  A network of $n$ wireless communication links is considered in a Rayleigh
fading environment. It is assumed that each link can be active and transmit
with a constant power $P$ or remain silent. The objective is to maximize the
number of active links such that each active link can transmit with a constant
rate $\lambda$. An upper bound is derived that shows the number of active links
scales at most like $\frac{1}{\lambda} \log n$. To obtain a lower bound, a
decentralized link activation strategy is described and analyzed. It is shown
that for small values of $\lambda$, the number of supported links by this
strategy meets the upper bound; however, as $\lambda$ grows, this number
becomes far below the upper bound. To shrink the gap between the upper bound
and the achievability result, a modified link activation strategy is proposed
and analyzed based on some results from random graph theory. It is shown that
this modified strategy performs very close to the optimum. Specifically, this
strategy is \emph{asymptotically almost surely} optimum when $\lambda$
approaches $\infty$ or 0. It turns out the optimality results are obtained in
an interference-limited regime. It is demonstrated that, by proper selection of
the algorithm parameters, the proposed scheme also allows the network to
operate in a noise-limited regime in which the transmission rates can be
adjusted by the transmission powers. The price for this flexibility is a
decrease in the throughput scaling law by a multiplicative factor of $\log \log
n$.
