Structural efficiency of percolation landscapes in flow networks

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.