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.

翻译：三元渗流将键渗流转化为由有效一维单峰映射控制的动力学问题。我们证明，超稳定环的几何结构为局部非线性提供了直接且不依赖于具体映射的探测手段：具体而言，从映射最大值点到吸引$2^n$环上一个特定的次最大点（该点与$2^n$超稳定性时最大值的一个原像重合）的距离按$|Δp|^γ$标度变化，其中$γ= 1/z$，而$z$为最大值点的非平坦阶数。该预测在典型单峰映射族和异质三元系综中得到验证，李雅普诺夫谱进一步证实了一维约化的有效性。激活核的导数条件确定了局部非线性阶数$z$（进而在标准单峰映射假设下确定了相应的$z$-逻辑斯蒂普适类），并给出了实现$z>2$的条件。该诊断方法在标准正则性假设下可直接对轨道数据进行操作，为高阶网络中的普适性分类提供了实用工具。