Patterns in Complex Networks via Spectral Analysis

Complex networks represent a variety of real-world systems in biology, ecology, society and technology. The study of structural properties of such systems has a tremendous impact in our understanding of their function, organisation and dynamics. Here I present a series of results toward the structural characterisation of complex networks. I start by analysing the centrality of nodes in complex networks and we introduce a measure which accounts for the participation of a node in all subgraphs in the network. This method is used to obtain a universal classification of networks into four topological classes. Then, I will develop a method to characterise the communicability between nodes in a network. The method is illustrated by ranking webpages in WWW and it is compared to other algorithms such as PageRank, SALSA, etc. Using the communicability approach I develop a method to identify overlapped communities in networks. I finalise by extending these ideas to account for general matrix functions.