[]{data-label="fig:bias_vir"}](f7.eps){width="45.00000%"}

Based on our various comparisons, we summarize our findings as follows.

-   The measured halo mass functions can differ from that in GR by about $5-10\%$ for $10\%$ deviation from GR ($|\zeta-1|=0.1$). Although the difference must be symmetric with respect to $\zeta=1$ in the limit $\zeta\rightarrow 1$, it already shows significant asymmetry for $|\zeta-1|\sim 10\%$. For example, the difference between the halo mass function of $\zeta=0.9$ and that of GR is about twice of the corresponding case of $\zeta=1.1$.

-   For a $10\%$ deviation from GR($|\zeta-1|=0.1$), the deviation of halo concentration can differ by $10-20\%$, which shows that modified gravity can significantly affect the structure formation on small scale and the internal properties of the dark matter halos.

-   The halo bias is less sensitive to the nature of gravity. For all the $\zeta\ne1$ values used in our simulation ($0.8\leq \zeta\leq 1.5$), the halo bias differ from that of GR by less than $10\%$.

We note that all the above findings are obtained based on the halos that are defined to be the spherical over-density regions within which the average over-density are $\Delta_{200}=200$ times the mean density $\rho_{\rm mean}$ of the universe. This definition is widely used in literature 