  We present several results relating to the contraction of generic tensor
networks and discuss their application to the simulation of quantum many-body
systems using variational approaches based upon tensor network states. Given a
closed tensor network $\mathcal{T}$, we prove that if the environment of a
single tensor from the network can be evaluated with computational cost
$\kappa$, then the environment of any other tensor from $\mathcal{T}$ can be
evaluated with identical cost $\kappa$. Moreover, we describe how the set of
all single tensor environments from $\mathcal{T}$ can be simultaneously
evaluated with fixed cost $3\kappa$. The usefulness of these results, which are
applicable to a variety of tensor network methods, is demonstrated for the
optimization of a Multi-scale Entanglement Renormalization Ansatz (MERA) for
the ground state of a 1D quantum system, where they are shown to substantially
reduce the computation time.
