  Using ideas from Generalized Degrees of Freedom (GDoF) analyses and extremal
network theory, this work studies the extremal gain of optimal power control
over binary (on/off) power control, especially in large interference networks,
in search of new theoretical insights. Whereas numerical studies have already
established that in most practical settings binary power control is close to
optimal, the extremal analysis shows not only that there exist settings where
the gain from optimal power control can be quite significant, but also bounds
the extremal values of such gains from a GDoF perspective. As its main
contribution, this work explicitly characterizes the extremal GDoF gain of
optimal over binary power control as $\Theta\left(\sqrt{K}\right)$ for all $K$.
In particular, the extremal gain is bounded between $\lfloor \sqrt{K}\rfloor$
and $2.5\sqrt{K}$ for every $K$. For $K=2,3,4,5,6$ users, the precise extremal
gain is found to be $1, 3/2, 2, 9/4$ and $41/16$, respectively. Networks shown
to achieve the extremal gain may be interpreted as multi-tier heterogeneous
networks. It is worthwhile to note that because of their focus on asymptotic
analysis, the sharp characterizations of extremal gains are valuable primarily
from a theoretical perspective, and not as contradictions to the conventional
wisdom that binary power control is generally close to optimal in practical,
non-asymptotic settings.
