In this paper, we address the logarithmic corrections to the leading powerlaws that govern thermodynamic quantities as a second-order phase transitionpoint is approached. For phase transitions of spin systems on d-dimensionallattices, such corrections appear at some marginal values of the orderparameter or space dimension. We present new scaling relations for theseexponents. We also consider a spin system on a scale-free network whichexhibits logarithmic corrections due to the specific network properties. Tothis end, we analyze the phase behavior of a model with coupled orderparameters on a scale-free network and extract leading and logarithmiccorrection-to-scaling exponents that determine its field- and temperaturebehavior. Although both non-trivial sets of exponents emerge from thecorrelations in the network structure rather than from the spin fluctuationsthey fulfil the respective thermodynamic scaling relations. For the scale-freenetworks the logarithmic corrections appear at marginal values of the nodedegree distribution exponent. In addition we calculate scaling functions, whichalso exhibit nontrivial dependence on intrinsic network properties.
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