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Area of Science:

  • Complex networks
  • Network geometry
  • Percolation theory

Background:

  • Network geometry is an emerging field for analyzing complex network structure and dynamics.
  • Previous work showed discontinuous percolation on Farey graphs (a type of hyperbolic manifold).

Purpose of the Study:

  • Investigate the role of network geometry in percolation transitions on planar hyperbolic manifolds.
  • Explore critical properties of link percolation on diverse hyperbolic manifolds.

Main Methods:

  • Renormalization group analysis.
  • Study of hyperbolic manifolds built by gluing m-polygons.
  • Analysis of link percolation on deterministic and random hyperbolic manifolds.

Main Results:

  • Percolation transition nature depends on the power-law exponent (γ) of the polygon size distribution.
  • Hybrid percolation transitions observed for γ in (3,4).
  • Continuous percolation transitions observed for γ in (2,3].

Conclusions:

  • Network geometry significantly impacts percolation universality classes.
  • The study extends understanding of percolation on hyperbolic structures.
  • Tail exponents of size distributions are crucial for determining transition types.