  A network of $n$ wireless communication links is considered. Fading is
assumed to be the dominant factor affecting the strength of the channels
between nodes. The objective is to analyze the achievable throughput of the
network when power allocation is allowed. By proposing a decentralized on-off
power allocation strategy, a lower bound on the achievable throughput is
obtained for a general fading model. In particular, under Rayleigh fading
conditions the achieved sum-rate is of order $\log n$, which is, by a constant
factor, larger than what is obtained with a centralized scheme in the work of
Gowaikar et al. Similar to most of previous works on large networks, the
proposed scheme assigns a vanishingly small rate for each link. However, it is
shown that by allowing the sum-rate to decrease by a factor $\alpha<1$, this
scheme is capable of providing non-zero rate-per-links of order $\Theta(1)$. To
obtain larger non-zero rate-per-links, the proposed scheme is modified to a
centralized version. It turns out that for the same number of active links the
centralized scheme achieves a much larger rate-per-link. Moreover, at large
values of rate-per-link, it achieves a sum-rate close to $\log n$, i.e., the
maximum achieved by the decentralized scheme.
