When higher-order interactions enhance synchronization: the case of the Kuramoto model on random hypergraphs

Abstract

Synchronization is a fundamental phenomenon in complex systems, observed across a wide range of natural and engineered contexts. The Kuramoto model provides a foundational framework for understanding synchronization among coupled oscillators, traditionally assuming pairwise interactions. However, many real-world systems exhibit group and many-body interactions, which can be effectively modeled through hypergraphs. Previous studies suggest that higher-order interactions shrink the attraction basin of the synchronous state, making it harder to reach and potentially impairing synchronization, despite enriching the dynamics. In this work, we show that this is not always the case. Through a numerical study of higher-order Kuramoto models on random hypergraphs, we find that while strong higher-order interactions do generally work against synchronization, weak higher-order interactions can actually enhance it when combined with pairwise ones. This result is further corroborated by a cost-benefit analysis: under a constrained budget of both pairwise and higher-order interactions, a mixed allocation involving both consistently achieves higher synchronization than relying on either interaction type alone. These findings provide new insights into the role of higher-order interactions in shaping collective dynamics and point to design principles for optimizing synchronization in complex systems.

Figures (26)

• Figure 1: a) Meaning of the order parameter $R$: high values correspond to synchronized states, whereas low values indicate incoherent dynamics. b) Pictorial representation of the main findings of this work. (1) While higher-order interactions generally hamper synchronization by making the attraction basin of the synchronous state "deeper but smaller", adding weak higher-order interactions to pairwise-coupled networks enhances synchronization. (2) Under a finite budget for interactions, regardless of the relative cost of higher-order interactions, the optimal configuration for synchronization always involves a combination of pairwise and higher-order interactions. Red (resp. pink) hyperedges denote higher-order interactions with larger (resp. smaller) coupling strength. Yellow (resp. blue) nodes indicate highly (resp. weakly) synchronized states, in accordance with panel a).
• Figure 2: a) Examples of $\mathcal{H}$-connected hypergraphs, i.e., whose projected networks (bottom row) are connected. b) Examples of $1$-connected hypergraphs, i.e., whose underlying networks are already connected.
• Figure 3: Average Kuramoto order parameter $R$ as a function of the coupling strengths of pairwise ($K_1$) and higher-order $3$-body interactions ($K_2$), computed over $N_{\mathcal{H}}=300$ randomly generated hypergraphs, using the $(1,1,-2)$ interaction. The grid $K_1,K_2$ is $51\times51$. Each hypergraph has $N_0=10$ nodes, $N_1 = 20$ links and $N_2 = 10$ triangles, with oscillator frequencies $\omega_{j}$ uniformly distributed in $[0, 0.3]$, and initial phases $[0, \theta_0^\mathrm{max}]$. Left panels show the cases of initial phases close to synchronization, i.e., $\theta_0^\mathrm{max} = 0.1$; right panels show the case of incoherent initial states, i.e., $\theta_0^\mathrm{max} = 2\pi$. On the right panels, the black line indicates, for each $K_1$, the $K_2$ yielding the maximum $R$: that value is never zero. Top panels show the case of $\mathcal{H}$-connected hypergraphs, while bottom panels show the case of $1$-connected hypergraphs: we observe no significant differences between the two.
• Figure 4: System behavior with different pairwise ($K_1$) and higher-order ($K_2$) coupling strengths, for initially incoherent $1$-connected hypergraphs from Fig. \ref{['fig:couplings_study_standard_coupl_R1']}. The top-center panel replicates the bottom right panel of Fig. \ref{['fig:couplings_study_standard_coupl_R1']} for comparison (color is the mean order parameter $R$). The top-left (resp. top-right) panel shows the mean value of $R$ (blue line) and its standard deviation (shaded area) as a function of the higher-order coupling strength $K_2$ (resp. $K_1$) while keeping the pairwise interaction strength $K_1$ (resp. $K_2$) fixed. The values of $K_1$, $K_2$ explored are also indicated in the top-center panel by a horizontal (resp. vertical) black line. Pink (resp. brown) triangles pointing upward/downward (resp. leftward/rightward) mark representative pairs $(K_1, K_2)$. The bottom panels display the distributions (integral normalized to $1$) of $R$ across the $N_{\mathcal{H}} = 300$ realizations of hypergraphs, initial conditions, and frequencies. For fixed $K_1 = 0.15$, a small $K_2$ (pink downward triangle) produces $R$ values relatively concentrated around the mean, whereas larger $K_2$ (pink upward triangle) leads to a broader distribution, spanning incoherent and strongly coherent regimes. For fixed $K_2 = 0.25$, increasing $K_1$ (brown leftward and upward triangles) causes the distribution to concentrate near $1$, indicating a consistently high level of synchronization.
• Figure 5: Average Kuramoto order parameter $R$ for different combinations of links and triangles, using the $(1,1,-2)$ interaction. $K_1$ and $K_2$ are the coupling strengths of $2$- and $3$-body interactions. The grid $(b_1,K_1)$ is $51\times51$. For each value of the link allocation fraction $b_1$, $N_{\mathcal{H}} = 500$ random hypergraphs were generated, with $N_0=10$ nodes, frequencies drawn uniformly at random from $[0, 0.3]$, and initial phases from $[0, \theta_0^\mathrm{max}]$---$\theta_0^\mathrm{max} = \frac{\pi}{2}$ in the left panels, and $\theta_0^\mathrm{max} = 2\pi$ in the right panels. The relative link cost is fixed at $c_1 = 1$; triangle costs $c_2$ are $1$ in top panels, $3$ in middle panels, and $5$ in bottom panels. For each $K_1$ (y-axis), the black line marks the $b_1$ (x-axis) yielding the highest $R$ (color). See \ref{['sec:methods']} for further details. We observe that combining links and triangles generally yields higher synchronization than using either alone.