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  3. Hysteretic Synchronization Driven By Triadic Interactions On Sparse Simplicial Complex Networks.
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  3. Hysteretic Synchronization Driven By Triadic Interactions On Sparse Simplicial Complex Networks.

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Hysteretic synchronization driven by triadic interactions on sparse simplicial complex networks.

Dong Feng1

  • 1Faculty of Civil Engineering, RWTH Aachen University, Mies-van-der-Rohe-Straße 1, Aachen 52074, Germany.

Chaos (Woodbury, N.Y.)
|June 1, 2026

View abstract on PubMed

Summary
This summary is machine-generated.

This study explores how group-based interactions, rather than simple two-way connections, influence the synchronization of oscillators. Researchers demonstrate that even in sparse networks, these three-way interactions create complex behaviors like bistability and hysteresis. The findings show that the arrangement of these connections significantly alters system stability and transition patterns.

Keywords:
collective dynamicsbistabilityhigher-order interactionsnonlinear oscillators

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Area of Science:

  • Nonlinear dynamics within simplicial complex networks
  • Collective behavior in triadic interactions systems

Background:

Complex systems often exhibit collective behaviors that cannot be explained by simple pairwise connections alone. Prior research has shown that higher-order interactions significantly alter the synchronization patterns of coupled oscillators. No prior work had resolved how sparse network architectures support these complex dynamical phenomena. That uncertainty drove the investigation into simplicial complexes where triangles exist alongside standard edges. It was already known that dense structures facilitate unique synchronization states through group-level forcing. This gap motivated a closer look at whether sparse connectivity provides a sufficient substrate for such effects. Prior studies frequently focused on dense regimes, leaving the sparse limit largely unexplored. This investigation addresses how triadic interactions function when higher-order structures remain relatively infrequent.

Purpose Of The Study:

This study aims to investigate how higher-order interactions reshape collective dynamics in sparse simplicial complex networks. The researchers seek to determine if sparse structures provide a sufficient substrate for triadic-interaction-driven bistability. This investigation addresses the limitation of previous models that relied on dense clique-dominated regimes. The team explores how triadic forcing influences phase synchronization when couplings are not reducible to pairwise links. They intend to clarify the role of structural-dynamical correlations in modulating hysteresis windows. The authors aim to provide a mechanism-oriented reduction using a mean-field closure approach. This work addresses the need to understand synchronization effects in networks where higher-order structures are present but not dominant. The study ultimately seeks to define the dynamical relevance of these interactions in sparse systems.

The researchers propose that triadic forcing acts as an emergent second-harmonic field. This mechanism allows oscillators to synchronize through three-way interactions, creating bistability and hysteresis even in sparse network environments.

The study utilizes a simplicial-complex extension of phase synchronization. This framework incorporates both edges and triangles to model interactions that are not reducible to simple pairwise links.

A mean-field closure is necessary to provide a mechanism-oriented reduction of the system. This approach simplifies the complex dynamics, allowing for the interpretation of triadic forcing as a second-harmonic field.

The researchers employ a structural-dynamical correlation. This data type links individual intrinsic frequencies to the degree of triangle participation, which modulates the hysteresis window and transition sharpness.

The team measures the hysteresis window and transition sharpness under parameter continuation. They observe that these phenomena emerge from the interplay between sparse simplicial structures and triadic interaction strength.

The authors imply that higher-order synchronization effects are dynamically relevant even in sparse regimes. This challenges the notion that dense clique-dominated structures are required for complex collective dynamics.

Main Methods:

The investigators employ a simplicial-complex extension to model oscillator interactions. Their review approach utilizes parameter continuation to observe system responses under varying conditions. A mean-field closure serves as the primary analytical tool for reducing complex dynamics. This method interprets triadic forcing as a second-harmonic field acting on individual phases. The team evaluates sparse regimes where higher-order structures exist but do not dominate. They establish a structural-dynamical correlation between intrinsic frequencies and triangle participation. This methodology allows for the systematic assessment of bistability and hysteresis. The researchers focus on quantifying how these higher-order effects reshape collective behavior.

Main Results:

The strongest finding indicates that sparse simplicial complexes support triadic-interaction-driven bistability and hysteresis. The researchers demonstrate that these phenomena occur even when higher-order structures are not dominant. A structural-dynamical correlation modulates the hysteresis window and transition sharpness based on triangle participation. The mean-field closure successfully interprets triadic forcing as an emergent second-harmonic field. The study confirms that higher-order synchronization effects remain dynamically relevant outside dense clique-dominated regimes. These results quantify the influence of group-level interactions on oscillator phase alignment. The analysis shows that intrinsic frequencies directly affect the observed transition patterns. The findings establish that sparse networks provide a sufficient substrate for complex collective dynamics.

Conclusions:

The authors demonstrate that sparse simplicial complexes provide a sufficient structural substrate for triadic-interaction-driven bistability. This research clarifies how higher-order synchronization effects remain dynamically relevant outside dense clique-dominated regimes. The findings suggest that intrinsic frequency distribution modulates the hysteresis window and transition sharpness. The researchers propose that triadic forcing functions as an emergent second-harmonic field acting on phases. This synthesis implies that higher-order structure significantly reshapes collective dynamics even when not dominant. The study confirms that structural-dynamical correlations link individual oscillator properties to triangle participation. These results provide a mechanism-oriented reduction for understanding complex network synchronization. The authors conclude that their mean-field closure effectively interprets the observed higher-order forcing effects.