Multiple first-order transitions in simplicial complexes on multilayer systems

The presence of higher-order interactions (simplicial complexes) on globally coupled systems yield abrupt first-order transitions to synchronization. We discover that simplicial complexes on multilayer systems can yield multiple basins of attraction, leading to multiple abrupt first-order transitions to (de)synchronization for associated coupled dynamics. Using the Ott-Antonsen approach, we develop an analytical framework for simplicial complexes on multilayer systems, reducing the high-dimensional evolution equation to a low-dimensional manifold, which thoroughly explains the origin and stability of all possible dynamical states, including multiple synchronization transitions. The study illustrating rich dynamical behaviors could be pivotal in comprehending the impacts of higher-order interactions on dynamics of complex real-world networks, such as brain, social, and technological, which have inherent multilayer architectures.

Figure 1

Schematic diagram for (left) multilayer system consisting of simplicial layers, and (right) emerging dynamical phenomena in absence ($D_x=0$) and in presence of multilayering ($D_x>0$). Intralayer connections are simplicial of orders 1, 2, and 3. Each node in the first layer is connected with all the nodes in the second layer with a multilayering strength $D_x$.

Figure 2

Multiple synchronization transitions or multistability along with analytical predictions [Eq. (8)]. (a) Open circles (blue) indicate adiabatic backward while closed circles (red) represent adiabatic forward direction, both combined making HT (see text). Open squares (green) indicate forward direction corresponding to the random initial conditions, also referred as WHT (see text). Solid lines are numerical solutions of Eq. (8). (b) Stability diagram: Solid (blue) and dashed (red) lines, respectively, indicate stable and unstable manifolds. The stable blue curve in the middle is stable only in $r$ and $\rho$ direction and unstable in third $\xi$ direction. (c) Basins of attraction for the entire $r-\rho$ space, depicting three stable regions for Eq. (7) for all possible initial conditions for $r$ and $\rho$. At $K_1^{\left(1\right)}=-1$, black, grey (purple), and white (yellow) correspond to the green open square, red closed circle, and blue open circle, respectively, in (a). $N=1000$, and $K_{2+3}^{\left(1\right)}=K_{2+3}^{\left(2\right)}=K_{2+3}=10$ for all the simulations results. Here $D_x=0.5$.

Figure 3

Effect of higher-order interactions strength: $K_1^{\left(1\right)}$ versus $r$. Subfigures illustrate the emergence and disappearance of a saddle-node bifurcation leading to the emergence of the third stable state in the system for different values of $K_{2+3}$. (c) Zoomed part of (b) corresponding to the regime where a new branch emerges. Here $D_x=1$ and $K_1^{\left(2\right)}=0$.

Figure 4

Loss of hysteresis as a result of an increase in $D_x$. $r^1$ versus $K_1^{\left(1\right)}$ for $K_{2+3}=10$. (a)–(f) Backward (red open circle) HT, forward (blue closed circle) HT and forward (green open square) WHT transitions for $D_x$ values 0, 0.1, 0.5, 1, 2, and 3, respectively. The black solid curves represent theoretical prediction using Ott-Antonsen method depicting all stable and unstable branches.

Figure 5

Synchronization profile for the second layer (red dashed line for stable and thick blue solid line for unstable manifold) as compared to the first layer (black thin line) for both stable and unstable manifolds for different values of $D_x$ and $K_{2+3}^{\left(2\right)}$. (a) and (b) depict robustness of $\rho$ against a change in $K_1^{\left(1\right)}$. (b) is an extended scale of (a). (c) depicts that only one stable state exists for higher values of interlayer couplings. (d) illustrates that for a lower $K_{2+3}^{\left(2\right)}$ value, pure simplicial layer desynchronizes earlier as compared to higher $K_{2+3}$ [(a)–(c)].