Multilayer Triadic Percolation Model

• The MTP model is a framework that generalizes triadic percolation by incorporating both intra- and interlayer regulatory interactions across multiple network layers.
• It employs a two-dimensional discrete-time dynamical system to capture time-dependent oscillations, bifurcation phenomena, and routes to chaos.
• The model enhances understanding of nonstationary giant component dynamics in complex systems such as brain networks, climate systems, and social structures.

The Multilayer Triadic Percolation (MTP) Model is a theoretical framework that generalizes the dynamics of triadic interactions to networks composed of multiple interacting layers. In this model, higher-order regulatory effects—where a regulator node modulates the link between two other nodes—are extended to multilayer structures in which both intra- and interlayer triadic regulations can occur. The model fundamentally differs from classical percolation by endowing the order parameter (typically the size of the giant component) with intrinsic time dependence and by producing new bifurcation phenomena absent in both standard (pairwise) percolation and single-layer triadic percolation (Sun et al., 10 Oct 2025).

1. Formulation of the Multilayer Triadic Percolation (MTP) Model

The MTP model is constructed on multilayer networks, each layer containing its own structural edges and its own network of triadic regulatory interactions. Importantly, the regulatory nodes not only control links within their native layer (intralayer regulation) but can also modulate the connectivity in other layers (interlayer regulation). This results in a time-discrete dynamical system characterized by an iterated two-dimensional map governing the evolution of the percolating components in each layer.

At each iteration, the process consists of:

• Percolation Step (in each layer A, B): Compute the fraction of nodes in the giant component $R_A^{(t)}$ and %%%%1%%%% using standard percolation generating function formalism as,

$$R_A^{(t)} = 1 - G_{0,A}(1 - S_A^{(t)} p_A^{(t-1)}), \ S_A^{(t)} = 1 - G_{1,A}(1 - S_A^{(t)} p_A^{(t-1)}),$$

and analogously for $R_B^{(t)}$.

• Regulatory Update: The probability of a link being active in each layer at time $t$ combines both "structural" (random damage via parameter $p$) and regulatory (positive/negative) inputs from both intra- and interlayer interactions:

$$p_A^{(t)} = p \cdot p_A^{\mathrm{intra}}(t) \cdot p_A^{\mathrm{inter}}(t),\ p_B^{(t)} = p \cdot p_B^{\mathrm{intra}}(t) \cdot p_B^{\mathrm{inter}}(t),$$

where, e.g.,

$$p_A^{\mathrm{intra}}(t) = G_{A_\mathrm{intra}}^-(1 - R_A^{(t)})[1 - G_{A_\mathrm{intra}}^+(1 - R_A^{(t)})],$$

and analogous expressions hold for $p_A^{\mathrm{inter}}$, depending on the nature (positive or negative) and the degree distributions of regulatory interactions.

The mapping thus becomes a two-dimensional discrete-time dynamical system,

$$(R_A^{(t)}, R_B^{(t)}) \to (R_A^{(t+1)}, R_B^{(t+1)})$$

whose fixed point structure and stability encode the percolation and dynamical regimes.

2. Dynamical Phenomena and Bifurcations

The added dimensionality due to multilayer coupling results in qualitative changes to the dynamical behavior of the giant component(s), compared to single-layer triadic percolation. In addition to the period-doubling route to chaos characteristic of the one-dimensional (single-layer) case (Sun et al., 2022), the MTP model features genuine two-dimensional bifurcation phenomena.

• Neimark–Sacker Bifurcation: The two-dimensional system exhibits Neimark–Sacker (torus) bifurcations when the eigenvalues $\Lambda_{\pm}$ of the Jacobian matrix $J$ at a fixed point cross the unit circle with nonzero argument:

$$J = \begin{pmatrix} A & B \ C & D \end{pmatrix}, \qquad \Lambda_{\pm} = \frac{A+D \pm \sqrt{(A+D)^2 - 4(AD-BC)}}{2}.$$

When $|\Lambda_{\pm}| = 1$ and the eigenvalues are complex (i.e., not real $\pm 1$), the system develops quasi-periodic oscillations or orbits of arbitrarily large period, which are fundamentally absent in single-layer settings.

• Period-two Oscillations Without Negative Regulation: The MTP model admits stable period-2 cycles even in the absence of negative regulatory interactions (i.e., when only positive regulation is present), a scenario where single-layer triadic percolation produces only discontinuous hybrid transitions rather than genuine oscillations. This indicates a broadening of the set of control parameters leading to temporally nonstationary order-parameter dynamics.
• Routes to Chaos and Pseudo-periodicity: Owing to the larger phase space structure, the model supports orbits exhibiting pseudo-periodicity and complex attractors not present in simpler models.

3. Comparison to Single-layer and Lower-order Models

The MTP model fundamentally generalizes both classical percolation (Hackett et al., 2015, Baxter et al., 2016) and its single-layer triadic extension (Sun et al., 2022, Millán et al., 2023). Whereas ordinary bond or site percolation yields stationary critical points (continuous or hybrid/discontinuous, depending on interdependencies), single-layer triadic percolation introduces time-dependence solely through regulatory feedback, generating period-doubling and chaos in its one-dimensional map. The introduction of multilayer regulatory structure in the MTP model leads to genuinely two-dimensional dynamics, including bifurcations of fundamentally new type (Neimark–Sacker) and the emergence of oscillations and multistability at parameter values that, in the single-layer case, would result only in static or discontinuous behaviors (Sun et al., 10 Oct 2025).

4. Interpretative Framework and Mathematical Structure

The MTP order parameter dynamics is characterized by two main coupled equations for the active fractions $R_A$ and $R_B$. The stability analysis of fixed points proceeds via the Jacobian matrix, with its trace $\tau$ and determinant $\Delta$ controlling the onset of various transitions:

Scenario	Condition (on $\Lambda_{\pm}$)	Dynamical regime
Discontinuous (hybrid) transition	$A + D \neq 0$, real eigenvalues	First-order-like jump
Period doubling	$\Lambda_{\pm} = -1$, real	Period-2 oscillation (logistic-like)
Neimark–Sacker bifurcation	$|\Lambda_{\pm}| = 1$, complex	Quasi-periodic orbits, pseudo-periodicity

This matrix analysis enables the explicit calculation of phase boundaries in terms of the system's parameters (regulatory strengths, proportion/type of regulation, network degree distributions, etc.).

5. Applications and Implications

The MTP model is particularly relevant for the modeling of systems where higher-order regulation and multilayer structure are both essential. Illustrative examples include:

• Brain Networks: MTP captures the regulatory influence of glial cells and interneurons in distinct layers over neuronal connectivity and synaptic plasticity, with dynamics reflecting observed time-dependent functional networks and intermittent activity patterns (Millán et al., 2023).
• Climate and Ecological Systems: Regulatory species or modes that operate across different functional layers (such as trophic levels in ecosystems or atmospheric/oceanic layers in climate networks) are natural settings for MTP dynamics.
• Infrastructure and Social Systems: In multilayered technological or social networks, cross-modal or cross-platform regulation can result in complex, time-varying connectivity with implications for vulnerability and resilience.

The observation that multilayer triadic regulation can enable oscillatory phases—even in networks where the single-layer model cannot—underscores the necessity of considering both the higher-order motifs and the multilayer architecture to accurately capture the phenomenology and critical transitions in real-world networks.

6. Significance in Percolation Theory and Directions for Research

The discovery of Neimark–Sacker bifurcations and novel oscillatory regimes in the MTP model extends the known universality classes of percolation dynamics and highlights the interplay of regulatory feedback and layer structure as a source of dynamical complexity. Analyses based on coupled generating function equations, dynamical systems theory (bifurcation and stability), and phase diagram computation provide a rigorous framework for further exploration.

A plausible implication is that, in empirical systems characterized by time-varying connectivity and regulation (e.g., blinking functional connectivity in neural systems or pseudo-periodic regime switches in engineered multilayered networks), the classification and prediction of critical thresholds and the nature of collective phases may require not only standard percolation-based metrics but also tools from dynamical systems and higher-order network science.

7. Key Phenomena and Mathematical Summary

• The MTP model defines a time-discrete, two-dimensional nonlinear map on $(R_A, R_B)$ space, coupling percolation and triadic regulation both within and across layers.
• Richer dynamical behavior than single-layer triadic percolation: Neimark–Sacker bifurcations, period-2 oscillations in the absence of inhibition, and nonstationary giant component sizes are observed.
• The Jacobian analysis around fixed points discriminates between hybrid discontinuities, period-doubling, and higher-order (torus) bifurcations.
• The critical region structure and transition types are controlled not only by the underlying network topologies but also by the statistical properties and architecture (intra-/inter-layer, positive/negative) of the regulatory interactions.
• These theoretical advances connect percolation theory to the paper of time-dependent, multistable, and complex phase behaviors in realistic multilayer networks.

In conclusion, the Multilayer Triadic Percolation Model establishes a new paradigm for understanding percolation dynamics in multiplex networks with higher-order regulation. It opens avenues for mathematically tractable modeling of systems where both the structure and function of the giant connected component are nonstationary and shaped by rich intra- and interlayer feedback.