Triadic percolation on multilayer networks

TL;DR

Triadic percolation on multilayer networks (MTP) generalizes higher-order regulatory interactions to two coupled layers, transforming percolation into a dynamical process with a two-dimensional state space. By deriving a two-step iterative map that couples layer-wise giant-component fractions through intra- and interlayer regulations, the work uncovers richer dynamical regimes than in single-layer models, including a Neimark--Sacker bifurcation and period-two oscillations, as well as non-monotonic phase boundaries. The analysis identifies three bifurcation pathways (discontinuous, period-doubling, and Neimark--Sacker) and shows that the Neimark--Sacker transition is unique to multilayer configurations and depends on the balance of regulatory interactions. These findings illuminate how multilayer structure and triadic regulation can generate time-varying connectivity in brain-like, climate, and ecological systems, with potential implications for adaptive control of network activity.

Abstract

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.

Figures (8)

• Figure 1: Schematic representation of a multilayer network with triadic regulatory interactions. The network is composed of the two layers A and layer B. Nodes in the two layers are not one-to-one interdependent. Indeed in this example layer A has $N_A=5$ nodes and layer B has $N_B = 8$ nodes. We distinguish two main types of interactions: structural intralayer links (gray lines) between pairs of nodes within the same layer; triadic regulatory interactions, either interlayer (dashed lines) or intralayer (solid lines), between regulator nodes and regulated structural links. A regulatory interaction can be either negative (red) or positive (green) depending on whether the regulator node down- or up-regulate the regulated structural link.
• Figure 2: Orbit diagrams of triadic percolation on single-layer networks (a) and multilayer networks (b). We characterize the orbit diagram via upper stability threshold $p_c^u$, re-stabilization threshold $p_c^s$ and lower stability threshold $p_c^l$. In panel (a), the model parameters are $c=20$, $c^+=1.8$, $c^-=2.5$. In panel (b), the model parameters are $c_A=c_B=30$, $c_{A_\text{inter}}^{+} = 10$, $c_{A_\text{intra}}^{+} = 10$, $c_{A_\text{inter}}^{-} = 0.1$, $c_{A_\text{intra}}^{-} = 1.3$, $c_{B_\text{inter}}^{+} = 20$, $c_{B_\text{intra}}^{+} = \infty$, $c_{B_\text{inter}}^{-} = 0$, $c_{B_\text{intra}}^{-} = 0$.
• Figure 3: Neimark--Sacker bifurcation of multilayer triadic percolation with the presence of both intralayer and interlayer regulations. (a) Orbit diagram of the order parameters $R_A$ and $R_B$. (b) Monte Carlo simulation of the time series of the dynamics at $p=0.72$. The corresponding value is indicated by the black dashed line in panel (a). (c) Theoretical time series of the dynamics at the same $p=0.72$. (d) Monte Carlo simulation of the time evolution of $(R_A, R_B)$ at $p=0.72$. A spiral periodic (quasi-periodic) orbit is shown. (e) Theoretical time evolution of $(R_A, R_B)$ at $p=0.72$. (f) The leading eigenvalue $\Lambda$ of the Jacobian evaluated at the fixed point $(R_A^\star, R_B^\star)$ (red line). The eigenvalues cross the unit circle transversely, signalling the onset of a Neimark–Sacker bifurcation. The model parameters are $c_{A_\text{intra}}^{+} = 5$, $c_{A_\text{intra}}^{-} = 1.5$, $c_{A_\text{inter}}^{+} = \infty$, $c_{A_\text{inter}}^{-} = 3$, $c_{B_\text{intra}}^{+} = \infty$, $c_{B_\text{intra}}^{-} = 0$, $c_{B_\text{inter}}^{+} = 3$, $c_{B_\text{inter}}^{-} = 0$. In panel (b), the Monte Carlo simulation is conducted on a quenched network with $N=5 \times 10^5$ nodes.
• Figure 4: Different types of bifurcation of triadic percolation on multilayer networks with exclusively interlayer triadic interactions. We show three different bifurcations: (a)-(d) discontinuous transition, (e)-(h) period-doubling transition, and (i)-(l) Neimark--Sacker bifurcation. In the second and third columns, we show the Monte Carlo simulation (the second row) and theory result (the third row) of the time series of the dynamics at $p=0.45$ (panel (b), (c)), $p=0.43$ (panel (f), (g)), and $p=0.50$ (panel (j), (k)). The corresponding $p$ values are indicated as the black dashed line in the first column. In the fourth column, we plot the leading eigenvalue $\Lambda$ of the Jacobian evaluated at the fixed point $(R_A^\star, R_B^\star)$ (red line). The eigenvalue crosses the unit circle at different angles indicating different natures of bifurcations: a discontinuous transition (panel (d)), a period-doubling bifurcation (panel (h)), and a special Neimark–Sacker bifurcation, where the eigenvalues form a purely imaginary pair (panel (l)). The model parameters are summarized as follows. The structural and regulatory networks have both Poisson degree distributions. In the first row, only interlayer regulations are considered, the parameters are $c_{A_\text{inter}}^{+} = 3$, $c_{A_\text{inter}}^{-} = 0$, $c_{B_\text{inter}}^{+} = 10$, $c_{B_\text{inter}}^{-} = 0$. In the second row, the parameters are $c_{A_\text{intra}}^{+} = 10$, $c_{A_\text{intra}}^{-} = 2.2$, $c_{A_\text{inter}}^{+} = \infty$, $c_{A_\text{inter}}^{-} = 2$, $c_{B_\text{intra}}^{+} = \infty$, $c_{B_\text{intra}}^{-} = 0$, $c_{B_\text{inter}}^{+} = 10$, $c_{B_\text{inter}}^{-} = 0$. In the third row, only interlayer regulations are considered, the parameters are $c_{A_\text{inter}}^{+} = 2$, $c_{A_\text{inter}}^{-} = 2.5$, $c_{B_\text{inter}}^{+} = 2$, $c_{B_\text{inter}}^{-} = 0$. All Monte Carlo simulations are conducted on quenched networks of size $N=5 \times 10^5$ nodes.
• Figure 5: Dynamics and orbit diagrams of triadic percolation on multilayer networks with exclusively interlayer regulations. (a) The iterative map defined in Eqs. \ref{['eq:second_iteration_map']} at $p=0.85$ (solid line), $p=0.625$ (dashed line) and $p=0.55$ (dotted line). At the critical value $p=0.625$, a non-trivial stable fixed point emerges. (b-d) The time series of the dynamics at $p=0.85$, with initial conditions $p_A^0=p_B^0=0.02$ (b), $p_A^0=p_B^0=0.9$ (c), and $p_A^0=0.02, p_B^0=0.9$ (d). In panels (e-f), we show the orbit diagram of the dynamics with initial conditions $p_A^0=0.02, p_B^0=0.9$ (e) and $p_A^0=p_B^0=0.9$ (f). The parameters used here are $c_A=c_B=4$, $c_{A_{\text{inter}}}^+=1$, $c_{B_{\text{inter}}}^+=3$, $c_{A_{\text{inter}}}^- = c_{B_{\text{inter}}}^-=0$.