Galam reshuffling introduced in opinion dynamics models is investigated underthe nearest neighbor Ising model on a square lattice using Monte Carlosimulations. While the corresponding Galam analytical critical temperature T_C\approx 3.09 [J/k_B] is recovered almost exactly, it is proved to be differentfrom both values, not reshuffled (T_C=2/arcsinh(1) \approx 2.27 [J/k_B]) andmean-field (T_C=4 [J/k_B]). On this basis, gradual reshuffling is studied asfunction of 0 \leq p \leq 1 where p measures the probability of spinreshuffling after each Monte Carlo step. The variation of T_C as function of pis obtained and exhibits a non-linear behavior. The simplest Solomon networkrealization is noted to reproduce Galam p=1 result. Similarly to the criticaltemperature, critical exponents are found to differ from both, the classicalIsing case and the mean-field values.
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