Zamora-López, GorkaBrasselet, Romain2023-01-242023-01-242019Zamora-López G, Brasselet R. Sizing complex networks. Commun Phys. 2019 Nov 14;2:144. DOI: 10.1038/s42005-019-0239-02399-3650http://hdl.handle.net/10230/55403Supplemental material files: supporting information; replication fileAmong 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.application/pdfengThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.info:eu-repo/semantics/articlehttp://dx.doi.org/10.1038/s42005-019-0239-0Applied mathematicsComplex networksinfo:eu-repo/semantics/openAccess