  We consider a communication scenario where a source communicates with a
destination over a directed layered relay network. Each relay performs analog
network coding where it scales and forwards the signals received at its input.
In this scenario, we address the question: What portion of the maximum
end-to-end achievable rate can be maintained if only a fraction of relay nodes
available at each layer are used?
  We consider, in particular, the Gaussian diamond network (layered network
with a single layer of relay nodes) and a class of symmetric layered networks.
For these networks we show that each relay layer increases the additive gap
between the optimal analog network coding performance with and without network
simplification (using k instead of N relays in each layer, k < N) by no more
than log(N/k)^2 bits and the corresponding multiplicative gap by no more than a
factor of (N/k)^2, asymptotically (in source power). To the best of our
knowledge, this work offers the first characterization of the performance of
network simplification in general layered amplify-and-forward relay networks.
Further, unlike most of the current approximation results that attempt to bound
optimal rates either within an additive gap or a multiplicative gap, our
results suggest a new rate approximation scheme that allows for the
simultaneous computation of additive and multiplicative gaps.
