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Area of Science:

  • Complex Systems
  • Network Science
  • Mathematical Physics

Background:

  • Network geometry profoundly influences network dynamics and critical phenomena.
  • Hyperbolic geometry in discrete manifolds affects percolation properties.
  • Nonamenable branching simplicial and cell complexes present unique geometric structures.

Purpose of the Study:

  • Investigate link percolation properties in 2D nonamenable branching simplicial and cell complexes.
  • Relate percolation in these complexes to interdependent percolation in multiplex networks.
  • Characterize the number and nature of phase transitions in these complex networks.

Main Methods:

  • Establishing a mathematical relation between branching cell complex percolation and multiplex network interdependent percolation.
  • Utilizing renormalization group theory to analyze phase transitions.
  • Analyzing the behavior of percolation probability and fractal exponents.

Main Results:

  • Branching cell complexes can exhibit more than two percolation phase transitions: upper, lower, and intermediate.
  • Intermediate transitions are characterized by discontinuities in percolation probability and fractal exponent.
  • The upper percolation transition can belong to diverse universality classes, including Berezinskii-Kosterlitz-Thouless (BKT) and discontinuous transitions.

Conclusions:

  • The geometry of nonamenable branching cell complexes leads to rich and complex percolation behavior.
  • The identified intermediate phase transitions offer new insights into network critical phenomena.
  • Renormalization group analysis reveals a variety of universality classes governing these transitions, extending existing theories.