Superstable Geometry in Triadic Percolation

TL;DR

The paper tackles universal classification in triadic percolation by using a geometry-based, map-agnostic diagnostic rooted in superstable cycles of the effective one-dimensional map. By analyzing the distance $|R_n(p_n)-R_m|$ from the maximum $R_m$ to the next-to-maximum point on the attracting $2^n$-cycle, it derives the scaling $|Δp|^{γ}$ with $γ=1/z$, where $z$ is the nonflat order of the map peak. The authors derive the reduced map $R^{(t)} = H_p(R^{(t-1)})$, identify $R_m$ via $\Phi_p'(R_m)=0$, locate $2^n$-superstable points through minima of the Lyapunov exponent, and provide Algorithm 1 to extract $γ$ directly from orbit data. They validate the approach on canonical unimodal families and heterogeneous triadic ensembles, showing $γ=1/z$ with $z=2$ (and $z=4$ in targeted cases), and corroborate the effective one-dimensionality with Lyapunov spectra. The work offers a practical diagnostic for universality in higher-order networks, robust to finite sampling and applicable directly to trajectory data, with implications for designing regulator statistics to realize nonstandard universality classes.

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.

Figures (5)

• Figure 1: Benchmarks on canonical unimodal maps. (a,b) Alternating-branch tracking for $z=2$ and $z=4$. (c,d) Log--log fits return $\gamma=0.5$ (quadratic) and $\gamma=0.25$ (quartic), verifying $\gamma=1/z$. $a_i$ is the slope from a fit of the data points $(\log \Delta p_{ni},\log d_{ni})$ over the chosen fit range (equivalently, the corresponding $i$-range) for the tracked branch/sequence.
• Figure 2: Period doubling and chaos in triadic percolation. (a) Poisson structure ($\langle k\rangle=30$) with Poisson regulators ($\langle \hat{\kappa}^+\rangle=1.8$, $\langle \hat{\kappa}^-\rangle=2.5$). (b) Scale-free structure (degree exponent $\gamma_{\mathrm{deg}}=2.5$, $k_{\min}=4$, $k_{\max}=100$) with Poisson regulators ($\langle \hat{\kappa}^+\rangle=10$, $\langle \hat{\kappa}^-\rangle=2.8$). (c,d) Lyapunov spectra: the leading exponent $\lambda>0$ in chaotic windows; a two-dimensional embedding yields $\lambda_1 = \lambda_{\max}\approx\lambda$ and a strongly negative subleading exponent $\lambda_2\ll 0$, indicating effectively one-dimensional dynamics. Here, $\lambda$ is the one-dimensional Lyapunov exponent, $\lambda_{\max}$ is the largest exponent estimated using Kantz's method Kantz1994, and $\lambda_1$ is the largest exponent obtained from QR decomposition.
• Figure 3: Superstable geometry in triadic percolation (Poisson regulators). (a) Alternating-branch selection produces a self-similar pattern anchored at the critical height $R_m$. (b) Power-law fits of $|\mathcal{R}_n(p_n)-R_m|$ versus parameter distance yield $\gamma\simeq 0.5$, consistent with a quadratic local maximum. At each $2^n$–superstable parameter $p_n$ the tracked point $\mathcal{R}_n(p)$ coincides with the unique preimage that maps to $R_m$ in one iterate; away from $p_n$ it is its smooth continuation. $a_i$ is the slope from a least-squares fit of the data points $(\log \Delta p_{ni},\log d_{ni})$ over the chosen fit range (equivalently, the corresponding $i$-range) for the tracked branch/sequence.
• Figure 4: Hill-type effective kernels. (a,b) Orbit diagrams and alternating-branch tracking for the kernels $p_L(x)=p\,(1-x)^{1/2}\bigl(1-(1-x)^{3/2}\bigr)$ and $p_L(x)=p\,(1-x)^{1/2}\bigl(1-(1-x)^{7/2}\bigr)$. (c,d) Broad-range fits can show intermediate slopes, but near the accumulation region the local exponent converges to $\gamma\to 0.5$, indicating local quadratic control. $a_i$ is the slope from a least-squares fit of the data points $(\log \Delta p_{ni},\log d_{ni})$ over the chosen fit range (equivalently, the corresponding $i$-range) for the tracked branch/sequence.
• Figure 5: Examples of triadic-percolation dynamics under simplified regulatory actions controlling the system’s nonlinearity (“cap”) set to quadratic or quartic. In all three cases, the structural network is generated from a Poisson distribution, with different $c$. (a) $c=7.5$: dynamics are regulated solely by negative regulators, with $p_L^{(t)} = p \bigl(1 - G_0^-(R^{(t)})\bigr)$. When $G_0^-$ is taken as a regular network of degree $d$, the system’s nonlinearity---and thus its local universality class in the sense of the $z$-logistic unimodal maps---is controlled by $d$. Since there is no period doubling or chaotic regime, the geometry of superstable points does not apply; however, the orbit diagram near the transition suggests a universality different from the classical ($d=2$) when $d=4$. (b) Same as in (a), except the regulator is a mixture of two regular networks: $p_L^{(t)} = p \bigl(1 - \bigl((1-\theta) + \theta\,G_0^-(R^{(t)})\bigr)\bigr)$ with $\theta=0.3$. (c) System with $c=22$ and $p_L^{(t)} = p_1 - p_2 \bigl(R^{(t)}\bigr)^{d}$ for $p_1=0.58$ and $p_2=0.42$.