Superstable geometry in triadic percolation

Area of Science:

• Complex Networks
• Dynamical Systems
• Statistical Physics

Background:

• Triadic percolation models network dynamics.
• Unimodal maps describe these dynamics effectively.
• Understanding local nonlinearity is key to network universality.

Purpose of the Study:

• To develop a map-agnostic method for probing local nonlinearity in triadic percolation.
• To connect network universality classes to the order of nonlinearity.
• To provide a practical tool for analyzing higher-order networks.

Main Methods:

• Analyzing the geometry of superstable cycles in the effective one-dimensional unimodal map.
• Calculating the scaling of cycle points relative to the map maximum.
• Using Lyapunov spectra to confirm the one-dimensional reduction.

Main Results:

• The distance to cycle points scales with a power law |Δp|^{γ}, where γ=1/z.
• The nonflat order 'z' of the map's maximum determines this scaling.
• This method successfully classifies universality across different unimodal families and triadic ensembles.

Conclusions:

• Cycle geometry offers a direct probe of local nonlinearity, independent of the specific map.
• The nonflat order 'z' can be determined from network properties, linking to universality classes.
• This diagnostic tool facilitates the classification of universality in complex, higher-order networks.