Localization of principal eigenvector in complex networks

Abstract

Networks provide a simple framework to understand and predict the properties of complex real-world systems by modeling them in terms of interacting units [28]. This framework is successful in explaining various mechanisms behind the emergence of collective behaviors of systems arising due to the local interaction patterns of their components. Much of the current effort has given to understand the relationship between structure as well as the dynamics of a complex system. As structure always affects the functionality of a system. The eigenvector corresponding to the largest eigenvalue of the adjacency matrix, referred to as the principal eigenvector (PEV), provides information about both the structural and dynamical behavior of the underlying systems [9]. For various linear dynamical processes on networks, for instance, epidemic spreading, population dynamics of Ribonucleic acid (RNA) neutral networks, rumor spreading, brain network dynamical models the steadystate vector has been shown to be approximated using the PEV of the adjacency matrix [9]. Hence, analyzing the behavior of PEV can help us to understand how an individual entity is infected or how information spreads in a network in the steadystate. Particularly, the localization behavior of PEV has been helpful in gaining insight into the propagation or restriction of information in the underlying systems. For instance, analyzing localization behavior of PEV have been shown to provide insight into the criticality in brain network dynamics and disease spreading near epidemic threshold in networks [4, 11]. Localization of PEV refers to a state where a few components of the vector take very high values while the rest of the components take small values independent of the network size. In other words, localization of PEV in networks says few nodes contribute more to the dynamical process, and the rest of the others have very less contribution in steady-state. Similarly, PEV is said to be delocalized when all the entries in PEV receive almost the same weight independent of the network size. Therefore, for a delocalized PEV, all the nodes have the same amount of contribution to a corresponding dynamical system. The topic of eigenvector localization is worthy of investigation and arise sufficient interest to the researchers working on network theory and statistical physics [20]. Mainly, emergence or absence of eigenvector localization in networks is of broad importance ranging from spreading dynamics to the numerical computation [20]. Discovering and analyzing network properties leading to increase or decrease the localization of PEV of adjacency matrix can help in understanding the behavior of a linear dynamical process in steady-state and which we have studied in the current thesis.