Synchronization induced by directed higher-order interactions

Abstract

Non-reciprocal interactions play a crucial role in many social and biological complex systems. While directionality has been thoroughly accounted for in networks with pairwise interactions, its effects in systems with higher-order interactions have not yet been explored as deserved. Here, we introduce the concept of M-directed hypergraphs, a general class of directed higher-order structures, which allow to investigate dynamical systems coupled through directed group interactions. As an application we study the synchronization of nonlinear oscillators on 1-directed hypergraphs, finding that directed higher-order interactions can destroy synchronization, but also stabilize otherwise unstable synchronized states.

Figures (11)

• Figure 1: An undirected $2$-hyperedge can be seen as the composition of three directed hyperedges. It is important to note that, in each of the directed $2$-hyperedges, the nodes acting as source of the interaction commute, i.e., a permutation of them does not alter the nature of the interaction, as denoted by the adjacency tensors.
• Figure 2: Directionality induced (de)synchronization. a) Structure of the weighted hypergraph as a function of $p$, controlling the transition of the hyperedges from directed to undirected (the structure is schematically represented for $N=8$ nodes). Each undirected $2$-hyperedge can be seen as the combination of three directed hyperedges, two of which have a weight $p\in[0,1]$. When $p=0$, a triplet of nodes interacts only through a single directed hyperedge, whereas when $p=1$, the hypergraph is symmetric. b) Synchronization diagram in the plane $(p,\sigma_1)$ for a system of Rössler oscillators with $x$-$x$ cubic coupling. The white area indicates the region of stability, while the orange one the region where synchronization is unstable. The horizontal dashed lines represent two values of $\sigma_1$ for which the system transits from a synchronized to an unsynchronized state as a function of $p$ (green line), and the other way around (blue line). Panels c)-f) show the locus of eigenvalues of $\mathcal{M}$ as a function of $p$, for a weighted hypergraph with $N=20$ nodes at two different values of $\sigma_1$ (color coding is such that the directed case $p=0$ is represented in yellow, and the symmetric one $p=1$, in blue). In the background, the white area indicates the region identified by a negative MSF, the black line the boundary of this region, and the gray area the region where the MSF is positive. Panels d) and f) represent a zoom of the area close to the origin of panels c) and e), respectively. Panels c) and d) show a setting where the directed topology drives the system unstable, indeed assuming a symmetric hypergraph the synchronization manifold will result stable. Panels e) and f) display a case for which the directed topology admits a stable synchronization state, while the symmetric hypergraph triggers the instability. The coupling strength for panels c) and d) is set as $\sigma_1=0.2$, while for panels e) and f) as $\sigma_1=0.07$. In both cases $r_2=10$.
• Figure 3: Directionality induced (de)synchronization with an alternative symmetrization method. a) Synchronization diagram in the plane $(q,\sigma_1)$ for a system of Rössler oscillators with $x$-$x$ cubic coupling. The white area indicates the region of stability, while the orange one the region where synchronization is lost. The horizontal dashed lines represent two values of $\sigma_1$ for which the system transits from a synchronized to an unsynchronized state as a function of $p$ (green line), and the other way around (blue line). Panels b)-e) display the locus of eigenvalues of $\mathcal{M}$ as a function of $q$, for a hypergraph with $N=20$ nodes at two different values of $\sigma_1$ (color coding is such that the directed case $q=0$ is represented in yellow, and the symmetric one $q=1/3$, in blue). In the background, the white area indicates the region where the MSF is negative, the black line the boundary of this region, and the gray area the region with positive MSF. Panels c) and e) represent a zoom of the area close to the origin of panels b) and d), respectively. Panels b) and c) display a setting where the directed topology drives the system unstable, starting from a symmetric hypergraph for which the synchronization manifold is stable. Panels d) and e) show a case for which the directed topology admits a stable synchronization state, while the symmetric hypergraph drives to instability. The coupling strength for panels b) and c) is fixed to $\sigma_1=0.195$, while for panels d) and e) to $\sigma_1=0.03$. In both cases $r_2=0.7$.
• Figure 4: From topology to dynamics: difference between the undirected and directed $1$-hyperedge. Top panel: the derivative of the state variables associated to each node $i$, $j$, $k$ receives a contribution from the higher-order interaction. Bottom panel: only the derivative of $\vec{x}_i$ receives a contribution from the source nodes $j$ and $k$, while the derivatives of the state variable of the source nodes, $\vec{x}_j$ and $\vec{x}_k$, do not.
• Figure 5: Symmetrization of a $1$-directed $2$-hyperedge via the increase of the weight of the hyperdeges associated to the other directions. Starting from a fully directed hyperedge ($p=0$), the strength of the couplings in the other directions grows until all directions of interaction have the same weight ($p=1$).