Critical manifolds in non-linear response of complex materials

We describe two geometric structures, the shortest path and the minimum cut, and show that these structures emerge at special threshold points in the highly non-linear electrical response of complex networks. Algorithms which find the shortest path and the minimum cut directly as well as methods for finding the full non-linear response of complex networks are outlined. Scaling laws for the behavior of the shortest path and minimum cut in random networks are then surveyed. Finally, applications of the shortest path and minimum cut to grain boundary controlled polycrystalline materials are elucidated.