Comment on "Non-Mean-Field Behavior of the Contact Process on Scale-Free Networks"

Recently, Castellano and Pastor-Satorras [1] utilized the finite size scaling (FSS) theory to analyze simulation data for the contact process (CP) on scale-free networks (SFNs) and claimed that its absorbing critical behavior is not consistent with the mean-field (MF) prediction. Furthermore, they pointed out large density fluctuations at highly connected vertices as a possible origin for non-MF critical behavior. In this Comment, we propose a scaling theory for relative density fluctuations in the spirit of the MF theory, which turns out to explain simulation data perfectly well. We also measure the value of the critical density decay exponent, which agrees well with the MF prediction. Our results strongly support that the CP on SFNs still exhibits a MF-type critical behavior.Comment: 1 page, 2 figures, typos are correcte

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

We revisit the Kuramoto model to explore the finite-size scaling (FSS) of the order parameter and its dynamic fluctuations near the onset of the synchronization transition, paying particular attention to effects induced by the randomness of the intrinsic frequencies of oscillators. For a population of size $N$, we study two ways of sampling the intrinsic frequencies according to the {\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 frequency interval. In the `{\em regular}' case, the $N$ frequencies are generated in a deterministic manner that minimizes the oscillator number fluctuations, leading to quasi-uniformly spaced frequencies in the population. We find that the two samplings yield substantially different finite-size properties with clearly distinct scaling exponents. Moreover, the hyperscaling relation between the order parameter and its fluctuations is valid in the regular case, but is violated in the random case. In this last case, a self-consistent mean-field theory that completely ignores dynamic fluctuations correctly predicts the FSS exponent of the order parameter but not its critical amplitude