  We investigate the structural organization of the point-to-point electric,
diffusive or hydraulic transport in complex scale-free networks. The random
choice of two nodes, a source and a drain, to which a potential difference is
applied, selects two tree-like structures, one emerging from the source and the
other converging to the drain. These trees merge into a large cluster of the
remaining nodes that is found to be quasi-equipotential and thus presents
almost no resistance to transport. Such a global "tree-cluster-tree" structure
is universal and leads to a power law decay of the currents distribution. Its
exponent, $-2$, is determined by the multiplicative decrease of currents at
successive branching points of a tree and is found to be independent of the
network connectivity degree and resistance distribution.
