Fig. 2: Directionality induced (de)synchronization.

a The weighted hypergraph as a function of the parameter 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 ∈ [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, σ₁) for a system of Rössler oscillators with x-x cubic coupling, being σ₁ the coupling strength of the pairwise interactions. The white area indicates the region of stability, while the orange one the region where the synchronous solution is unstable. The horizontal dashed lines represent two values of σ₁ 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 the effective Laplacian matrix \({{{{{{{\mathcal{M}}}}}}}}\) as a function of p, for a weighted hypergraph with N = 20 nodes at two different values of σ₁ (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 Master Stability Function (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 symmetric topology drives the system unstable, whereas with a directed hypergraph the synchronization manifold results stable. Panels e and f display a case for which the symmetric topology admits a stable synchronization state, while the directed hypergraph triggers the instability. The coupling strength for panels c and d is set to σ₁ = 0.02, while for panels e and f to σ₁ = 0.007. In both cases we set the ratio r₂ = σ₂/σ₁ = 10, where σ₂ is the coupling strength of the three-body interactions.