Gaussian Half-Duplex Diamond Networks: Ratio of Capacity the Best Relay Can Achieve

This paper considers Gaussian half-duplex diamond n-relay networks, where a source communicates with a destination by hopping information through one layer of n non-communicating relays that operate in half-duplex. The main focus consists of investigating the following question: What is the contribution of a single relay on the approximate capacity of the entire network? In particular, approximate capacity refers to a quantity that approximates the Shannon capacity within an additive gap which only depends on n, and is independent of the channel parameters. This paper answers the above question by providing a fundamental bound on the ratio between the approximate capacity of the highest-performing single relay and the approximate capacity of the entire network, for any number n. Surprisingly, it is shown that such a ratio guarantee is f = 1/(2+2cos(2π/(n+2))), that is a sinusoidal function of n, which decreases as n increases. It is also shown that the aforementioned ratio guarantee is tight, i.e., there exist Gaussian half-duplex diamond n-relay networks, where the highest-performing relay has an approximate capacity equal to an f fraction of the approximate capacity of the entire network.