Error and attack tolerance of small-worldness in complex networks

Complex networks may undergo random and/or systematic failures in some of their components, i.e. nodes and edges. These failures may influence various network properties. In this article, for a number of real-world as well as Watts–Strogatz model networks, we investigated the profile of the network small-worldness as random failures, i.e. errors, or systematic failures, i.e. attacks, occurred in the nodes. In errors nodes are randomly removed along with all their tipping edges, while in attacks the nodes with highest degrees are removed from the network. Interestingly, in many cases, the small-worldness of violated networks increased as more nodes underwent an attack. This indicates an important role of the hub nodes in controlling the small-worldness of Watts–Strogatz networks. The profile of changes in the small-worldness as a result of errors/attacks was independent of network size, while it was influenced by average degree and rewiring probability of Watts–Strogatz model. We also found that the pattern of the changes of the small-worldness in real-world networks is completely different than that of the Watts–Strogatz networks. Therefore, although Watts–Strogatz model is often used for constructing networks with small-world property, the resulting networks have different properties compared to real-world ones in terms of robustness in the small-worldness index against errors/attacks.