Error and attack tolerance of small-worldness in complex networks

Abstract

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.

Introduction

Available tools of network theory are nowadays widely used to represent real-world systems’ both structural features and dynamics (Boccaletti et al., 2006, Newman et al., 2006). Complex networks are everywhere and we can find them in different disciplines, ranging from biology to medicine, sociology and engineering (Boccaletti et al., 2006, Strogatz, 2001). Many of real-world networks show common non-trivial structural properties such as scale-free (Barabasi & Albert, 1999) and small-world attributes (Watts & Strogatz, 1998), and their function largely depends on these features (Boccaletti et al., 2006, Newman et al., 2006). Watts and Strogatz showed that many real-world networks have a structure with some properties similar to pure random networks and some others similar to regular networks (Watts & Strogatz, 1998). They have small characteristic path length comparable to that of pure random graphs, and at the same time, their transitivity, i.e. clustering coefficient, is close to that of regular networks that is much higher as compared to pure random graphs (Watts, 2003b, Watts and Strogatz, 1998). This property makes it easier to have processes such as navigation (Kleinberg, 2000) and synchronization (Barahona & Pecora, 2002) in such networks. Later on, scholars revealed this property, i.e. small characteristic path length and large clustering coefficient, in many real-world networks such as human brain functional network (Sporns and Honey, 2006, Sporns and Zwi, 2004), food webs (Montoya & Sole, 2002), protein–protein interaction network (Goldberg & Roth, 2003), and many social networks (Newman and Park, 2003, Strogatz, 2001, Watts, 2003b). One may calculate the degree of small-worldness in a network by comparing its properties with those of properly randomized networks (Humphries and Gurney, 2008, Humphries et al., 2006).

Networks may undergo errors or attacks in their edges and/or nodes and consequently lose some of their components. Therefore, it is important to study the tolerance of critical network properties to errors – failures of randomly chosen nodes and/or edges of the networks and attacks – systematic failures of components that play a critical role in the network, e.g. the hub nodes with high degrees (Albert et al., 2000, Buldyrev et al., 2010). For example, accidental or systematic attacks may lead to large blackouts in power networks (Arianos, Bompard, Carbone, & Xue, 2009), and thus, the network should be designed in a way to be resilient against such attacks. Many complex networks display surprising robustness against failures (Albert et al., 2000). Biological networks are prototype examples that have shown to be robust against errors that might happen in their structure (Albert et al., 2000, Alon, 2003, Barkai and Leibler, 1997, Melian and Bascompte, 2002, Williams and Martinez, 2000). In general, it has been shown that scale-free networks, i.e. networks whose node-degree distribution is heavy-tailed, are robust against errors, but, at the same time, they are vulnerable to attacks (Albert et al., 2000, Cohen et al., 2001, Crucitti et al., 2003, Doyle et al., 2005). The influence of other types of attacks such as cascading failures has also been studied on various network structures (Buldyrev et al., 2010, Motter and Lai, 2002, Smart et al., 2008, Zhao et al., 2004), that gives insight into understanding the way these attacks change the networks properties. Errors and attacks may also influence the dynamical process happening on the network such as evolution of cooperation (Perc, 2009) and synchronization (Jalili, submitted for publication).

Often a number of network parameters are monitored as random or systematic failures occur in a network and the network robustness against such errors and attacks is linked to the profile of the change in these parameters. Efficiency of networks is one of those measures investigated in this context (Crucitti et al., 2003, Sun et al., 2007). Another frequently used indicator of network robustness is the size of the largest connected component (Albert et al., 2000, Callaway et al., 2000, Cohen et al., 2001). In this article we investigate the profile of network small-worldness in response to errors/attacks in the nodes of the networks. We extend the definition of the small-worldness proposed by Humphries and Gurney (Humphries and Gurney, 2008, Humphries et al., 2006), by considering local and global efficiency measures (Latora & Marchiori, 2001). Then, the networks’ small-worldness is monitored as errors/attacks happening in their nodes. As network structure we consider artificially constructed networks based on the model proposed by Watts and Strogatz (Watts & Strogatz, 1998) as well as a number of real-world networks. Interestingly we find different behaviour in real-world networks as compared to Watts–Strogatz networks in terms of robustness of the small-worldness index against failures.