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The susceptible-infected-susceptible (SIS) model on networks exhibits a Griffiths phase due to eigenvector localization. A critical condition for phase transition is derived, linking infection spread to star subgraph lifespan.

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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.