Percolation on branching simplicial and cell complexes and its relation to interdependent percolation

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.