Network Simplification: the Gaussian N-Relay Diamond Network

We consider the Gaussian $N$-relay diamond network, where a source wants to communicate to a destination through a layer of $N$-relay nodes. We investigate the following question: What fraction of the capacity can we maintain by using only $k$ out of the $N$ relays? We show that in every Gaussian $N$-relay diamond network, there exists a subset of $k$ relays which alone provide approximately a fraction $\frac{k}{k+1}$ of the total capacity. The result holds independent of the number of available relay nodes $N$, the channel configurations and the operating SNR. The approximation is within $3\log N+3k$ bits/s/Hz to the capacity.