We propose a novel model for a glass-forming liquid which allows to switch in a continuous manner from a standard three-dimensional liquid to a fully connected mean-field model. This is achieved by introducing k additional particle-particle interactions which thus augments the effective number of neighbors of each particle. Our computer simulations of this system show that the structure of the liquid does not change with the introduction of these pseudo neighbours and by means of analytical calculations, we determine the structural properties related to these additional neighbors. We show that the relaxation dynamics of the system slows down very quickly with increasing k and that the onset and the mode-coupling temperatures increase. The systems with high values of k follow the MCT power law behaviour for a larger temperature range compared to the ones with lower values of k. The dynamic susceptibility indicates that the dynamic heterogeneity decreases with increasing k whereas the non-Gaussian parameter is independent of it. Thus we conclude that with the increase in the number of pseudo neighbours the system becomes more mean-field like. By comparing our results with previous studies on mean-field like system we come to the conclusion that the details of how the mean-field limit is approached are important since they can lead to different dynamical behavior in this limit.
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