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Synchronization of phase oscillators on complex hypergraphs.

Sabina Adhikari1, Juan G Restrepo1, Per Sebastian Skardal2

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Higher-order interactions in coupled phase oscillators can lead to explosive synchronization. This study develops a framework using hypergraphs to predict and understand these complex collective behaviors.

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Area of Science:

  • Complex Systems
  • Network Science
  • Nonlinear Dynamics

Background:

  • Collective behavior in coupled systems is often studied using pairwise interactions.
  • Higher-order interactions, represented by hypergraphs, can significantly alter system dynamics.
  • Understanding these complex interactions is crucial for fields like neuroscience and social network analysis.

Purpose of the Study:

  • To investigate the impact of structured higher-order interactions on the collective behavior of coupled phase oscillators.
  • To develop a generalizable framework for analyzing synchronization in hypergraph-based systems.
  • To identify conditions leading to phenomena like bistability and explosive synchronization.

Main Methods:

  • Utilized a hypergraph generative model to represent higher-order interactions.
  • Applied dimensionality reduction techniques to derive a simplified system of differential equations.
  • Analyzed a specific case with hyperedges of sizes 2 (links) and 3 (triangles).

Main Results:

  • Derived coupled nonlinear algebraic equations for order parameters in the example case.
  • Demonstrated that strong coupling via triangles can induce bistability and explosive synchronization transitions.
  • Identified specific hypergraph properties that correlate with the emergence of bistability.

Conclusions:

  • The developed framework provides a robust method for studying synchronization in complex hypergraph networks.
  • Results highlight the critical role of higher-order interactions in shaping collective dynamics.
  • The approach is extensible to more complex hypergraph structures and dynamic correlations.