K-core decomposition of Internet graphs: hierarchies, self-similarity and measurement biases

We consider the $k$-core decomposition of network models and Internet graphs at the autonomous system (AS) level. The $k$-core analysis allows to characterize networks beyond the degree distribution and uncover structural properties and hierarchies due to the specific architecture of the system. We compare the $k$-core structure obtained for AS graphs with those of several network models and discuss the differences and similarities with the real Internet architecture. The presence of biases and the incompleteness of the real maps are discussed and their effect on the $k$-core analysis is assessed with numerical experiments simulating biased exploration on a wide range of network models. We find that the $k$-core analysis provides an interesting characterization of the fluctuations and incompleteness of maps as well as information helping to discriminate the original underlying structure

Interbank markets and multiplex networks: centrality measures and statistical null models

The interbank market is considered one of the most important channels of contagion. Its network representation, where banks and claims/obligations are represented by nodes and links (respectively), has received a lot of attention in the recent theoretical and empirical literature, for assessing systemic risk and identifying systematically important financial institutions. Different types of links, for example in terms of maturity and collateralization of the claim/obligation, can be established between financial institutions. Therefore a natural representation of the interbank structure which takes into account more features of the market, is a multiplex, where each layer is associated with a type of link. In this paper we review the empirical structure of the multiplex and the theoretical consequences of this representation. We also investigate the betweenness and eigenvector centrality of a bank in the network, comparing its centrality properties across different layers and with Maximum Entropy null models.Comment: To appear in the book "Interconnected Networks", A. Garas e F. Schweitzer (eds.), Springer Complexity Serie

Correlation between centrality metrics and their application to the opinion model

In recent decades, a number of centrality metrics describing network properties of nodes have been proposed to rank the importance of nodes. In order to understand the correlations between centrality metrics and to approximate a high-complexity centrality metric by a strongly correlated low-complexity metric, we first study the correlation between centrality metrics in terms of their Pearson correlation coefficient and their similarity in ranking of nodes. In addition to considering the widely used centrality metrics, we introduce a new centrality measure, the degree mass. The m order degree mass of a node is the sum of the weighted degree of the node and its neighbors no further than m hops away. We find that the B_{n}, the closeness, and the components of x_{1} are strongly correlated with the degree, the 1st-order degree mass and the 2nd-order degree mass, respectively, in both network models and real-world networks. We then theoretically prove that the Pearson correlation coefficient between x_{1} and the 2nd-order degree mass is larger than that between x_{1} and a lower order degree mass. Finally, we investigate the effect of the inflexible antagonists selected based on different centrality metrics in helping one opinion to compete with another in the inflexible antagonists opinion model. Interestingly, we find that selecting the inflexible antagonists based on the leverage, the B_{n}, or the degree is more effective in opinion-competition than using other centrality metrics in all types of networks. This observation is supported by our previous observations, i.e., that there is a strong linear correlation between the degree and the B_{n}, as well as a high centrality similarity between the leverage and the degree.Comment: 20 page