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Synchronization on small-world networks
H. Hong, M. Y. Choi, Beom Jun Kim
TL;DR
The paper examines collective phase and frequency synchronization in coupled oscillators on small-world networks using order parameters, dynamic simulations, and finite-size scaling. Synchronization emerges with even a small shortcut fraction, while phase synchronization and relaxation times saturate near P ≈ 0.5, approaching random-network behavior.
Problem
The paper investigates how shortcut rewiring affects phase and frequency synchronization in coupled oscillators on small-world networks.
Source: pp. 1, 4Method
The authors introduce phase and frequency order parameters and analyze them with dynamic simulations, relaxation-time measurements, and finite-size scaling.
Source: pp. 1–3Results
Phase and frequency synchronization emerge for any nonzero shortcut fraction considered, the phase transition is mean-field type, and phase synchronization and relaxation times saturate for P ≳ 0.5.
Source: pp. 1–4Takeaways & Limitations
A relatively small shortcut fraction can produce synchronization behavior and synchronization times nearly matching those of a random network with P = 1.
Source: pp. 1, 2, 4Finite-size scaling does not precisely determine the critical coupling for frequency synchronization, and conclusions near P = 0 require more extensive simulations.
Source: pp. 3–4
Abstract
from arXiv · showhide
We investigate collective synchronization in a system of coupled oscillators on small-world networks. The order parameters which measure synchronization of phases and frequencies are introduced and analyzed by means of dynamic simulations and finite-size scaling. Phase synchronization is observed to emerge in the presence of even a tiny fraction $P$ of shortcuts and to display saturated behavior for $P \gtrsim 0.5$. This indicates that the same synchronizability as the random network (P=1) can be achieved with relatively small number of shortcuts. The transient behavior of the synchronization, obtained from the measurement of the relaxation time, is also discussed.