Superstable geometry in triadic percolation.

Abstract

Triadic percolation turns bond percolation into a dynamical problem governed by an effective one-dimensional unimodal map. We show that the geometry of superstable cycles provides a direct, map-agnostic probe of local nonlinearity: specifically, the distance from the map's maximum to a distinguished next-to-maximum point on the attracting 2^{n} cycle (which coincides with a preimage of the maximum at 2^{n} superstability) scales as |Δp|^{γ}, with γ=1/z, where z is the nonflat order of the maximum. This prediction is verified across canonical unimodal families and heterogeneous triadic ensembles, with Lyapunov spectra corroborating the one-dimensional reduction. A derivative condition on the activation kernel fixes the local nonlinearity order z (and thus, under standard unimodal-map hypotheses, the associated z-logistic universality class) and gives conditions under which z>2 can be realized. The diagnostic operates directly on orbit data under standard regularity assumptions, providing a practical tool to classify universality in higher-order networks.