same reasons, no significant evolution is measurable in the [[[agn]{}]{}]{} runs when extending the analysis to $z\sim1$ (in agreement with previous analysis by @2011MNRAS.416..801F and @battaglia.etal.2013). However, the value of the [[[agn]{}]{}]{} slope, $\beta_{{M_{\rm g}}}$, decreases at $z=1.5$ and even more at $z=2$ to a maximum difference of $\sim13\%$ with respect to $z=0$. The cause of this variation relies on the scarse ability to retain the gas component by the lowest mass systems after the intense AGN activity that took place at $z>2$. On the other hand, systems that have a total gravitational mass of $M_{500}> 5 \times 10^{13} h^{-1} M_{\odot}$ at this high redshift had enough gravitational potential to accelerate the accretion of the necessary gas mass that brings the $z\sim 2$ most massive objects close to the $z \sim 0$ scaling relation. In other words, at redshifts $z > 1$ the gas fraction within $R_{500}$ evolves at different rates depending on the cluster mass. This causes the flattening of the $M-{M_{\rm g}}$ relation at $z=2$ with respect to $z=0$.

Finally, we confirm previous results that compared radiative and non-radiative runs and find that the slope of the radiative runs is shallower [@2010ApJ...715.1508S; @battaglia.etal.2013]. This is due to the 