Wheel graph strategy for PEV localization of networks

¹ Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore - Khandwa Road, Simrol, 453552, Indore, India
² Discipline of Biosciences and Biomedical Engineering, Indian Institute of Technology Indore - Khandwa Road, Simrol, 453552, Indore, India

Received: 5 November 2019
Accepted: 20 February 2020

Abstract

Investigation of eigenvector localization properties of complex networks is not only important for gaining insight into fundamental network problems such as network centrality measure, spectral partitioning, development of approximation algorithms, but is also crucial for understanding many real-world phenomena such as disease spreading, criticality in brain network dynamics. For a network, an eigenvector is said to be localized when most of its components take value near to zero, with a few components taking very high values. In this article, we devise a methodology to construct a principal eigenvector (PEV) localized network from a given input network. The methodology relies on adding a small component having a wheel graph to the given input network. By extensive numerical simulation and an analytical formulation based on the largest eigenvalue of the input network, we compute the size of the wheel graph required to localize the PEV of the combined network. Using the susceptible-infected-susceptible model, we demonstrate the success of this method for various models and real-world networks considered as input networks. We show that on such PEV localized networks, the disease gets localized within a small region of the network structure before the outbreaks. The study is relevant in controlling spreading processes on complex systems represented by networks.