Journal of Sensors

Volume 2019, Article ID 6963290, 11 pages

https://doi.org/10.1155/2019/6963290

## Cross-Layer Optimized Energy-Balanced Topology Control Algorithm for WSNs

The School of Electronics & Information Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China

Correspondence should be addressed to Yongwen Du; nc.utjzl.liam@newgnoyud

Received 13 November 2018; Revised 19 August 2019; Accepted 18 September 2019; Published 5 November 2019

Academic Editor: Jesús Lozano

Copyright © 2019 Yongwen Du et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Because of the complexity of the environment and the limited resources of nodes, there will be an imbalanced energy consumption and a short life among nodes in the wireless sensor network. In this paper, by introducing the concept of game theory and supermodel game theory, we solve the challenge of a wireless sensor network topology control method based on cross-layer information design. The cross-layer information such as node degree, network connectivity, and MAC layer interference is integrated into the design of utility function to establish a new topological game model. Then, based on this topology control model, we propose a cross-layer optimized energy-balanced topology control algorithm (COETC). Compared with other algorithms, our COETC algorithm not only guarantees the network connectivity and robustness while reducing the required node transmitting power but also achieves good energy balance and high energy efficiency. Ultimately, our method effectively prolongs network lifetime and improves network performance.

#### 1. Introduction

A wireless sensor network (WSN) is a multihop self-organizing network intended for data collection. WSNs are widely used in IoT systems because of their low cost, small size, and rapid deployment characteristics. However, due to the complexity of network working environments and node resource limitations, it is difficult to supplement the energy of WSNs after their energy is exhausted [1]. Therefore, energy efficiency and energy balance have always been the main factors that restrict WSN development. Topology control technology is a key aspect of the WSN field, which is mainly applied to the link and network layers. It is also important to improve the efficiency of routing and MAC protocols, reduce communication interference, and make network energy consumption balanced. Therefore, effectively controlling and optimizing the network topology is a key challenge in WSNs [2].

Topology control technology provides an optimized network topology, which affects network routing and MAC protocols. The network layer, MAC layer, and physical layer in WSNs can all affect the result of topology control, and they also have a mutually restricted relationship. The transmitting power and transmission rate of the physical layer can affect access control in the MAC layer and routing decisions in the network layer. The MAC layer channel allocation limits the network bandwidth and affects routing decisions in the network layer, which is responsible for both routing decisions and packet transmission. Therefore, the design of an effective topology control method needs to consider not only the energy consumption factors but also the cross-protocol layer for a variety of performance parameters.

When the sensor nodes perform data forwarding, they will exhibit selfish behavior due to energy saving considerations, and competition will occur between nodes [3]. On this basis, game theory was introduced into the study of WSN topology control. Game theory provides a powerful tool [4] for describing the phenomena of competition and individual coping strategies between intelligent rational decision-makers, and it has been used in systems involving action and payoff. Topkis proposed a supermodular game in 1978 [5], which used a game theory based on lattice theory. The model has the following properties [6]: (1) It considers the complementarity of the strategies between the parties in the game, while making the existence of the Nash equilibrium and related static results clearly visible. (2) It makes up for the shortcomings of the traditional game theory, which makes game theory no longer need convexity and differentiability of objective function and expands the application scope of noncooperative game theory.

In order to accurately describe the topological game behavior between nodes, this paper introduces the supermodel game theory into the WSN topology control technology and integrates multiple performance parameters across the protocol layer into the utility function of the model. First, this paper establishes a new topology game model. Secondly, COETC is proposed, which is a multilayer optimized topology game algorithm for wireless sensor networks. Finally, experiments show that the network topology constructed by the COETC algorithm can accurately describe the competition and contradictory behavior between nodes. Under the premise of ensuring network connectivity and robustness, COETC solves the problem of energy imbalance between nodes, which improves energy efficiency and effectively prolongs network lifetime.

#### 2. Related Work

So far, many topology control algorithms have been proposed for WSN, which are mainly divided into layering, power control, and game-type topology control algorithms. For example, in [7], a low-power multilevel hierarchical WSN topology control algorithm is designed. The algorithm extends the network level and improves WSN maintainability using a combination of static and dynamic addresses. In [8], a low-energy adaptive clustering hierarchy (LEACH) topology control method was established for WSNs using time slots, in which a cluster-head-selecting approach reduced cluster size differences and the responsibility mechanism of the active node leads to a more balanced energy consumption in the cluster. In [9], Kubisch et al. implemented dynamic power control to set the upper and lower limits of the node degree, thus resulting in a network topology with lower total energy consumption. The power control algorithm proposed in [10] uses a Borel Cayley graph to construct a network topology with a short average link and low energy consumption. However, it does not consider the robustness of the network topology and the residual energy of nodes, both of which affect network operation.

Traditional game theory methods are used in current research. Komali et al. [11, 12] formulated energy-efficient topology control as a potential noncooperative game. This approach guaranteed the existence of at least one Nash’s equilibrium (NE) and proposed a distributed noncooperative game topology control algorithm based on game theory. The authors of [13] designed a topology control algorithm based on a link power consumption game to run the minimum MLPT algorithm for the maximum power of the node. To also consider network lifetime, researchers have proposed two game-based topology control algorithms: the virtual game-based energy-balanced (VGEB) algorithm [14] and the energy welfare topology control (EWTC) algorithm [15], both of which were developed to improve network lifetime via energy-balanced network topologies. In [16], the adaptive cooperative topological control algorithm (CTCA) based on game theory considered the smallest potential lifetime and degree as primary and secondary utility functions, respectively. In [17], a distributed energy-balanced topology control algorithm (DEBA) based on an ordinal potential game was proposed by designing a payoff function that considered both network connectivity and the energy balance of nodes. In [18], Wang et al. proposed ATGG, an adaptive topology game algorithm for energy balance in a wireless sensor network. According to the average life of nodes, nodes adjust their own power to help nodes with the shortest life to reduce transmitting power, which can prolong the entire network life. In [19], a novel topological control game algorithm, TCAMLPM, was proposed based on Markov’s life prediction model, a distributed topology control game algorithm for WSN which ensures the algorithm to converge to the Nash equilibrium by making use of the best response strategy. In [20], by comprehensively considering the benefits between node coverage and residual energy, CTCL, a node scheduling algorithm, is proposed, which introduces a noncooperative game theory. Although some of the abovementioned algorithms based on game theory achieve network topology control and improve network performance, they cannot guarantee the connectivity and robustness of the network. Additionally, the algorithms do not fully consider the remaining energy, energy balance, and energy efficiency of the nodes. Thus, the existing algorithms based on game theory can achieve network topology control to some extent and improve network performance, but they cannot effectively guarantee the network’s connectivity and robustness. Simultaneously, due to the complexity of the WSN deployment environment, the factors that affect network operation make it more difficult to quantify the node revenue; therefore, the existing topology control algorithms based on game theory cannot accurately describe the phenomena of balancing the competition between nodes with network energy consumption. The above method requires that the objective function is convex and differentiable, which affects the result of the algorithm.

#### 3. Supermodular Game Theory

A supermodular game is based on a rich mathematical foundation of lattice theory and comparative statics. The strategy space of every player is a partially ordered set, and the utility of playing higher strategy increases when the opponents also play higher strategy [21]. Suppose is a real valued function in lattice , is a partially ordered set, and is a subset of . If , then is the upper bound of , where and . If , then is the lower bound of . If the set of the upper (lower) bounds of has a least (greatest) element, then this least upper bound (greatest lower bound) of is the supremum of .

The strategy game consists of players, the possible strategy of the players, and consequences of applying strategy. The following definition is given for the strategy game: where (1) represents the player set, and is the number of players in the game; (2) represents the policy space, and is the Cartesian product of the set of policies , where represents an optional set of policies for node , usually abbreviated as . In general, we use to describe a strategy combination, where represents the strategy choice of node and represents the strategy choices of nodes other than node . Finally, (3) represents a utility function , where denotes the maximum utility function that node can achieve in the policy combination .

A strategy game is a supermodular game if the set of feasible joint strategies is a sublattice of . The utility function of the supermodular game belongs to the supermodular function, whose definition is as follows: for set , , can be called a lattice, where represents the upper bound [5] of set and represents the lower bound [5] of set . For an objective function defined on lattice , , it can be understood that the increment of the function realized from the point change to the point (or ) is smaller than the change from the point (or ). For the corresponding increment of point , then can be the supermodular function on .

*Definition 1. *In a strategy game , if the strategy of any game player is the best strategy response to the strategy combination of the remaining game participants, then there must be a , where means that , and the th strategy for the game player is valid. Then, is called the “Nash equilibrium (NE)” [22] of the game.

Theorem 1. *If the game is supermodular, there exists the largest and the smallest Nash equilibrium in pure strategies [23].*

Lemma 1. *The condition that the function on the set is a supermodular function is that must have a difference on [23].*

Lemma 2. *If is the interval on set and the existence function is quadratic and continuously different in an open interval containing , then the necessary and sufficient condition for to become a supermodular function at is , where vector and in the dimension Euclid spaces [23].*

*Definition 2. *For the strategy game , in the case where the strategy space of any participant is a real interval and the utility function is twice continuously differentiable on , when , the strategy game is a supermodular game [24].

Theorem 2. *A supermodular game with a limited space for any participant strategy will converge to a pure strategy Nash equilibrium point when a better response strategy is adopted [24].*

#### 4. System Model

In this section, we first construct a topology control game model. Then, we prove that the game model belongs to the supermodular game and has a pure strategy Nash equilibrium.

##### 4.1. Topological Game Model

This paper uses the strategy game to realize the cooperative optimization of various WSN performance goals. To maximize the utility function, the participants in this game model adjust their power selfishly, which is typical behavior in a noncooperative topology control game. The three elements of the game model are as follows: (1)Participant set: all the nodes in the network can be regarded as game participants, specifically expressed as , where is the total number of nodes in the network(2)Strategy space: the optional power set of the node can be regarded as the game strategy space, where indicates the current power selection of node and represents the number of candidate powers of node (3)Utility function: the environments in which WSNs are used are complex, and it is difficult to quantify the node revenue. Therefore, the design of node utility functions needs to consider the influence of multiple factors. To accurately describe the competition phenomenon between nodes and the energy balance situation, for node , this paper comprehensively considers the utility function design problem from the following aspects: (a)Network connectivity at the link layer: for network topology, maintaining connectivity is a basic network aspect that must be guaranteed. Therefore, when constructing the utility function, it is necessary to consider network connectivity. The connectivity factor is defined as , indicates that the network is in the connected state, and indicates that the network is in a unconnected state. , when transmitting power . Obviously, is monotonically nondecreasing(b)Node degree: by optimizing the node degree, the total energy consumption of the network can be reduced and the overall performance of the network can be improved. Therefore, the node degree of the node in the power transmission state is introduced into the design of utility function, and represents the total number of one-hop neighbor nodes of node .(c)Degree of interference at the MAC layer: interference between nodes also affect network performance. The interference between nodes will increase with the growth of power, and at the same time, the probability of packet retransmission will also increase. Similarly, network operating efficiency will decrease with the increase of the network energy consumption. The definition of the degree of interference at the MAC layer is , where node is the one-hop neighbor of node , and represents the total number of one-hop neighbor nodes of node in the power transmission .(d)Residual energy balance: when the residual energy distribution of the network nodes is uneven, some nodes will experience premature death, which will affect network operation. Therefore, the residual energy needs to be considered in the design of the utility function. The residual energy equalization factor of the defined node is , where and represent the residual energy and the initial energy of node , and represents the total number of one-hop neighbor nodes of node in the power transmission , respectively.

In summary, for , considering factors such as network node degree, node residual energy, and network interference, the utility function is designed as follows: where and are weighting factors to ensure that the revenue when the network is connected is greater than the income in the unconnected case, and both are positive numbers, which in the simulation analysis part of this paper, we will determine how to set weighting factors and . Although the node selects a larger transmit power, it can increase the node degree to enhance the network connectivity. However, a larger transmit power will inevitably generate a stronger network interference, increase network energy consumption, and reduce network operating efficiency. These factors exist between contradictions and constraints.

##### 4.2. Model Analysis

Theorem 3. *The topological game model using equation (2) as the utility function is a supermodular game and must have a Nash equilibrium solution.*

*Proof 1. *The policy set of node is a subset of the real set .
(1)If the network is in the unconnected state, , equation (2) can be simplified to
For equation (3), and , the first-order partial derivative is obtained for power:
where is one-hop neighbor of node . The second-order partial derivative is obtained:
(a)For node is not a neighbor of node (b)For node is a neighbor of node In the network environment, is positively correlated with , . Therefore, in equation (6), .
(2)And if the network is in the connected state, , equation (2) can be simplified to
For equation (7), and , the first-order partial derivative is obtained for power:
where is the one-hop neighbor of node . The second-order partial derivative is obtained:
(a)For node is not a neighbor of node (b)For node is a neighbor of node Obtained by Equation (6), in Equation (10), .

In summary, , which indicates that the utility function of the topological game model has incremental differences. Thus, according to Definition 2, the topological game model is a supermodular game, and there is at least one pure strategy Nash equilibrium.

#### 5. Cross-Layer Optimized Energy-Balanced Topology Control Algorithm

In this section, we design the cross-layer optimized energy-balanced topology control algorithm (COETC) based on a supermodular game using the topological game model constructed in Section 4.1.

The node must meet certain preconditions when running the algorithm: (1) the network node ID must be unique; (2) the node cannot be moved after deployment; (3) the transmit power of all nodes is continuously adjustable and the maximum transmit power of the nodes can vary; (4) the network layer can obtain global information concerning the network; and (5) when all nodes have the maximum transmit power characteristics, the network topology graph formed by default is connected (this must support bidirectional link communications).

The operation of the COETC algorithm is divided into four phases: a neighbor discovery phase, a topology establishment phase, a power adjustment phase, and a topology maintenance phase.

##### 5.1. Neighbor Discovery Phase

In this phase, each node needs to discover its local neighbor nodes and build a list of neighbor information through information interchanges. Each source node initializes its own transmit power to the maximum transmit power , by broadcasting the “Hello” message and receiving ACK messages returned by the target nodes . The source node obtains the neighbor node information by parsing the ACK message and storing the parsed information in a neighbor table , generating its own policy set through the information exchange nodes.

The “Hello” message includes the source node’s ID, its current transmit power , and its remaining energy . The ACK message includes the target node’s ID, its minimum power , and its remaining energy that can ensure normal communication with the source node, thus ensuring normal communications among network nodes. The minimum power is calculated by the free space model following [25].

##### 5.2. Topology Establishment Phase

To improve the efficiency of the algorithm, node first needs to sort the policy set in descending order based on the obtained minimum power , where represents the number of candidate powers of node , and simultaneously calculate and count the number of neighboring nodes in its current transmit power range. Then, the topology game process is launched according to the node ID order and the influence of the current power of the nodes on network connectivity is determined in turn.

When the network is in an unconnected state under the current power, the connectivity factor is set to 0, and the utility function under the current power level is used. The value is recorded in the neighbor table and the node must return to the previous power level that can maintain connectivity. When the network is in the connected state under the current power, the utility value function under the current power level is recorded in the neighbor table and then sequentially decreased. The low power level and the network connectivity under the current node power are determined, thereby finding the optimal power combination that satisfies the network connectivity conditions.

##### 5.3. Power Adjustment Phase

After the topology establishment phase is completed, the utility values of the respective power levels satisfying the network connectivity condition are stored in the node’s neighbor table. Then, the node selects the power level by comparing the size of the utility value. To ensure that the game converges to the Nash equilibrium, the algorithm uses a better response strategy update scheme [26] to perform power adjustment. This scheme can ensure that the supermodular game with limited tactical space will converge to a pure strategy Nash equilibrium point.

For any node , given the power of the other participants, the preferred response of participant is , where . During the game process, the node chooses a power level lower than the current transmitting power to communicate. Then, it observes whether the corresponding comprehensive utility function value increases. If it does increase, that power level is more suitable for use as the transmit power; otherwise, the node uses the current transmitting power to determine its maximum utility value.

When the transmit power of a node changes, its communication radius, neighboring node set, and its related links will also change, which changes the network topology. As shown in Figure 1, when the transmit power of node increases, a new node may be included in its communication range. Then, the nearest neighbor node of node is changed from the original node to the current node . Therefore, node can reduce its transmit power appropriately while still guaranteeing full network connectivity.