Transmission Power Control in Multi-Hop Communications of THz Communication Using a Potential Game Approach

Abstract

Terahertz (THz)-band communications are a possible candidate for fast communication. Transmission power needs to be optimised in order to satisfy the requirements of such a network of nodes. Multi-hop communication can be used in THz communications to add relays when obstacles or other interference is evident. This paper investigates the domain of multi-hop THz communications, acknowledging the possibility of interference from other devices affecting the communication process and utilises interference cancellation. It formulates the Transmission Power Control (TPC) problem within a game-theoretic framework, specifically as a potential game, with the assurance of convergence to a Nash equilibrium in the majority of scenarios. Bounds of the Price of Anarchy (PoA) and the Price of Stability (PoS) are provided, given assumed factors. Also, asymptotic Lyapunov stability is shown. The findings from simulations conducted to assess the effectiveness of this approach are then discussed. In particular, the final utilities when utilising interference cancellation show an improvement compared to without-interference cancellation between 19.30 and 67.30% for five simulated players.

1. Introduction

The current capabilities of communication channels, including their maximum bitrate, attempt to meet the needs of modern applications. Advancements in spectral efficiency and noise reduction have facilitated an annual growth in data rates of around 32% [1]. However, these improvements necessitate further investments in technology, costs, and energy consumption. As the demand for data and capacity continues to rise, it is crucial to explore alternative approaches for future networks.

One promising avenue is the exploration of new frequencies for data transmission, particularly in the terahertz (THz) range, which spans from 300 GHz to 30 THz. This wide frequency band enables the transmission of extremely high data rates from a single source, thereby significantly improving both energy efficiency and network capacity to meet the evolving needs of communication systems.

A rigorous survey regarding THz communications is given in [2]. Moreover, the work in [3] offers an in-depth analysis and technological overview of THz communications and sensing, covering key aspects such as benefits, applications, propagation characteristics, channel modeling, measurement campaigns, antennas, transceiver technologies, beamforming techniques, networking, the convergence of communication and sensing, and experimental testbeds. Furthermore, 6G is addressed with THz communications as the reader can see in [4].

THz waves present notable advantages over microwave and optical signals. They offer significantly greater bandwidth compared to microwave signals and exhibit much lower attenuation due to atmospheric disturbances such as rain and fog than optical links. Specifically, the 300 to 900 GHz frequency range is highly promising for fixed wireless links thanks to its low atmospheric attenuation and the availability of early-stage electronic circuits designed for these frequencies. However, it is important to consider that precipitation droplets and atmospheric molecules can still cause substantial scattering and attenuation of THz waves, potentially limiting the maximum achievable data rates for these wireless links [5].

Multi-hop communication in THz systems [6,7] offers several benefits, including extended range by using intermediate nodes to relay signals, which counters the high attenuation of THz waves over a distance. It also improves connectivity in obstructed or non-line-of-sight environments, enhances network reliability through multiple data pathways, and reduces power consumption by enabling shorter transmission distances, which is particularly beneficial for battery-operated devices.

Multi-hop THz communication presents several challenges and considerations. The complexity of these networks demands sophisticated routing protocols and advanced network management strategies to ensure efficient data transmission. Additionally, managing interference becomes more difficult with multiple nodes transmitting, particularly at THz frequencies where signal interference can significantly impact performance. Synchronizing intermediate nodes for efficient data relay further adds to the complexity of implementing effective multi-hop THz communication systems.

In this paper, the implementation of multi-hop THz communications is explored, where multiple relay nodes are used to extend communication distances and improve signal quality. However, multi-hop communication in the THz band faces unique challenges, particularly due to interference from other devices operating in the same frequency range. Hence, interference cancellation is considered. This interference can significantly impact the quality of the communication links and the overall system performance.

To address this, the problem of Transmission Power Control (TPC) is formulated using a game-theoretic approach, specifically as a potential game. In this framework, each node acts as a rational player aiming to maximize its own utility, defined in terms of its Signal-to-Interference-plus-Noise Ratio (SINR). The use of a potential game guarantees desirable properties, such as the existence of a potential function that aligns with the overall system utility. This ensures that the system almost surely converges to a Nash equilibrium, where no player can improve its utility by unilaterally changing its transmission power level.

The proposed game-theoretic TPC approach is evaluated through simulations, which demonstrate its effectiveness in achieving efficient power allocation and mitigating interference in a multi-hop THz communication environment. The results show that the system converges to a Nash equilibrium while maintaining high SINR levels and minimizing interference. Also, the PoA and PoS are bounded to a constant. Finally, Lyapunov stability is employed to investigate the stability properties of the proposed system.

This paper continues as follows: Section 2 provides the contributions summary, Section 3 gives the related work, Section 4 gives a background on game theory and potential games, Section 5 provides the SINR-based TPC as a potential game, Section 6 shows the results of the approach, Section 7 gives a discussion, and Section 8 gives the conclusions and future work.

2. Contributions

This formulation surpasses traditional TPC methods by leveraging a game-theoretic approach, with the utility function directly based on the SINR. This ensures that the system optimizes a key metric for wireless communication, naturally improving signal quality while minimizing interference. The model guarantees convergence to the Nash equilibrium through a potential function that aligns individual utilities with global optimization goals. Unlike traditional methods, this approach provides quantifiable performance guarantees, such as bounded PoA and Po, ensuring that the Nash equilibrium is close to the socially optimal strategy. Additionally, it incorporates SIC, enabling efficient interference management and enhancing SINR without penalizing neighboring nodes.

The formulation’s robustness is further demonstrated through Lyapunov stability analysis, which confirms the system’s ability to handle perturbations and converge to equilibrium under realistic interference and noise conditions. The use of gradient descent for finding Nash equilibria ensures computational efficiency, making the model scalable and adaptable to complex, dynamic networks. Unlike traditional methods, which often rely on heuristics or lack formal stability guarantees, this approach integrates theoretical proofs with practical applicability. By addressing potential limitations transparently, the formulation provides a reliable and comprehensive framework for modern wireless networks, balancing efficiency, scalability, and stability in diverse scenarios. This formulation outperforms other game-theoretic methods for transmission power control by employing a potential game structure that guarantees convergence to a Nash equilibrium while aligning individual incentives with the system’s overall performance. Moreover, general game theoretic models may be intractable when computing the Nash equilibrium. Unlike general game-theoretic approaches, where individual utility optimization may lead to inefficiencies or unpredictable outcomes, this model uses a potential function that ensures the Nash equilibrium corresponds to a near-optimal global solution.

3. Related Work

3.1. TPC and Methods Other than SIC in Non-THz Communications

TPC is a quite extensively researched study from the period of wireless sensor networks (WSN) and continues to the day [8,9,10]. It has been used for link quality estimation and optimization, and topology control [11,12], localization and energy efficiency [13], in conjunction with the rate for vehicular networks [14], as well as 5G [15,16]. In this section, we provide the reader with THz communication-related works.

Also, there are numerous interference cancellation methods apart from SIC as well as other methods that are given below.

In [17], an information-theoretic analysis of the simplest multi-hop network, the two-hop source-relay-destination system, leads to the development of a novel transmission technique known as structured self-interference cancellation (or “structured cancellation” for short). In this strategy, the source intentionally avoids transmitting at certain signal levels, while the relay designs its transmission to learn the residual self-interference channel. By observing the portion of its own transmitted signal that appears at the vacant signal levels left by the source, the relay can cancel the self-interference. The study demonstrates that, in certain complex scenarios, the structured cancellation method not only outperforms half-duplex systems but also surpasses full-duplex systems that rely on time-orthogonal training for estimating the residual self-interference channel.

In [18], the paper explores a multi-source video multicast in Internet-connected wireless mesh networks (WMNs). The primary focus is on designing a shareable integrated multicast framework that enables video source multicasts to utilize common Internet shortcuts or WMN paths, minimizing the potential overhead of WMNs and excessive Internet traffic. Several algorithms are introduced, collectively forming a video multicast framework that manages a controlled number of shareable multicasts, subject to the availability of Internet resources. These algorithms include the resource-efficient source group algorithm, the efficient integrated architecture algorithm, and the interference-controlled multicast tree algorithm.

In [19], the authors propose and analyze parallel low-rate transmissions alongside alternative rate transmissions to investigate the benefits of Multi-Rate Multi-Channel (MRMC) in enhancing the performance and coverage tradeoff, while operating within the constraints of limited channel resources. These new transmission schemes are then applied to optimize the WMN multicast experience. By integrating reliable interference-controlled connections, the authors design a novel MRMC multicast algorithm, LC-MRMC, which efficiently utilizes channel and rate resources, significantly expanding wireless multicast coverage with high throughput and low-delay performance.

In [20], the study addresses the challenge of achieving a high Packet Delivery Ratio (PDR), energy efficiency, and cost-effective clustering and routing in WSNs. To reduce deployment costs, a heterogeneous setup is considered, featuring energy-constrained normal sensors for environmental monitoring, alongside high-energy, TPC-enabled multi-radio super nodes. These super nodes function as cluster heads and are responsible for collecting data from the normal sensors. By equipping super nodes with additional radios, static channel assignment is enabled, resulting in high PDR with minimal network overhead. This TPC-based multi-channel heterogeneous setup offers cost-effectiveness, high PDR, and energy efficiency. The problem is decoupled into two phases: configuring the super nodes and normal sensors, which are optimized using the Grey Wolf Optimization (GWO) algorithm.

In [21], the authors propose an innovative approach in which TinyML models are seamlessly integrated into access points (APs) for adaptive channel optimization and transmission power control. This decentralized method enables APs to autonomously optimize their settings based on local data, reducing dependence on central controllers and improving scalability. By utilizing lightweight TinyML models, APs can make swift decisions and quickly adapt to changes in the network. Through dynamic adjustments to channel selection and transmission power based on signal strength, interference, and traffic load, this approach minimizes interference and enhances overall network performance.

3.2. Power Control in THz Communications

In [22], a lone UAV is designated to serving ground users via the THz frequency band. However, the inherent unpredictability of the THz channel poses new challenges in optimizing UAV positioning, user power levels, and bandwidth distribution. Consequently, it is imperative to devise a fresh framework for deploying UAVs within THz wireless systems. This task is formally framed as an optimization challenge with the aim of minimizing overall transmission delays for both uplink and downlink communication between the UAV and ground users. This objective is achieved through the simultaneous optimization of UAV deployment, transmit power, and user bandwidth. Given the critical significance of communication latency, particularly in emergency contexts, addressing this nonconvex delay minimization problem demands the development of an alternating algorithm. This algorithm iteratively tackles three subproblems: location optimization, power control, and bandwidth allocation. Again, this is a cross-layer approach.

This paper [23] introduces a secure UAV-relay mobile system operating in sub-THz bands for cognitive networks. The study accounts for path loss effects and examines the performance of a UAV relay that receives data from a base station (SBS) and transmits it to a secondary user (SU), while minimizing interference from primary users (PUs) and eavesdroppers. To achieve the goal of maximizing the secrecy rate in UAV communication, an optimization problem is formulated, incorporating constraints such as UAV velocity, elevation, maximum transmission power, information causality, and interference temperature (IT). The primary optimization challenge is divided into two subproblems, and an alternative algorithm is proposed to solve it efficiently. Extensive analysis and performance results demonstrate that the proposed algorithm significantly enhances physical layer security in UAV-relay communications within the sub-THz spectrum.

In [24], the authors present a pioneering operational framework for the IEEE standard [25] 802.15, designed to curtail power transmission requirements while adhering to specified data rate benchmarks. This leads to a decreased reliance on battery capacity or antenna size to uphold consistent communication link robustness. Their method involves the meticulous optimization of power levels, modulation schemes, and channel allocations to effectively minimize total transmitted power while upholding a minimum quality-of-service standard. Additionally, our solution incorporates considerations for the influence of humidity on system performance, providing a holistic approach tailored to practical, real-world scenarios. In this paper, we see a coherent and complete approach that does take into account a number of parameters that influence the communication.

In [26], the paper presents a theoretical framework for device-to-device (D2D) communication operating in the THz band. They derive closed-form formulas for data rates, outage probability, and energy efficiency. Through simulations, they illustrate improvements in both data rates and energy efficiency, along with a decrease in D2D communication outage probability within the THz spectrum. Notably, they observe an 86% increase in energy efficiency with a transmission power of 19 dBm. Furthermore, optimal transmission power allocation across 50 resource blocks results in an 87% improvement in energy efficiency.

In [27], the study delves into THz wireless communication among UAVs, focusing keenly on the diverse fading channels pivotal for modeling THz communication networks. It meticulously analyzes the performance of UAV communications within THz networks, accounting for critical structural aspects, wireless channel parameters, and transmission characteristics essential for evaluating wireless technology effectiveness. The analysis encompasses a comprehensive range of fading channels, including Nakagami-m, Rician, Weibull, Rayleigh, and log-normal fading. Moreover, it introduces an optimization algorithm tailored to the THz channel, with the objective of minimizing transmission power by optimizing trajectory paths for both uplink and downlink transmissions between UAVs and users. The study derives equations governing UAV locations and transmission power optimization for individual users. Furthermore, it formulates analytical models addressing capacity, outage probability, and bit error rate (BER), considering factors influencing performance.

4. Game Theory and Potential Games

This paper builds on the framework from Spyrou et al. [28] to apply game theory in modeling interactions between nodes, drawing on concepts from [29]. In this context, a game represents any social interaction involving multiple nodes. Each node is considered rational, aiming to satisfy its preferences by selecting strategies that align with general rationality axioms. These preferences are captured by a utility function, which quantifies the node’s choices, with the primary objective being to maximize this utility function.

This work centers on strategic non-cooperative games, in which nodes function as self-interested players focused on safeguarding their individual interests. The core idea is that nodes will reach an optimal state without incurring additional costs to maximize their payoffs. The Nash equilibrium, introduced by John F. Nash in 1950 [30], is the key equilibrium concept in non-cooperative games. It represents a state where no node can improve its utility by changing its strategy alone.

Game theory has long been utilized in wireless communications, as noted in [31]. Over recent years, numerous potential games have emerged in the fields of communications and networking [32]. Specifically for Transmission Power Control (TPC), several game-theoretic approaches have been developed, addressing this challenge either independently or in combination with other techniques [12,33,34,35,36,37].

From [38], potential games gained recognition for their unique characteristics, especially the assurance of pure strategy equilibrium existence and the high likelihood of convergence through best response dynamics.

A game $〈N,A,u〉$ is an exact potential game if there exists a potential function such as

$$V:A\to \mathbb{R}$$

subject to $\forall iϵG,\forall {\sigma }_{-i}ϵA_{-i},\forall {\sigma }_i,{\sigma }_i^{\prime }ϵA_i$, such as,

$$V({\sigma }_i,{\sigma }_{-i})-V({\sigma }_i^{\prime },{\sigma }_{-i})=u_i({\sigma }_i,{\sigma }_{-i})-u_i({\sigma }_i^{\prime },{\sigma }_{-i})$$

(1)

Thus, once the current strategy $\sigma$, and node i moves from strategy ${\sigma }_i$ to strategy ${\sigma }_i^{\prime }$, In this case, the savings precisely coincide with the change in the potential function value $V\left(A\right)$.

A specific type of potential game is the ordinal potential game, where an ordinal potential function is essential. It aligns in sign with a player’s utility function when that player makes a unilateral change. Like exact potential games, the potential function $V\left(A\right)$ exhibits similar behavior but focuses on tracking the “direction” of improvement for a node rather than the exact level of gain. Essentially, when a node takes a beneficial action, the potential function decreases.

More formally:

A game $〈N,A,u〉$ is an ordinal potential game if there exists an ordinal potential function

$$V:A\to \mathbb{R}$$

(2)

subject to $\forall iϵG,\forall {\sigma }_{-i}ϵA_{-i},\forall {\sigma }_i,{\sigma }_i^{\prime }ϵA_i$ such as,

$$V({\sigma }_i,{\sigma }_{-i})-V({\sigma }_i^{\prime },{\sigma }_{-i})>0\iff u_i({\sigma }_i,{\sigma }_{-i})-u_i({\sigma }_i^{\prime },{\sigma }_{-i})>0.$$

(3)

5. SINR-Based TPC as a Potential Game

In this section, we will provide the reader with how the potential game of the SINR approach is ensured and the respective proof is given.

5.1. System Model

A THz multi-hop communication network is considered to comprise multiple nodes, denoted by N. The network operates in an environment where high-density node deployments and obstacles introduce challenges, including interference and signal degradation. Hence, intermediate nodes are strategically deployed to mitigate communication obstacles and extend the range of transmissions.

Amplify-and-Forward (AF) and Decode-and-Forward (DF) relaying can be applied in THz multi-hop systems, with the choice depending on the network topology, application requirements, and available resources. In this paper, a game-theoretic framework for relaying is proposed to dynamically select transmission strategies based on network conditions and resource constraints. Note that the directional antennas may be utilized for beamforming in THz communications, which also assist mitigation applications.

The SINR for the transmission between nodes $x_1$ and $x_2$ remains:

$$SIN{R}_{x_1,x_2}={\displaystyle \frac{P_{R_x}}{I_N+S_N}},$$

(4)

where $P_{R_x}$ is the received signal power from $x_1$ to $x_2$, $I_N$ is the aggregate interference power from nearby transmitting nodes, and $S_N$ is the aggregate noise power.

Incorporating the effect of path loss, fading, and directional gains:

$$P_{R_x}=P_T{G}_T{G}_R{L}_f^{-1},$$

(5)

where $P_T$ is the transmitted power from $x_1$, $G_T,G_R$ is the transmit and receive antenna gains, and $L_f$ is the path loss, which may depend on distance d and frequency f (e.g., $L_f=c_1{d}^{\alpha }{f}^{\beta }$). The interference power ($I_N$) is given by,

$$I_N=\sum _{j\in \mathcal{I}}{P}_{T_j}{G}_{T_j}{G}_{R_j}{L}_{f_j}^{-1},$$

(6)

where $\mathcal{I}$ represents the set of interfering nodes.

The thermal noise power plus other environmental factors is given by,

$$S_N=k_{B}TB,$$

(7)

where $k_B$ is the Boltzmann constant, T is the system temperature, and B is the bandwidth.

Here, $SIN{R}_{x_1,x_2}$ represents the SINR of the transmission from node $x_1$ to node $x_2$. The calculation of $SIN{R}_{x_1,x_2}$ is given by (4). In this equation, $P_{R_x}$ represents the received signal power, $I_N$ is the aggregate power of the interferers, and $S_N$ is the aggregate noise.

5.2. Interference Cancellation

If a collision occurs at the simultaneous arrival of two or more packet transmissions at the receiver, the conventional signal extraction process, only the packet with the strongest signal is decipherable, considering all other signals as interference. When the signal of interest is not the strongest, it becomes irretrievable. In contrast, Successive Interference Cancellation (SIC) empowers the retrieval of even weaker signals of interest.

With SIC, the bits of the strongest signal are prioritized for decoding initially. Subsequently, the original signal corresponding to those bits is reconstructed and subtracted from the combined signal. This iterative method facilitates the recovery of multiple packets, hence earning the moniker successive interference cancellation [39]. The aim here is to decode the signal of interest and cancel out the interference caused by the interfering nodes in order to obtain the SINR of the current node, as can be seen in Figure 1. The A, B, and C circles correspond to nodes, which operate within a transmission range and the boxes the operations that take place during the operation.

Interference cancellation is employed in multi-hop wireless communication to enhance performance because it directly addresses the adverse effects of interference, which can significantly degrade signal quality, throughput, and overall network efficiency. This technique allows for a more effective use of available spectrum by reducing the interference that degrades the desired signal. Other methods such as beamforming, and frequency reuse are also viable, while interference cancellation offers a targeted solution for improving the SINR.

In comparison to traditional RF bands, interference cancellation in THz communication presents several unique challenges. THz signals experience greater atmospheric absorption, scattering, and attenuation than RF signals, leading to higher path loss and reduced range. This results in more frequent re-transmissions and increased interference. While traditional RF bands benefit from well-established interference management techniques and longer propagation distances, THz communication, with its shorter wavelengths, offers higher spatial resolution and enables advanced beamforming. However, this requires more precise and dynamic interference cancellation methods to address the higher density and more complex channel conditions inherent in THz communication. Additionally, THz signals are highly sensitive to environmental factors, which complicates the implementation of interference cancellation. These challenges necessitate the implementation of more adaptive algorithms to ensure reliable performance.

5.3. Potential Game Formulation

Initially, the game that is played in this approach consists of N players, $A=\{P_1,P_2,\dots ,P_n\}$ strategies, where each node selects the best receiver power according to the transmission power with which the data are sent, and $U_1,U_2,\dots ,U_n$ utility functions. The utility function of each node is given by the formula

$$U_i=SINR(i,j)$$

(8)

The potential function of the approach is given using the following formula,

$$V=\sum _{i\in N}{U}_i.$$

(9)

We utilise a mechanism, the same one provided with the example above as given in [40], to ensure that the strategies of the remaining utilities of the players are 0. To this end, we can use the following lemma when using the SINR:

Lemma 1. 

If there is a node $x_1$ that changes the transmission power level from $p_1$ to $p_2$, where $p_1<p_2$, the interference in neighboring nodes can be mitigated, such as ${\displaystyle \sum _{x_3\in N}}SIN{R}_{x_1,x_3}^{p_1}={\displaystyle \sum _{x_3\in N}}SIN{R}_{x_1,x_3}^{p_2}$.

Proof. 

The proof follows from the SIC interference cancellation and the fact that the SINR is reconstructed in such a way that the two sums are equal. Essentially, the denominator of the $SIN{R}_{x_1,x_2}={\displaystyle \frac{P_{R_x}}{I_N+S_N}}$ diminishes the interference; thus making the SINR equal to the one before the change in the transmission power. □

To that end, the following proposition is provided that follows the aforementioned lemma and is at the heart of the approach, namely the proof of the potential game of the TPC.

$$\begin{array}{c}\hfill \begin{array}{c}\hfill V(s_i,s_{-i})-V(s_i^{\prime },s_{-i})=\\ \hfill \sum _{i\in N}{U}_i(s_i,s_{-i})-\sum _{i\in N}{U}_i(s_i^{\prime },s_{-i})=\\ \hfill =U_i(s_i,s_{-i})-U_i(s_i^{\prime },s_{-i})+\\ \hfill +(\sum _{m\in N,m\ne i}{U}_m(s_m,s_{-m})-\sum _{m\in N,m\ne i}{U}_m(s_m^{\prime },s_{-m}))\end{array}\end{array}$$

(10)

As was shown in the Lemma 1, after the interference cancellation, the term

$$\sum _{m\in N,m\ne i}{U}_m(s_m,s_{-m})-\sum _{m\in N,m\ne i}{U}_m(s_m^{\prime },s_{-m})=0.$$

(11)

hence, the game is a potential game with the potential function given by (9).

The algorithm of the approach follows. The Signal-to-Noise Ratio (SNR) for a specified player is computed, taking into account the transmission powers of all players, the channel gains, and the noise power. Similarly, the utility (SINR) for a given player is determined, essentially performing the same computation as the SINR calculation. Thereafter, the potential function is evaluated, by summing up the utilities (SINRs) of all players. Moving forward, the Nash equilibrium is found employing gradient descent to identify the Nash Equilibrium transmission powers.

Note that the Nash equilibrium that is found by the potential game approach may not be the global optimum. Hence, we have the fact that comes as a result of the theorem:

Proposition 1. 

Let $U_i(s_i,s_{-i})$ denote the utility of node i in the system where the utility is based on the SINR. Let $s^{\ast }$ be the strategy profile that maximizes the sum of utilities (the socially optimal strategy), and let $s^{Nash}$ be the Nash equilibrium strategy profile.

If α is a constant such that for every node i,

$$U_i(s^{\ast },s_{-i})\le \alpha U_i(s^{Nash},s_{-i}),$$

(12)

then PoA is bounded by α, i.e.,

$$PoA\le \alpha .$$

(13)

Proof. 

Let $s^{\ast }$ be the optimal strategy profile that maximizes the social welfare, which is the sum of all utilities:

$$OPT=\sum _{i\in N}{U}_i(s^{\ast },s_{-i}).$$

(14)

Similarly, let $s^{Nash}$ be the Nash equilibrium strategy profile, where no node has an incentive to deviate unilaterally. The total utility at the Nash equilibrium is given by:

$$Nash=\sum _{i\in N}{U}_i(s^{Nash},s_{-i}).$$

(15)

By the definition of Nash equilibrium, each player i maximizes its own utility given the strategies of the other players. Hence, each player’s utility at the Nash equilibrium may be worse than in the optimal strategy profile due to the lack of coordination, but it is assumed that for each player:

$$U_i(s^{\ast },s_{-i})\le \alpha U_i(s^{Nash},s_{-i}),$$

(16)

where $\alpha \ge 1$ is a constant bounding, and the performance of the Nash equilibrium is compared to the optimal strategy for each individual player.

We now sum this inequality over all players:

$$\sum _{i\in N}{U}_i(s^{\ast },s_{-i})\le \alpha \sum _{i\in N}{U}_i(s^{Nash},s_{-i}).$$

(17)

This yields,

$$OPT\le \alpha ·Nash.$$

(18)

Rearranging, we obtain,

$$PoA={\displaystyle \frac{OPT}{Nash}}\le \alpha .$$

(19)

Thus, the PoA is bounded by the constant $\alpha$. □

Next, we give some evidence in the PoS by providing the following proposition:

Proposition 2. 

Let $s^{\ast }$ be the optimal strategy profile that maximizes the total utility (social welfare) in the system, and let $s^{Nash}$ be the Nash equilibrium strategy profile that results in the highest total utility among all possible Nash equilibria.

If β is a constant such that:

$$U_i(s^{\ast },s_{-i})\ge \beta U_i(s^{Nash},s_{-i})$$

(20)

for all players i, then the PoS is bounded by:

$$PoS\le \beta .$$

(21)

Proof. 

Let $s^{\ast }$ be the strategy profile that maximizes the total utility (the social optimum), and let $s^{Nash}$ be the best Nash equilibrium, i.e., the Nash equilibrium that yields the highest total utility among all possible equilibria.

The total utility at the social optimum is:

$$OPT=\sum _{i\in N}{U}_i(s^{\ast },s_{-i}),$$

(22)

and the total utility at the best Nash equilibrium is:

$$Nas{h}_{\mathrm{best}}=\sum _{i\in N}{U}_i(s^{Nash},s_{-i}).$$

(23)

By definition, the PoS is the ratio of the total utility at the best Nash equilibrium to the total utility at the optimal solution:

$$PoS={\displaystyle \frac{OPT}{Nas{h}_{\mathrm{best}}}}.$$

(24)

Assuming that the utility of each player at the best Nash equilibrium is no less than a factor $\beta$ of their utility at the social optimum, we have:

$$U_i(s^{\ast },s_{-i})\ge \beta U_i(s^{Nash},s_{-i}).$$

(25)

Summing this inequality across all players gives:

$$\sum _{i\in N}{U}_i(s^{\ast },s_{-i})\ge \beta \sum _{i\in N}{U}_i(s^{Nash},s_{-i}).$$

(26)

This leads to:

$$OPT\ge \beta ·Nas{h}_{\mathrm{best}}.$$

(27)

Rearranging the inequality, we obtain:

$$PoS={\displaystyle \frac{OPT}{Nas{h}_{\mathrm{best}}}}\le {\displaystyle \frac{1}{\beta }}.$$

(28)

Thus, the Price of Stability is bounded by ${\displaystyle \frac{1}{\beta }}$, indicating that the best Nash equilibrium can be at most $\beta$-times worse than the social optimum:

$$PoS\le {\displaystyle \frac{1}{\beta }}.$$

(29)

□

Next, we analyze the Lyapunov stability of our approach. Here, we assume that the interference and the received power are measured in dBs and that the interference is not positive. Notably moderate to high interference is from −30 to 0 dB. We select the potential function V as the Lyapunov function. Hence, we have the following theorem.

Theorem 1. 

The system is asymptotically Lyapunov stable with respect to the function V.

Proof. 

There are three conditions to ensure the Lyapunov asymptotic stability of the system.

The first one is that $V\left(p\right)=0$. This is the case since $\forall i\in N,V\left(P_{R_x}\right)=0$ since the nominator of (4) is 0 and the summation of the equation provides the same result.

The second condition is that $V\left(P_{R_x}\right)>0,x\ne 0$. This is also satisfied since the nominator and denominator of (4) are negative, which give us a positive number. This is also true for the summation.

The third condition is that ${\displaystyle \frac{\partial V}{\partial P_i,j}}<0$. Here, the partial derivative of V for node j is

$${\displaystyle \frac{\partial V}{\partial P_N,j}}={\displaystyle \frac{1}{I_N+S_N}}$$

(30)

Since the $I_N$ is negative and much larger than $S_N$, the resulting derivative is negative. Hence, asymptotic stability is ensured. Note that the condition ${\displaystyle \frac{\partial V}{\partial P_i,j}}\le 0$ is not satisfied; thus, stability cannot be ensured. □

Although the system is asymptotically stable, the lack of strict adherence to ${\displaystyle \frac{\partial V}{\partial P_i,j}}\le 0$ highlights potential areas where stability could fail under extreme scenarios.

6. Results

The evaluated metrics, such as SINR = and the potential function, align with the proposed work’s motivation by directly assessing the impact of interference cancellation in wireless communication systems. The code compares two scenarios, namely one with SIC and the other without SIC. By tracking the evolution of transmission powers and utilities (SINR) over iterations, the metrics provide insight into the system’s stability and performance under interference. The game formulations dictate that the devices attempt to maximise their utilities in a competitive manner and the unilateral deviation of the Potential Game ensures the existence of a Nash equilibrium.

Interference cancellation in the provided work is demonstrated through the SIC model, where each player decodes only their own signal while treating interference from others as noise. This is reflected in the utility function, where SINR is calculated as the ratio of the player’s own signal to the interference and noise. The SIC model leads to higher SINR values compared to the non-SIC model, where interference is not canceled.

The implemented code simulates the performance of a multi-player communication network over 300 iterations, where players adjust their transmission powers to optimize their Signal-to-Interference-plus-Noise Ratio (SINR). There are five players with random channel gains ranging from $0.5$ to $2.0$. Two utility calculation models are implemented, namely, SIC, where players decode only their signals, and non-SIC, where interference from all signals is considered. Players update their transmission powers iteratively using a gradient-based approach, influenced by a step size of $0.1$ and bounded within 5 to 15 dB. Random fluctuations are added to transmission powers to introduce variability, and convergence is defined by changes in power levels being below ${10}^{-3}$.

The potential function depicted in Figure 2 shows the potential function with SIC and non-SIC. It is clear that the potential function is higher when using the SIC-based potential game with the value being $1.72$ as opposed to non-SIC, which is $1.26$.

Thereafter, the transmission power evolution of the players with SIC and non-SIC is illustrated in Figure 3. Nash equilibrium transmission powers that correspond to near-optimal receiver powers are those where no player finds it beneficial to unilaterally deviate from their chosen power level, considering the strategies of the other players. There are cases in which the transmission power level of certain nodes exhibit a higher value with SIC as opposed to non-SIC. Moreover, the opposite is also shown. More specifically, the transmission power levels of each player/node are given in

Table 1. Final transmission powers for each player (linear scale).

Player	SIC (Final Transmission Power)	No SIC (Final Transmission Power)
Player 1	11.4764	12.3867
Player 2	12.3635	10.8294
Player 3	10.7411	10.8689
Player 4	12.1847	11.2238
Player 5	9.7110	11.0607

. The difference is mainly caused by the interference whereby the players increase their transmission power levels to accomplish a higher SINR.

The final transmission powers at equilibrium differ, reflecting the sensitivity of the equilibrium to initial conditions and the interaction dynamics among the players. This outcome underscores the complex interplay between the initial power configuration, the iterative optimization process, and the convergence towards a Nash Equilibrium. While the potential function provides a valuable measure of the overall system utility, the Nash Equilibrium captures the strategic stability of the system under the given conditions.

In Figure 4, the reader can observe the final utilities based on the SINR over the iterations. It is evident that the SIC method achieves a higher utility. Should these results be cross-examined with the transmission powers, it is clear that players increase their transmission powers to accomplish higher utilities (e.g., player 2), while in other cases, the players select a smaller transmission power level since it is not in their incentive to increase it because it will cause more interference (e.g., player 5).

In Figure 5, the reader can observe the evolution of the utilities of each player with SIC and non-SIC. In

Table 2. Final utilities (SINR) for each player.

Player	SIC (Final SINR)	Non-SIC (Final SINR)
Player 1	0.2668	0.2235
Player 2	0.4556	0.2773
Player 3	0.2880	0.2282
Player 4	0.3765	0.2506
Player 5	0.3332	0.2832

, the reader can see the final values of the utilities of each player. Players achieve higher SINR values overall with SIC, as interference is mitigated more effectively through the successive decoding process. Player 2 has the highest SINR of $0.4556$, likely benefiting from favorable channel conditions or a lower interference environment. SINR values with non-Sic are uniformly lower than in the SIC scenario due to the cumulative impact of interference from all players. The highest SINR in this case is $0.2832$ for Player 5, reflecting the inability of players to decode and cancel out other signals effectively.

Overall, the SIC mechanism demonstrates its advantage by enabling better SINR performance for all players, as shown by consistently higher values in the SIC column compared to No SIC. This improvement emphasizes the effectiveness of SIC in enhancing signal quality in multi-user environments.

Also, a Deep Reinforcement Learning (DRL) algorithm has been implemented to show the differences with the proposed method. The potential function over the iterations can be seen in Figure 6. Here, the observation is that the potential function is not that smooth compared to the game theoretic algorithm even though it may seem that it surpasses it. Moreover, a general comment is that the method takes too long to produce results.

Moreover, another plot using the DRL, Figure 7, illustrates the utility of each player. It is evident that two of the players accomplish quite high utility while the remaining three accomplish very small utility. This may be a result of the learning process of the algorithm, which does not allocate the power in a good manner, and each player does not maximise their utility based on their self interests.

7. Discussion

The deployment of SIC in real-world environments presents a series of significant challenges, primarily related to hardware requirements. High-precision components, such as analog-to-digital converters (ADCs), digital-to-analog converters (DACs), and mixers, are crucial for accurate interference cancellation. However, these components come with high costs and substantial energy demands, particularly for wideband signals. Additionally, nonlinearities in Radio Frequency (RF) chains, such as amplifier imperfections and phase noise, complicate the cancellation process, making it difficult to maintain optimal performance. Achieving real-time, low-latency SIC requires considerable computational power, which in turn increases power consumption. These hardware-related challenges make SIC implementation complex and energy-intensive, especially for mobile devices where performance and battery life must be carefully balanced.

In dense node networks, interference management becomes progressively more difficult as the number of devices increases, threatening communication quality. The high cost of deploying infrastructure, particularly for backhaul in both urban and remote settings, adds another layer of complexity. Scaling the network while ensuring reliable communication demands efficient load balancing and synchronization across nodes. Energy consumption is another critical concern, particularly in dense networks where nodes must operate efficiently to minimize power usage. To address these challenges, emerging solutions such as AI-driven interference management, innovations in energy-efficient hardware, and reconfigurable intelligent surfaces (RISs) show promise in enhancing the practicality and efficiency of SIC and dense network deployments in real-world scenarios.

In the simulations, several trends related to utility values and system scaling can be observed. One interesting trend is that players who achieve higher utilities typically have better channel conditions, as indicated by the channel gain matrix. Players with stronger self-channel gains experience higher SINR, leading to higher utility. In contrast, players with weaker channel conditions or higher interference from other players tend to have lower utility values. As the transmission powers evolve over time, players adjust their transmission power to maximize their utility, which results in fluctuations depending on the interference levels and channel conditions. In particular, the presence of SIC leads to more effective suppression of interference, allowing players to achieve higher utilities compared to the non-SIC case.

As the system scales with an increased number of players, the interference between players becomes more significant, and the system becomes more difficult to manage. This is reflected in the increasing complexity of the optimization process, as each player’s strategy must take into account the interference from all other players. The potential function, which sums the utilities of all players, also exhibits some fluctuations as the system moves toward equilibrium. Larger systems with more players may take longer to converge and require more computational resources. The convergence rate and the final transmission powers depend on the number of players, and as more players are added, it may become harder to achieve a global optimum due to the increasing interference and competition for resources.

8. Conclusions

This paper investigates the field of multi-hop Terahertz (THz) communications, focusing on its integration into modern communication networks while addressing its inherent challenges, particularly the susceptibility to interference from neighboring devices operating within the same frequency spectrum. As multi-hop THz communications are expected to play a critical role in future wireless networks due to their potential for ultra-high-speed data transmission, managing interference becomes a pivotal challenge in ensuring reliable and efficient communication over multiple hops.

A key issue addressed in this context is the Transmission Power Control (TPC) problem, which directly influences the Signal-to-Interference-plus-Noise Ratio (SINR) and, consequently, the overall network performance. To effectively address this challenge, the paper introduces an innovative approach that leverages game theory to model TPC as a potential game. This game-theoretic formulation offers several advantages, including the ability to represent the network as a system of rational players (nodes) that individually aim to maximize their utility (SINR). Through this approach, it is guaranteed that the system will converge to a Nash equilibrium, a stable state in which no node has the incentive to unilaterally alter its transmission power. The convergence to Nash equilibrium, coupled with the potential game framework, ensures that the network collectively optimizes power allocation, thus enhancing overall performance. We also give bounds for the PoA and PoS.

The effectiveness of this proposed approach is thoroughly validated through a series of comprehensive simulations, which provide evidence of its ability to improve the performance and reliability of multi-hop THz communication networks. The results show that the system not only achieves higher power allocations but also effectively accomplishes higher utilities with SIC, leading to enhanced SINRs and a more efficient use of network resources. The simulations further highlight the potential of this approach in future multi-hop THz deployments, particularly in scenarios where interference management is critical for maintaining high data rates and low latency.

In terms of future work, several key directions are proposed to extend and refine the current methodology. One of the primary objectives is to introduce discrete strategies into the current TPC problem, transitioning from continuous to discrete action spaces, which may more accurately reflect real-world constraints in THz communications. In discrete strategies, the problem becomes more challenging due to the nature of the mathematical formulation. The utilization of Schur concavity may be utilized to provide an optimal solution. Also, as the strategies increase, the problem may become intractable or insufficient for practical implementations.

Additionally, the use of concavity in the utility functions will be explored to provide tighter bounds on the solution space, ensuring more robust guarantees for system performance. Another important aspect of future research is the investigation of compact strategy spaces, which could simplify the analysis and computation of equilibria. Finally, the paper aims to investigate the conditions for equilibrium uniqueness in the current framework, ensuring that the Nash equilibrium is not only stable but also unique, which would further solidify the predictability and reliability of the proposed TPC solution.

Moreover, a more inclusive utility function will be considered, incorporating energy-efficiency and fairness in the given problem. This may give significant insights into the continuation of the proposed scheme. Moreover, an additional stability analysis, considering varying interference intensities and node densities, could further validate the robustness and adaptability of their approach under diverse network conditions.