Abstract
The large-scale structure of complex systems is intimately related to their functionality and evolution. In particular, global transport processes in flow networks rely on the presence of directed pathways from input to output nodes and edges, which organize in macroscopic connected components. However, the precise relation between such structures and functional or evolutionary aspects remains to be understood. Here, we investigate which are the constraints that the global structure of directed networks imposes on transport phenomena. We define quantitatively under minimal assumptions the structural efficiency of networks to determine how robust communication between the core and the peripheral components through interface edges could be. Furthermore, we assess that optimal topologies in terms of access to the core should look like “hairy balls” so to minimize bottleneck effects and the sensitivity to failures. We illustrate our investigation with the analysis of three real networks with very different purposes and shaped by very different dynamics and time-scales–the Internet customer-provider set of relationships, the nervous system of the worm Caenorhabditis elegans, and the metabolism of the bacterium Escherichia coli. Our findings prove that different global connectivity structures result in different levels of structural efficiency. In particular, biological networks seem to be close to the optimal layout.
Highlights
Despite profound differences, natural and artificial networked systems share striking similarities
This core is usually connected to peripheral components: the in-component (IN), composed of all vertices that can reach the strongly connected component (SCC) but cannot be reached from it, and the out-component (OUT), made of all vertices that are reachable from the SCC but cannot reach it
Changing perspective from nodes to edges, this picture is complemented by the edge percolated map [18], where the number of relevant structures increases to five: edges connecting nodes within the IN, OUT, and SCC form respectively the edge in component (ICE), the edge out component (OCE), and the edge strongly connected component (SCE); and edges bridging the peripheral IN and OUT components to the SCC form the in and the out interfaces (ITF and OTF respectively)
Summary
Natural and artificial networked systems share striking similarities. Transport has been studied as one of the main functions influenced by topology [7,8] and functional design principles of global flux distributions have been discussed for biological networks [9,10,11] Despite these efforts, the ‘‘form follows function’’ assertion still remains to be fully understood from a complex network science perspective, a major difficulty being the fact that present network patterns are the result of nonstationary and adaptive evolutionary histories that can greatly vary depending on the network. In some of these models [16], spontaneous structures are able to form that exhibit the typical architectures of transport systems
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