The discovery of clear criteria that can deterministically describe the eruptive state of a solar active region would lead to major improvements on space weather predictions. Using series of numerical simulations of the emergence of a magnetic flux rope in a magnetized coronal, leading either to eruptions or to stable configurations, we test several global scalar quantities for the ability to discriminate between the eruptive and the non-eruptive simulations. From the magnetic field generated by the three-dimensional magnetohydrodynamical simulations, we compute and analyse the evolution of the magnetic flux, of the magnetic energy and its decomposition into potential and free energies, and of the relative magnetic helicity and its decomposition. Unlike the magnetic flux and magnetic energies, magnetic helicities are able to markedly distinguish the eruptive from the non-eruptive simulations. We find that the ratio of the magnetic helicity of the current-carrying magnetic field to the total relative helicity presents the highest values for the eruptive simulations, in the pre-eruptive phase only. We observe that the eruptive simulations do not possess the highest value of total magnetic helicity. In the framework of our numerical study, the magnetic energies and the total relative helicity do not correspond to good eruptivity proxies. Our study highlights that the ratio of magnetic helicities diagnoses very clearly the eruptive potential of our parametric simulations. Our study shows that magnetic-helicity-based quantities may be very efficient for the prediction of solar eruptions.
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