Susceptible-infected-susceptible model on networks with eigenvector localization

Area of Science:

• Network science
• Epidemiology
• Statistical physics

Background:

• The susceptible-infected-susceptible (SIS) model is a fundamental epidemiological model used to study disease spread on networks.
• A longstanding debate exists regarding the absence of a threshold for the SIS model on networks with a finite second-order moment of degree distribution.
• Eigenvector localization in network adjacency matrices leads to a Griffiths phase, characterized by slow activity decay around highly connected nodes due to dynamical fluctuations.

Purpose of the Study:

• To re-evaluate the understanding of the SIS model by incorporating eigenvector localization and its impact on phase transitions.
• To derive a critical condition for the transition from the Griffiths (inactive) phase to the active phase in the SIS model.
• To investigate the role of star subgraphs and their lifespan in determining the epidemic threshold.

Main Methods:

• Analysis of the SIS model on networks with a finite second-order moment of degree distribution.
• Investigation of eigenvector localization properties of the network's adjacency matrix.
• Derivation of the critical condition for the Griffiths to active phase transition based on the lifespan of star subgraphs.

Main Results:

• A critical condition for the Griffiths to active phase transition is derived: on average, an infected node must infect another within the characteristic lifespan of its nearest-neighbor star subgraph.
• The infection density of a node is proportional to its degree and the exponentially growing lifespan of its associated star subgraph.
• Eigenvector localization enhances infection spread among highly connected nodes while suppressing it among low-degree nodes.

Conclusions:

• The study provides a new perspective on the SIS model by demonstrating how eigenvector localization influences epidemic dynamics and phase transitions.
• The derived critical condition offers a mechanism for understanding the vanishing threshold in the thermodynamic limit.
• The lifespan of star subgraphs emerges as a crucial factor in epidemic spread and network behavior.