Towards an Extremal Network Theory -- Robust GDoF Gain of Transmitter Cooperation over TIN

Significant progress has been made recently in Generalized Degrees of Freedom (GDoF) characterizations of wireless interference channels (IC) and broadcast channels (BC) under the assumption of finite precision channel state information at the transmitters (CSIT), especially for smaller or highly symmetric network settings. A critical barrier in extending these results to larger and asymmetric networks is the inherent combinatorial complexity of such networks. Motivated by other fields such as extremal combinatorics and extremal graph theory, we explore the possibility of an extremal network theory, i.e., a study of extremal networks within particular regimes of interest. As our test application, we study the GDoF benefits of transmitter cooperation in a $K$ user IC over the simple scheme of power control and treating interference as Gaussian noise (TIN) for three regimes of interest -- a TIN regime where TIN was shown to be GDoF optimal for the $K$ user interference channel, a CTIN regime where the GDoF region achievable by TIN is convex without time-sharing, and an SLS regime where a simple layered superposition (SLS) scheme is shown to be optimal in the $K$ user MISO BC, albeit only for $K\leq 3$. As our first result, we show that under finite precision CSIT, TIN is GDoF optimal for the $K$ user IC throughout the CTIN regime. Furthermore, under finite precision CSIT, appealing to extremal network theory we obtain the following results. In both TIN and CTIN regimes, we show that the extremal GDoF gain from transmitter cooperation over TIN is bounded regardless of the number of users: the gain is exactly a factor of $3/2$ in the TIN regime, and $2-1/K$ in the CTIN regime, for arbitrary number of users $K>1$. However, in the SLS regime, the gain is $\Theta(\log_2(K))$, i.e., it scales logarithmically with the number of users.