Robustness of relay synchronization driven by layer-specific higher-order interactions

Relay synchronization (RS) enables outer layers to achieve coherence via an intermediate layer even in the absence of direct connections, yet its behavior under heterogeneous topologies and higher-order interactions (HOI) remains unclear. Here, we construct a three-layer framework with explicit outer-layer heterogeneity, introduce three layer-specific HOI schemes, and systematically evaluate RS attainability and robustness across several typical network architectures. Results show that all schemes enhance RS, with improvements jointly governed by HOI strength and spatial placement. Increasing HOI strength generally accelerates RS, with outer-layer HOI most effective for synchronization efficiency. In contrast, robustness exhibits strong layer-selective, only relay-layer HOI yields multi-fold gains in topological tolerance. Mechanistically, edge disruption and triangle mismatch metrics analyses reveal that higher-order motifs are more perturbation sensitive than pairwise links, acting as topological amplifiers. Accordingly, we propose a universal design principle for layer-specific HOI organization: outer-layer HOI optimize RS efficiency, while relay-layer HOI enhance structural robustness. This provides a general framework for structurally guided control of multilayer systems, enabling the simultaneous realization of high RS performance and strong robustness.