Classes of small-world networks

TLDR

Small‑world networks are of recent interest because they can overcome limitations of random networks or regular lattices for modeling interaction networks in complex systems. The study aims to empirically examine the statistical properties of diverse real‑world networks. The authors analyze a variety of real‑world networks to assess their statistical characteristics. The analysis reveals three classes of small‑world networks—scale‑free, broad‑scale, and single‑scale—distinguished by their connectivity distributions, and indicates that constraints on link addition in broad‑scale and single‑scale networks likely drive the emergence of these classes.

Abstract

Small-world networks are the focus of recent interest because they appear to circumvent many of the limitations of either random networks or regular lattices as frameworks for the study of interaction networks of complex systems. Here, we report an empirical study of the statistical properties of a variety of diverse real-world networks. We present evidence of the occurrence of three classes of small-world networks: (a) scale-free networks, characterized by a vertex connectivity distribution that decays as a power law; (b) broad-scale networks, characterized by a connectivity distribution that has a power-law regime followed by a sharp cut-off; (c) single-scale networks, characterized by a connectivity distribution with a fast decaying tail. Moreover, we note for the classes of broad-scale and single-scale networks that there are constraints limiting the addition of new links. Our results suggest that the nature of such constraints may be the controlling factor for the emergence of different classes of networks.