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We revisit the Kuramoto model to explore the finite-size scaling (FSS) of theorder parameter and its dynamic fluctuations near the onset of thesynchronization transition, paying particular attention to effects induced bythe randomness of the intrinsic frequencies of oscillators. For a population ofsize $N$, we study two ways of sampling the intrinsic frequencies according tothe {\it same} given unimodal distribution $g(\omega)$. In the `{\em random}'case, frequencies are generated independently in accordance with $g(\omega)$,which gives rise to oscillator number fluctuation within any given frequencyinterval. In the `{\em regular}' case, the $N$ frequencies are generated in adeterministic manner that minimizes the oscillator number fluctuations, leadingto quasi-uniformly spaced frequencies in the population. We find that the twosamplings yield substantially different finite-size properties with clearlydistinct scaling exponents. Moreover, the hyperscaling relation between theorder parameter and its fluctuations is valid in the regular case, but isviolated in the random case. In this last case, a self-consistent mean-fieldtheory that completely ignores dynamic fluctuations correctly predicts the FSSexponent of the order parameter but not its critical amplitude.

Finite-size scaling, dynamic fluctuations, and hyperscaling relation in the Kuramoto model