  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.
