Classifying Complex Networks using Unbiased Local Assortativity

Assortativity is a network-level measure which quantifies the tendency of nodes to mix with similar nodes in a network. Local assortativity has been introduced as a measure to analyse the contribution of individual nodes to network assortativity. In this paper we argue that there is a bias in the formulation of local assortativity which favours low-degree nodes. We show that, after the bias is removed, local assortativity of a node can be interpreted as a scaled difference between the average excess degree of the node neighbours and the expected excess degree of the network as a whole. Finally, we study the local assortativity profiles of a number of model and real world networks, demonstrating that four classes of complex networks exist: (i) assortative networks with disassortative hubs, (ii) assortative networks with assortative hubs, (iii) disassortative networks with disassortative hubs, and (iv) disassortative networks with assortative hubs. Introduction Many complex systems are amenable to be described as networks, with a given number of nodes and connecting edges. These include ecological systems, author collaborations, metabolism of biological species, and interaction of autonomous systems in the Internet, among others (Solé and Valverde, 2004; Albert and Barabasi, 2002; Albert et al., 1999; Newman, 2003; Faloutsos et al., 1999). It has been a recent trend to study common topological features of such networks. Network diameter, clustering coefficients, modularity and community structure, information content are some features analysed in recent literature in this regard (Faloutsos et al., 1999; Alon, 2007; Lizier et al., 2009; Prokopenko et al., 2009). One such measure which has been analysed extensively is assortativity (Solé and Valverde, 2004; Newman, 2002; Albert and Barabasi, 2002; Newman, 2003; Callaway et al., 2001; Palsson, 2006; Maslov and Sneppen, 2002; Zhou et al., 2008; Bagler and Sinha, 2007; Vázquez, 2003). Having originated in ecological and epidemiological literature (Albert and Barabasi, 2002), the term ‘assortativity’ refers to the correlation between the properties of adjacent network nodes. While similarity between adjacent nodes can be measured in a number of ways, the property that is of interest to us is node degree. Based on degree-degree correlations, assortativity has been defined as a correlation function, and the level of assortative mixing has been measured quantitatively for a number of networks, including social, biological and technical networks (Solé and Valverde, 2004). The networks that have a positive correlation coefficient are called assortative: similar nodes tend to mix with each other in such networks. The networks characterised by a negative correlation coefficient are called disassortative: dissimilar nodes tend to connect predominantly in these networks. The precise local contribution of each node to the global level of assortative mixing can also be quantified (Piraveenan et al., 2008, 2009b, 2010). This quantity has been called “local assortativity”. Local assortativity measures the local contribution of each node to the global correlation coefficient which is the network assortativity. Local assortativity profiles (as distributions of local assortativity over nodes’ degrees) can also be constructed for various networks, and these profiles, in turn, can be used to classify networks (Piraveenan et al., 2008, 2009a). Two such classes of disassortative networks have been proposed in Piraveenan et al. (2008). In this paper, we demonstrate that the formulation proposed for local assortativity in Piraveenan et al. (2008) has a bias, which favours low-degree nodes over hubs. This bias needs to be removed before networks can be analysed in terms of local assortativity. Therefore, our objective is twofold: (i) to propose an unbiased formulation of local assortativity, and (ii) to characterise classes of networks in terms of this unbiased formulation. After presenting the unbiased formulation for local assortativity, we show that the classification of disassortative real-world networks that was proposed in Piraveenan et al. (2008) still holds, and in addition, there are two similar classes among assortative networks as well. The unbiased formulation also provides a clearer interpretation of what it means for a node to be locally assortative. Definitions and Terminology We need to introduce a number of definitions before removing the bias from the formulation of local assortativity. Consider a network with N nodes and M links. Assortativity for