From loop quantum gravity to cosmology: The two-vertex model

We study the notion of volume and its dynamics in the loop-quantum-gravity truncation known as the two-vertex model. We also show that its $U\left(N\right)$-symmetry reduction provides the old effective dynamics of loop quantum cosmology with an arbitrary perfect barotropic fluid content. A suitable modification of the Poisson bracket structure of the $U\left(N\right)$-symmetric model leads to the loop quantum cosmology improved dynamics.