Spectral Techniques in Complex Networks

Silvia Gago

Most physical, biological, chemical, technological and social systems have a network structure. Examples of complex networks range from cell biology to epidemiology or to the Internet. In the recent years, several models of complex networks have been proposed, as the random graph of Erdős and Rényi, the small-world model of Watts and Strogatz or the scale-free networks of Barabási and Albert. The topological structure of such networks can be fully described by the associated adjacency matrices and their spectral density. The rich information about the topological structure and diffusion processes can be extracted from the spectral analysis of the networks. For instance, the power-law behavior of the density of eigenvalues is a notable feature of the spectrum of scale-free networks. Dynamical network processes, like synchronization can be determined by the study of their Laplacian eigenvalues. Furthermore, the eigenvalues are related to many basic topological invariants of networks such as diameter, mean distance, betweenness centrality, etc. Spectral techniques are also used for the study of several network properties: community detection, bipartition, clustering, design of highly synchronizable networks, etc.