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

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

On the influence of topological characteristics on robustness of complex networks

In this paper, we explore the relationship between the topological characteristics of a complex network and its robustness to sustained targeted attacks. Using synthesised scale-free, small-world and random networks, we look at a number of network measures, including assortativity, modularity, average path length, clustering coefficient, rich club profiles and scale-free exponent (where applicable) of a network, and how each of these influence the robustness of a network under targeted attacks. We use an established robustness coefficient to measure topological robustness, and consider sustained targeted attacks by order of node degree. With respect to scale-free networks, we show that assortativity, modularity and average path length have a positive correlation with network robustness, whereas clustering coefficient has a negative correlation. We did not find any correlation between scale-free exponent and robustness, or rich-club profiles and robustness. The robustness of small-world networks on the other hand, show substantial positive correlations with assortativity, modularity, clustering coefficient and average path length. In comparison, the robustness of Erdos-Renyi random networks did not have any significant correlation with any of the network properties considered. A significant observation is that high clustering decreases topological robustness in scale-free networks, yet it increases topological robustness in small-world networks. Our results highlight the importance of topological characteristics in influencing network robustness, and illustrate design strategies network designers can use to increase the robustness of scale-free and small-world networks under sustained targeted attacks