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Using ideas from Generalized Degrees of Freedom (GDoF) analyses and extremalnetwork theory, this work studies the extremal gain of optimal power controlover binary (on/off) power control, especially in large interference networks,in search of new theoretical insights. Whereas numerical studies have alreadyestablished that in most practical settings binary power control is close tooptimal, the extremal analysis shows not only that there exist settings wherethe gain from optimal power control can be quite significant, but also boundsthe extremal values of such gains from a GDoF perspective. As its maincontribution, this work explicitly characterizes the extremal GDoF gain ofoptimal 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 extremalgain is found to be $1, 3/2, 2, 9/4$ and $41/16$, respectively. Networks shownto achieve the extremal gain may be interpreted as multi-tier heterogeneousnetworks. It is worthwhile to note that because of their focus on asymptoticanalysis, the sharp characterizations of extremal gains are valuable primarilyfrom a theoretical perspective, and not as contradictions to the conventionalwisdom that binary power control is generally close to optimal in practical,non-asymptotic settings.

The Extremal GDoF Gain of Optimal versus Binary Power Control in $K$ User Interference Networks Is $\Theta(\sqrt{K})$