Analog network coding in general SNR regime: Performance of network simplification

A communication scenario where a source communicates with a destination over a directed layered relay network is considered. 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 and a class of symmetric layered networks. For these networks we provide upper bounds on additive and multiplicative gaps between the optimal analog network coding performance when all N relays in each layer are used and when only k such relays are are used, k < N (network simplification). We show that asymptotically (in source power), the additive gap increases at most logarithmically with ratio N/k and the number of layers, and the corresponding multiplicative gap increases at most linearly with ratio N/k and is independent of the number of layers in the layered network. 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.