Toward an Extremal Network Theory—Robust GDoF Gain of Transmitter Cooperation Over TIN

Abstract

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 identified by Geng et al. where TIN was shown to be GDoF optimal for the K user interference channel, a CTIN regime identified by Yi and Caire where the GDoF region achievable by TIN is convex without time-sharing, and an SLS regime identified by Davoodi and Jafar where a simple layered superposition (SLS) scheme is shown to be optimal in the K user MISO BC, albeit only for K ≤ 3. The SLS regime includes the CTIN regime, and the CTIN regime includes the TIN regime. 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 the TIN regime as well as the CTIN regime, we show that the extremal GDoF gain from transmitter cooperation over TIN is bounded regardless of the number of users. In fact, 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 Θ(log 2 (K)), i.e., it scales logarithmically with the number of users.Finding the capacity limits of wireless networks is one of the grand challenges of network information theory. While exact capacity characterizations remain elusive, much progress has been made on this problem within the past decade through Degrees of Freedom (DoF) [1] and Generalized Degrees of Freedom (GDoF) [2] studies. This includes both new achievable schemes, including those inspired by the idea of interference alignment (IA) [3][4][5][6], and new outer bounds, such as those based on the aligned images (AI) approach [7]. With these advances as stepping stones, a worthy goal at this stage is to bring the theory closer to practice by adapting the models and metrics to increasingly incorporate practical concerns. As a step in this direction, this work is motivated by three practical concerns -robustness, simplicity, and scalability.By robustness we refer specifically to the channel state information at the transmitters (CSIT). GDoF characterizations under perfect CSIT provide important theoretical benchmarks, but often ...

On the Optimality of Treating Inter-Cell Interference as Noise in Uplink Cellular Networks

“…The significance of cycles is that they are useful for describing polyhedral-TIN regions-essential building blocks for characterizing TINA GDoF regions, see, e.g., [7,[12][13][14]. We now define a polyhedral-mc-TIN region of interest.…”

“…Lemma 1. (Lemma 2.1 [12]) Within the context of the above deterministic model, consider the output signals given byȲ…”

“…,X K , which are independent of the exact realizations of channel coefficients G. The bound in (19) tells us that a maximal difference of entropies, in the GDoF sense, is created throughX i , where i is the index yielding a maximum difference in strengths (λ i − ν i ) + . For more insights related to this lemma, readers are referred to [12] and [6].…”

“…For brevity, we adopt the compact notation of [12] to represent differences of entropies of the type in Lemma 1. Using this compact notion, inequality (19) is expressed as…”

“…While [7] and [8] make no implicit assumptions regarding the availability of CSIT, the robustness of TIN renders these results valid under finite precision CSIT. More recently, Chan et al [12] employed AI bounds to show that by explicitly limiting CSIT to finite precision, the GDoF optimality of TIN is extended to the CTIN regime, where GDoF benefits due to IA disappear. In this paper, we generalized this recent observation to cellular networks.…”