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

TL;DR

This paper analyzes the maximum possible advantage of optimal power control over binary control in large interference networks, revealing that the extremal GDoF gain scales as  ext{ } ext{ }\sqrt{K} for K users, with bounds and exact values for small K.

Contribution

It explicitly characterizes the asymptotic extremal GDoF gain as  ext{ } ext{ }\sqrt{K} for all K, providing bounds and exact values for small networks, advancing theoretical understanding.

Findings

Extremal GDoF gain scales as  ext{ } ext{ }\sqrt{K} for large K.

Bounds on the extremal gain are between  ext{ } ext{ }\lfloor  ext{ } ext{ }\sqrt{K} floor and 2.5 ext{ } ext{ }\sqrt{K}.

Exact extremal gains for K=2 to 6 are 1, 3/2, 2, 9/4, and 41/16 respectively.

Abstract

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.