Hysteretic synchronization driven by triadic interactions on sparse simplicial complex networks

Higher-order interactions can fundamentally reshape collective dynamics when couplings are not reducible to pairwise links. This paper studies a simplicial-complex extension of phase synchronization in which oscillators interact through both edges and triangles, with emphasis on sparse regimes where the higher-order structure is present but not dominant. We show that sparse simplicial complexes already provide a sufficient structural substrate for triadic-interaction-driven bistability and hysteresis under parameter continuation. A structural-dynamical correlation linking intrinsic frequencies to triangle participation is further shown to modulate the hysteresis window and transition sharpness. A mean-field closure is used as a mechanism-oriented reduction that interprets triadic forcing as an emergent second-harmonic field acting on phases. The framework, therefore, clarifies how genuinely higher-order synchronization effects remain dynamically relevant outside dense clique-dominated regimes.