  We compactify the classical moduli variety of compact Riemann surfaces by
attaching moduli of (metrized) graphs as boundary. The compactifications do not
admit the structure of varieties and patch together to form a big connected
moduli space in which $\sqcup_{g} M_{g}$ is open dense.
  The metrized graphs, which are often studied as "tropical curves", are
obtained as Gromov-Hausdorff collapse by fixing diameters of the hyperbolic
metrics of the Riemann surfaces. This phenomenon can be also seen as an
archemidean analogue of the tropicalization of Berkovich analytification of
$M_{g}$ (cf., [ACP]).
