly important to prevent large scale damages. In this publication, we examine the collective nature of the simultaneous failure of several transportation links. In particular, we focus on the difference between single link failures and the collective failure of several links. We demonstrate that collective effects can amplify or attenuate the impacts of multiple link failures - and even lead to a reversal of flows on certain links. A simple classifier is introduced to predict the overall strength of collective effects that we demonstrate to be generally stronger if the failing links are close to each other. Finally, we establish an analogy between link failures in supply networks and dipole fields in discrete electrostatics by showing that multiple failures may be treated as superpositions of multiple electrical dipoles for lattice-like networks. Our results show that the simultaneous failure of multiple links may lead to unexpected effects that cannot be easily described using the theoretical framework for single link failures