Percolation landscapes, the layouts of global connectivity structures in self-organized complex networks, are the result of basic architectural transitions which ensure global communication essential to develop collective behavior. In directed networks, the characteristic topology in the percolated phase denotes a global flux that organizes into a bow-tie architecture of distinct linked components. The traditional node percolation map recognizes a core structure that usually acts as a processing unit transferring input into output, a picture that is complemented by the edge percolation map uncovering interfaces bridging core and periphery. This recursive pattern is further shaped by system dependent specificities that at the same time limit the performance and are the re flection of functional demands and evolutionary forces. In particular, interfaces encode information about the structural efficiency and robustness of the network as a global transport system and the potential risk of bottleneck effects. We define and discuss these concepts and use theoretical arguments to propose the hairy ball as the conformation of maximal structural efficiency. The analysis of real networks reveals that biological systems, exposed to long-term evolutionary pressure, may be much closer to optimality than information technologies systems, at an early stage of development dominated by competitive forces.
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