Complex networks characterized by global transport processes rely on the presence of directed paths from input to output nodes and edges, which organize in characteristic linked components. The analysis of such network-spanning structures in the framework of percolation theory, and in particular the key role of edge interfaces bridging the communication between core and periphery, allow us to shed light on the structural properties of real and theoretical flow networks, and to define criteria and quantities to characterize their efficiency at the interplay between structure and functionality. In particular, it is possible to assess that an optimal flow network should look like a hairy ball, so to minimize bottleneck effects and the sensitivity to failures. Moreover, the thorough analysis of two real networks, the Internet customer-provider set of relationships at the autonomous system level and the nervous system of the worm Caenorhabditis elegans --that have been shaped by very different dynamics and in very different time-scales--, reveals that whereas biological evolution has selected a structure close to the optimal layout, market competition does not necessarily tend toward the most customer efficient architecture.
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