Triadic percolation on multilayer networks

Triadic interactions are special types of higher-order interactions that occur when regulator nodes modulate the interactions between other two or more nodes. In presence of triadic interactions, a percolation process occurring on a single-layer network becomes a fully-fledged dynamical system, characterized by period-doubling and a route to chaos. Here, we generalize the model to multilayer networks and name it as the multilayer triadic percolation (MTP) model. We find a much richer dynamical behavior of the MTP model than its single-layer counterpart. MTP displays a Neimark-Sacker bifurcation, leading to oscillations of arbitrarily large period or pseudo-periodic oscillations. Moreover, MTP admits period-two oscillations without negative regulatory interactions, whereas single-layer systems only display discontinuous hybrid transitions. This comprehensive model offers new insights on the importance of regulatory interactions in real-world systems such as brain networks, climate, and ecological systems.

💡 Research Summary

The authors introduce a novel framework called Multilayer Triadic Percolation (MTP) that extends the concept of triadic percolation—where regulator nodes switch structural links on or off—to networks composed of multiple interacting layers. In a single‑layer setting, triadic percolation already transforms the static percolation problem into a dynamical system that can exhibit period‑doubling cascades and a route to chaos, akin to the logistic map. The present work asks what new phenomena emerge when the underlying substrate consists of two (or more) layers that are coupled both structurally and through triadic regulatory interactions.

Model construction
The system comprises two layers, A and B. Each layer contains a set of nodes, a set of structural edges, and a set of directed signed regulatory interactions (the “triadic” part) that connect regulator nodes to structural edges. In addition, there are bipartite regulatory networks linking nodes in one layer to edges in the other layer, thereby providing inter‑layer triadic regulation. Positive regulators turn a link on, negative regulators turn it off, and a stochastic parameter p accounts for random link failures.

The dynamics proceeds in discrete time steps:

1. Percolation step – given the current set of active links, the giant component (GC) of each layer is identified; nodes belonging to the GC are marked as active.
2. Regulation step – each structural link is updated according to a Boolean rule: it stays active if at least one positive regulator (from either layer) is active, it is deactivated if any negative regulator (from either layer) is active, and finally it survives the random failure with probability p.

Mathematically the process can be written as a two‑dimensional map. The fraction of active nodes in layer i at time t, R_i(t), follows the standard percolation equations expressed through generating functions G_{0,i} and G_{1,i} that depend on the link‑retention probability p_i(t‑1). The link‑retention probabilities p_i(t) themselves are functions g_i of the current active‑node fractions R_A(t) and R_B(t), incorporating the statistics of intra‑layer and inter‑layer regulatory interactions (through generating functions G^{±}{i,intra} and G^{±}{i,inter}). Consequently the full dynamics is captured by

p_A(t)=g_A(R_A(t),R_B(t)), p_B(t)=g_B(R_A(t),R_B(t)),
R_A(t)=f_A(p_A(t‑1)), R_B(t)=f_B(p_B(t‑1)).

Because the map is two‑dimensional, its linear stability analysis yields three generic bifurcation scenarios: (i) a discontinuous (hybrid) transition where the fixed point disappears abruptly, (ii) a period‑doubling cascade leading to higher‑order periodic orbits, and (iii) a Neimark‑Sacker (Hopf‑like) bifurcation that creates a closed invariant curve and quasi‑periodic oscillations.

Key findings

• Neimark‑Sacker bifurcation: When the strength of inter‑layer positive regulation exceeds a critical value, the Jacobian at the fixed point acquires a pair of complex conjugate eigenvalues that cross the unit circle, generating sustained oscillations of arbitrarily large period or true quasi‑periodicity. This phenomenon has no analogue in single‑layer triadic percolation.
• Period‑two oscillations without negative regulators: Surprisingly, even when all regulatory interactions are excitatory (no negative κ⁻), the system can settle into a stable two‑step cycle alternating between a silent state (R_A=R_B=0) and an active state (R_A,R_B>0). This arises from the interplay of intra‑ and inter‑layer feedback loops and is impossible in the single‑layer case, where negative regulation is required for bistable oscillations.
• Rich cascade of period‑doubling: By increasing the average number of regulators per link (κ^{±}) or the mean degree of the underlying structural networks, the model reproduces the classic Feigenbaum cascade: period‑2 → period‑4 → period‑8 … → chaos, but now embedded in a two‑dimensional phase space. The route to chaos follows the logistic‑map universality class, yet the critical parameter values shift depending on the balance between intra‑ and inter‑layer regulation.
• Hybrid (discontinuous) transitions: In regions where regulatory feedback is weak, the system behaves similarly to ordinary percolation on multiplex networks, showing an abrupt collapse of the giant component when the effective link‑retention probability falls below a threshold.

Numerical simulations were performed on random multilayer networks with Poisson degree distributions (average degree ≈3) and Poisson‑distributed regulator counts (average κ^{+}=κ^{-}=1.5). Scanning the space of intra‑layer and inter‑layer regulator strengths confirmed the analytical predictions: the Neimark‑Sacker boundary is highly sensitive to inter‑layer regulation, while the period‑doubling region expands with stronger intra‑layer regulation. Lyapunov exponent calculations demonstrated positive values in the chaotic regime, confirming genuine deterministic chaos rather than stochastic fluctuations.

Implications for real systems
The authors argue that many empirical systems naturally map onto the MTP framework. In the brain, glial cells act as regulators of synaptic connections, providing both intra‑cortical (within a region) and inter‑cortical (between regions) triadic modulation; the observed low‑frequency oscillations and high‑frequency chaotic bursts could be interpreted as manifestations of Neimark‑Sacker and period‑doubling dynamics, respectively. Climate dynamics, where atmospheric and oceanic layers exchange heat and moisture, can be modeled as inter‑layer positive regulation, potentially explaining quasi‑periodic phenomena such as ENSO. Ecological networks, with species acting as regulators of interaction strengths across habitats, may exhibit hybrid collapses and bistable cycles akin to the transitions described here.

Conclusions and outlook
MTP reveals that adding a second layer to triadic percolation dramatically enriches the dynamical repertoire: new bifurcations, oscillations without inhibitory feedback, and a broader chaotic regime emerge. The work sets the stage for several future directions: extending the analysis to more than two layers (higher‑dimensional maps), calibrating the model against empirical time‑series data, and exploring control strategies that manipulate regulator strengths to steer the system toward desired dynamical states (e.g., preventing catastrophic collapses or inducing beneficial rhythmic activity). Overall, the study provides a powerful theoretical lens for understanding how higher‑order regulatory interactions shape the resilience and temporal complexity of multilayer complex systems.