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This study introduces a unified model for cascading failures in complex systems, merging k-core percolation and interdependent networks. The research reveals a rich phase diagram with novel two-stage transitions, offering deeper insights into system resilience.

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Area of Science:

  • Complex systems science
  • Statistical physics
  • Network science

Background:

  • Cascading failures are critical in complex systems.
  • Existing models like k-core percolation and interdependent networks study these failures separately.
  • A unified approach is needed to capture combined effects.

Purpose of the Study:

  • To develop and analyze a general model combining k-core percolation and interdependent networks.
  • To investigate the phase diagram of cascading failures in this unified model.
  • To analytically derive critical exponents and phase boundaries.

Main Methods:

  • Analytical solution of the combined model.
  • Extensive simulations for validation.
  • Phase diagram analysis by tuning k-core threshold and network coupling.

Main Results:

  • A rich phase diagram with novel features, including first- and second-order transitions.
  • Identification of two tricritical lines enclosing a two-stage transition region.
  • Discovery of a transition sequence (first-order → second-order → two-stage → first-order) with increasing k-core threshold.

Conclusions:

  • The unified model provides a more comprehensive understanding of cascading failures.
  • The novel transition phenomena highlight complex system behaviors not seen in isolated models.
  • Analytical derivations offer predictive power for system stability and resilience.