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Validity of annealed approximation in a high-dimensional system.

Jaegon Um1, Hyunsuk Hong2, Hyunggyu Park3

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The annealed approximation is suitable for dense networks in the thermodynamic limit. However, it inaccurately predicts regular states in the disordered phase of coupled oscillator systems.

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Area of Science:

  • Complex systems
  • Statistical physics
  • Network science

Background:

  • The annealed approximation is a common simplification for analyzing disordered systems.
  • Dense networks with quenched disorder present challenges for analytical treatment.
  • Coupled oscillator models are used to study emergent dynamics in complex systems.

Purpose of the Study:

  • To evaluate the accuracy of the annealed approximation in high-dimensional dense networks with quenched disorder.
  • To identify discrepancies between annealed and dense network models in the disordered phase.
  • To understand the role of finite-size effects in determining steady-state patterns.

Main Methods:

  • Investigated dense networks of coupled oscillators with quenched link disorder.
  • Analyzed dynamic equations in the thermodynamic limit.
  • Focused on identical oscillators with competitive attractive and repulsive couplings.
  • Compared steady-state patterns in dense and annealed network models.

Main Results:

  • Dynamic equations for dense networks converge to complete-graph models in the thermodynamic limit.
  • Annealed and dense network models exhibit similar dynamics in the thermodynamic limit.
  • A significant discrepancy appears in the disordered phase, where finite-size effects are critical.
  • Dense networks show random irregular states, while the annealed approximation predicts a symmetric, regular incoherent state.

Conclusions:

  • The annealed approximation should be used cautiously in dense-network systems, especially in the disordered phase.
  • Finite-size effects are crucial for accurately describing the steady-state in disordered dense networks.
  • The annealed approximation may fail to capture the true complexity of emergent states in certain network configurations.