# Optimized evolution of networks for principal eigenvector localization

## TL;DR

This paper presents an optimization method to evolve networks with localized principal eigenvectors, revealing structural features that influence eigenvector localization and its dependence on multiple network properties.

## Contribution

The study introduces an edge rewiring optimization technique based on inverse participation ratio to produce networks with localized principal eigenvectors, highlighting structural features necessary for localization.

## Key findings

- Localized eigenvectors depend on specific network edges.
- Rewiring a single key edge causes delocalization.
- Multiple structural features influence eigenvector localization.

## Abstract

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 having diverse applications including 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. The evolution process yields a network having localized principal eigenvector. We analyze various properties of the optimized networks and those obtained at the intermediate stage. Our investigations reveal the existence of few special structural features of such optimized networks including the presence 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. Our investigation reveals that PEV localization is not a consequence of a single network property, and preferably requires co-existence of various distinct structural as well as spectral features.