Localization in complex networks

Abstract

Since the pioneering work of Anderson on localization of electronic wave function in disordered media, eigenvector localization has become a fascinating and active area of research [1]. In his original paper, Anderson argued that disorder introduced in the diagonal elements of a Hamiltonian matrix will lead to localization of the electronic wave function. Later this theory successfully explained the phenomenon of metal-insulator transition. Then after, the phenomenon of localization has been studied in various systems such as random regular graph (RRG) [8], Cayley tree [9], and random networks [10] [11] and it still remains an active field of research. Moreover, there exist systems with many interacting units that behave very unusu ally, and it becomes difficult to control or predict the system, such as the human brain and the world economy. These systems can be better understood in the frame work of networks. A network consists of nodes and links. The nodes correspond to the elements of a system and links represent the interactions between these el ements. Eigenvector localization serves well in understanding and getting insight into various structural and dynamical properties of networks.