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 $f = 1/(2+2\cos (2\pi /(n+2)))$ , 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.