  The failure of a single link can degrade the operation of a supply network up
to the point of complete collapse. Yet, the interplay between network topology
and locality of the response to such damage is poorly understood. Here, we
study how topology affects the redistribution of flow after the failure of a
single link in linear flow networks with a special focus on power grids. In
particular, we analyze the decay of flow changes with distance after a link
failure and map it to the field of an electrical dipole for lattice-like
networks. The corresponding inverse-square law is shown to hold for all regular
tilings. For sparse networks, a long-range response is found instead. In the
case of more realistic topologies, we introduce a rerouting distance, which
captures the decay of flow changes better than the traditional geodesic
distance. Finally, we are able to derive rigorous bounds on the strength of the
decay for arbitrary topologies that we verify through extensive numerical
simulations. Our results show that it is possible to forecast flow rerouting
after link failures to a large extent based on purely topological measures and
that these effects generally decay with distance from the failing link. They
might be used to predict links prone to failure in supply networks such as
power grids and thus help to construct grids providing a more robust and
reliable power supply.
