This paper studies how linking more than two units at a time changes the time a network takes to settle into synchronized motion. The authors focus on a standard model of coupled oscillators (the Kuramoto model) and ask how the time to reach the steady synchronized state — the first synchronization time, or FST — depends on both the strength of connections and the order of the interactions (pairs, triads, and higher). They combine an analytical reduction with large-scale simulations to map out these effects.

To make the problem tractable they introduce higher-order interactions as hyperedges that couple d nodes at once. They then apply the Ott–Antonsen ansatz, a mathematical trick that reduces the many-oscillator dynamics to a single equation for a global order parameter r(t) that measures how synchronized the network is. From that reduced equation they derive an integral formula for the FST (their Eq. 7) and a closed-form solution for the purely pairwise case. In a commonly used limit of very strong pairwise coupling they obtain a simple estimate showing the FST scales inversely with the pairwise coupling strength.

Numerical tests use large networks (N = 100,000) with a Cauchy distribution of natural frequencies and verify the analytical reduction. The reduced equation produces a trajectory that nearly overlaps the median behavior of many full-network realizations, so the derived τ (the FST) predicts the typical synchronization time well. The simulations use a small initial coherence (r0 = 10⁻²) and measure the time to enter a tight neighborhood (δ = 10⁻²) of the steady state. They fix the frequency spread parameter Δ = 1, which gives a pairwise transition threshold εc1 = 2.

The main findings are twofold. First, increasing the strength of pairwise coupling ε1 always speeds up synchronization: τ falls monotonically with ε1 and follows an ε1⁻¹ scaling in the large-ε1 regime. Second, the effect of higher-order interactions is non-monotonic. Adding triadic interactions (three-body links) tends to speed up convergence relative to the pairwise case. But adding still higher-order interactions (beyond triads) can slow down convergence, and in some parameter ranges the network takes longer to synchronize than it would with only pairwise links. The influence of the higher-order coupling strength εd also depends on ε1: for small ε1, increasing εd reduces τ, while for large ε1 the FST is largely insensitive to εd.