Optimized evolution of networks for principal eigenvector localization

Network science is increasingly being developed to get new insights about behavior and properties of complex systems represented in terms of nodes and interactions. One useful approach is investigating localization properties of eigenvectors which have diverse applications ranging from detection of influential nodes to disease-spreading phenomena in underlying networks. In this work, we evolve an initial random network with an edge rewiring optimization technique considering the inverse participation ratio as a fitness function to obtain a network having a localized principle eigenvector and analyze various properties of the optimized networks. We report few special structural features of such optimized networks including an existence of a set of edges which are necessary for the localization, and rewiring only one of them leads to a complete delocalization of the principal eigenvector. We analytically derive a condition on eigenvector entries for changes in the IPR values as a function of edge rewiring. The derivation provides an easy method of finding a pair of nodes, rewiring among them lead to an enhancement in the eigenvector localization.