  The recent discovery of high redshift dusty galaxies implies a rapid dust
enrichment of their interstellar medium (ISM). To interpret these observations,
we run a cosmological simulation in a 30$h^{-1}$ cMpc/size volume down to $z
\approx 4$. We use the hydrodynamical code dustyGadget, which accounts for the
production of dust by stellar populations and its evolution in the ISM. We find
that the cosmic dust density parameter ($\Omega_{\rm d}$) is mainly driven by
stellar dust at $z \gtrsim 10$, so that mass- and metallicity-dependent yields
are required to assess the dust content in the first galaxies. At $z \lesssim
9$ the growth of grains in the ISM of evolved systems
(Log$(M_{\star}/M_{\odot})>8.5$) significantly increases their dust mass, in
agreement with observations in the redshift range $4 \lesssim z < 8$. Our
simulation shows that the variety of high redshift galaxies observed with ALMA
can naturally be accounted for by modeling the grain-growth timescale as a
function of the physical conditions in the gas cold phase. In addition, the
trends of dust-to-metal (DTM) and dust-to-gas (${\cal D}$) ratios are
compatible with the available data. A qualitative investigation of the
inhomogeneous dust distribution in a representative massive halo at $z \approx
4$ shows that dust is found from the central galaxy up to the closest
satellites along polluted filaments with $\rm Log({\cal D}) \leq -2.4$, but
sharply declines at distances $d \gtrsim 30$ kpc along many lines of sight,
where $\rm Log({\cal D}) \lesssim -4.0$.
