Phase transition in the link weight structure of networks.

When transport in networks follows the shortest paths, the link weights are shown to play a crucial role. If the underlying topology with nodes N is not changed and if the link weights are independent from each other, then we show that, by tuning the link weights, a phase transition occurs around a critical extreme value index alphac of the link weight distribution alpha<alphac. If the extreme value index of the link weight distribution , transport in the network traverses many links whereas for , all transport flows over a critical backbone consisting of N-1 links. For connected Erdös-Rényi random graphs Gp(N) and square lattices, we have characterised the phase transition and found that alphac approximately =bN(-beta) with betaGp(N) and betalattice approximately = 0.62.