Dimension reduction of higher-order dynamical networks

Low-dimensional reductions provide a useful framework for studying high-dimensional dynamics on complex networks, but most existing approaches are restricted to pairwise interactions. Here, we develop a one-dimensional reduction for dynamical systems on networks with purely higher-order interactions. The reduction is formulated through an effective higher-order interaction strength (βΔ), associated with the triangular interactions of the underlying network and the dynamical system's effective state. We present a theoretical framework for the dimension-reduction approach and validate it across three dynamical models with exclusively higher-order interactions. We find that the reduction accuracy is mainly determined by the homogeneity of node states, that is the deviations in state values become very small. Numerical results on synthetic and real networks show that the reduced model captures the effective steady states and transitions of the full system with good accuracy.