New approaches and their applications in measuring mixing patterns of complex networks

Abstract

In this thesis, mixing patterns of complex networks are analysed. Synthesised canonical networks, scale-free networks, small-world networks and random networks along with existing, real-world networks are analysed using various approaches. Assortativity is a measure that quantifies the similarity among nodes that are connected. In this thesis, two new approaches to quantify node assortativity have been proposed. First approach presented eliminates the dependency of node assortativity calculation on average excess degree, which was present in currently used approache. The second approach to node assortativity proposed is calculated based on the contribution of nodes toward the network assortativity. Similarly, a new approach to quantify the heterogeneity of nodes' neighbors has been proposed. It is shown that standard deviations of degree differences between nodes could be used to quantify the heterogeneity of nodes. This measure, which is called ‘versatility’ in this thesis, is then used to classify networks and used to identify the impact of versatility on other measures of networks. Using versatility calculations, it was found that there are three classes of real world networks: (i) Networks where the versatility converges to a non-zero value with node degrees (ii) Networks where the versatility converges to zero with node degrees (iii) Networks where the versatility does not converge with degree. Also, two cases were identified - a) Networks where the majority of the nodes have low versatility values, and b) Networks where the majority of the nodes have medium versatility values. It was found that often (i) and (ii) correlate with (a) and (iii) correlates with (b). Another measure called ‘Area Under Curve’, to quantify the level of herd-immunity present in a network is also introduced. Using this measure, it is shown that assortative networks exhibited higher levels of herd immunity.