LAUSR

Among the many features of natural and man-made complex networks the small-world phenomenon is a relevant and popular one. But, how small is a small-world network and how does it compare to others? Despite its importance, a reliable and comparable quantification of the average pathlength of networks has remained an open challenge over the years. Here, we uncover the upper (ultra-long (UL)) and the lower (ultra-short (US)) limits for the pathlength and efficiency of networks. These results allow us to frame their length under a natural reference and to provide a synoptic representation, without the need to rely on the choice for a null-model (e.g., random graphs or ring lattices). Application to empirical examples of three categories (neural, social and transportation) shows that, while most real networks display a pathlength comparable to that of random graphs, when contrasted against the boundaries, only the cortical connectomes prove to be ultra-short.Small-world networks describe mathematically many natural and man-made networks such as neurons, power grids or social networks, but a measure of how small a small-world network is, remains a subject of debate. The authors identify the limiting cases with the shortest and longest average path for a given number of nodes and edges, which can be used as benchmarks to evaluate the average shortest path length for both empirical and model networks.