Hierarchy and vulnerability of complex networks

Abstract

In this paper we suggest a method for studying complex networks vulnerability. This method takes into account the network topology, the node dynamics and the potential node interactions. It is based on the PageRank and VulnerabilityRank algorithms. We identify the problem with these algorithms, i.e. they tend towards zero for very large networks. Thus, we propose another method to evaluate the amount of hierarchy in a given complex network, by calculating the relative variance of the system vulnerability. This measure can be used to express how much one network is being hierarchical, thus revealing its vulnerability. We use the proposed method to discover the vulnerability and hierarchical properties of four characteristic types of complex networks: random, geometric random, scale-free and small-world. As expected, the results show that networks which display scale-free properties are the most hierarchical from the analyzed network types. Additionally, we investigate the hierarchy and vulnerability of three real-data networks: the US power grid, the human brain and the Erdös collaboration network. Our method points out the Erdös collaboration network as the most vulnerable one.