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Dense networks that do not synchronize and sparse ones that do.

Alex Townsend1, Michael Stillman1, Steven H Strogatz1

  • 1Department of Mathematics, Cornell University, Ithaca, New York 14853, USA.

Chaos (Woodbury, N.Y.)
|September 3, 2020
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Summary

Researchers explored the Kuramoto model, finding that 75% connectivity might guarantee synchrony. They also improved the lower bound for critical connectivity and analyzed sparse networks, showing progress on destabilizing twisted states in specific ring networks.

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Area of Science:

  • Complex Systems
  • Nonlinear Dynamics
  • Network Science

Background:

  • The Kuramoto model describes coupled oscillators, with applications in physics and neuroscience.
  • Global synchrony in these networks depends on connectivity, but the exact critical threshold remains an open question.
  • Previous studies established upper bounds for guaranteed synchrony, with recent work at 78.89%.

Purpose of the Study:

  • To investigate the critical connectivity threshold for global in-phase synchrony in Kuramoto oscillator networks.
  • To improve existing lower bounds on this critical connectivity.
  • To analyze the conditions under which sparse networks can achieve global synchrony.

Main Methods:

  • Mathematical analysis of Kuramoto oscillator networks with varying connectivity densities.
  • Development of new analytical techniques to establish bounds on critical connectivity.
  • Investigation of network modifications, specifically adding edges to sparse ring networks.

Main Results:

  • Evidence suggests the critical connectivity for guaranteed synchrony may be as low as 75%.
  • The best known lower bound for critical connectivity was improved from 68.18% to 68.28%.
  • For sparse networks, a partial result shows that O(n log n) added edges can destabilize twisted states in specific ring networks (n=2^m).

Conclusions:

  • The precise critical connectivity for Kuramoto oscillator synchrony is still unknown but appears lower than previously thought.
  • Significant challenges remain in fully characterizing synchrony in both dense and sparse network regimes.
  • Further research is needed to determine the exact critical threshold and to fully understand sparse network synchronization.