# Susceptible-infected-susceptible model on networks with eigenvector localization

## TL;DR

This paper investigates how eigenvector localization in network adjacency matrices influences the SIS epidemic model, revealing a vanishing threshold and highlighting the role of highly connected nodes in sustaining infections.

## Contribution

It introduces a critical condition for phase transition based on star subgraph lifespan and elucidates the impact of eigenvector localization on infection dynamics in networks.

## Key findings

- Infection density depends on node degree and star lifespan.
- Eigenvector localization enhances infection in hubs and suppresses it in low-degree nodes.
- The average star lifespan divergence causes a vanishing epidemic threshold.

## Abstract

It is a longstanding debate on the absence of threshold for susceptible-infected-susceptible (SIS) model on networks with finite second order moment of degree distribution. The eigenvector localization of the adjacency matrix for a network gives rise to the inactive Griffiths phase featuring slow decay of the activity localized around highly connected nodes due to the dynamical fluctuation. We show how it dramatically changes our understanding of SIS model, opening up new possibilities for the debate. We derive the critical condition for Griffiths to active phase transition: on average, an infected node can further infect another one in the characteristic lifespan of the star subgraph composed of the node and its nearest neighbors. The system approaches the critical point of avoiding the irreversible dynamical fluctuation and the trap of absorbing state. As a signature of the phase transition, the infection density of a node is not only proportional to its degree, but also proportional to the exponentially growing lifespan of the star. And the divergence of the average lifespan of the stars is responsible for the vanishing threshold in the thermodynamic limit. The eigenvector localization exponentially reinforces the infection of highly connected nodes, while it inversely suppresses the infection of small-degree nodes.