  We examine the extent to which Gaussian relay networks can be approximated by
deterministic networks, and present two results, one negative and one positive.
  The gap between the capacities of a Gaussian relay network and a
corresponding linear deterministic network can be unbounded. The key reasons
are that the linear deterministic model fails to capture the phase of received
signals, and there is a loss in signal strength in the reduction to a linear
deterministic network.
  On the positive side, Gaussian relay networks are indeed well approximated by
certain discrete superposition networks, where the inputs and outputs to the
channels are discrete, and channel gains are signed integers.
  As a corollary, MIMO channels cannot be approximated by the linear
deterministic model but can be by the discrete superposition model.
