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 $|\Delta p|^{\gamma}$ with $\gamma = 1/z$, 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.