Cooperative Topology Control with Adaptation for Improved Lifetime
in Wireless Sensor Networks
Abstract
Topology control algorithms allow each node in a wireless multihop network to adjust the power at which it makes its transmissions and choose the set of neighbors with which it communicates directly, while preserving global goals such as connectivity or coverage. This allows each node to conserve energy and contribute to increasing the lifetime of the network. In this paper, in contrast to most previous work, we consider (i) both the energy costs of communication as well as the amount of available energy at each node, (ii) the realistic situation of varying rates of energy consumption at different nodes, and (iii) the fact that cooperation between nodes, where some nodes make a sacrifice by increasing energy consumption to help other nodes reduce their consumption, can be used to extend network lifetime. This paper introduces a new distributed topology control algorithm, called the Cooperative Topology Control with Adaptation (CTCA), based on a gametheoretic approach that maps the problem of maximizing the network’s lifetime into an ordinal potential game. We prove the existence of a Nash equilibrium for the game. Our simulation results indicate that the CTCA algorithm extends the life of a network by more than 50% compared to the best previouslyknown algorithm. We also study the performance of the distributed CTCA algorithm in comparison to an optimal centralized algorithm as a function of the communication ranges of nodes and node density.
I Introduction
In wireless ad hoc networks, especially ad hoc sensor networks, the battery life of each node plays a critical role in determining the functional lifetime of the entire network. When a node exhausts its limited energy supply, it may fail to reach nearby nodes leading to a disconnected network and disabling some essential communications. Without energy, the node will also fail to continue the environmental monitoring activities essential to the functional operation of the system. Adding redundant nodes in the network may extend the functional lifetime but it is ultimately a less costeffective approach. In this paper, we consider the problem of extending the lifetime of a network using a new adaptive gametheoretic approach.
Topology control is among the betterknown approaches to conserving energy and prolonging a network’s functional life. In a topology control algorithm, each node adjusts the power at which it makes its transmissions to reduce the energy consumption to only what is needed to ensure topological goals such as connectivity or coverage. Examples of topology control algorithms include Directed Relative Neighborhood Graph (DRNG) [1], Directed Local Spanning Subgraph (DLSS) [1], Step Topology Control (STC) [2] and Routing Assisted Topology Control (RATC) [3]. In most traditional algorithms, the topology of the network is determined at the very beginning of the life of the network where the only consideration for each node is to reduce its transmission power while keeping the graph connected. After the execution of one of these algorithms, each node will transmit at the selected power level until it eventually runs out of energy. However, depending on the location of a node in relation to others, some nodes may end up with a much larger communication radius, and therefore a much larger transmission power, than some others. This uneven distribution of the assigned transmission powers may result in an unbalanced energy consumption at the nodes, leading to some nodes exhausting their energy far sooner than some others. Such a scenario can end the functional life of the network earlier than necessary. This highlights two weaknesses of these algorithms: they are not adaptive to different rates of energy consumption on different nodes and they do not allow cooperation between nodes to extend the network lifetime. Each of these weaknesses is addressed by the algorithm proposed in this paper: Cooperative Topology Control with Adaptation (CTCA).
We illustrate the principle of cooperative topology control with a simple toy example shown in Fig. 1. Suppose Fig. 1(a) illustrates the result of a topology control algorithm, where no node can reduce its transmission power unilaterally without disconnecting the graph. In this figure, the presence of an edge from one node, say , to another node, say , implies that can transmit at a power level sufficient to reach node . The communication radius of each node is shown by the dashed arcs. Assuming all nodes start out with the same energy supply and make transmissions at the same rate, we note that node has the largest energy cost and thus has the shortest lifetime. Node , on the other hand, has the smallest transmission power, and therefore, has the longest lifetime. Traditional topology control algorithms discussed above will lead to the situation in Fig. 1(a) ending the functional life of the network when node ’s energy is exhausted even though node would have plenty of remaining energy. Fig. 1(b) illustrates a topology where node increases its transmission power so that it can now reach node directly. Now, node is able to reduce its transmission power to only be directly connected to node , as shown in Fig. 1(c). This involves node making a sacrifice by increasing the power at which it makes its transmissions in order to allow node to reduce its transmission power, thus extending the life of node and of the network.
The actual power consumption for sending and receiving a data packet varies significantly depending on the radio environment of the space where the sensor nodes are located and also the electronics of the devices. The logdistance path loss model based on the path loss as a logarithmic function of the distance has been confirmed both theoretically and by measurements in a large variety of environments [4, 5, 6]. In this model, the path loss at distance , is expressed as:
where the constant is an arbitrary reference distance and is called the path loss exponent. This implies that the energy consumed to make a transmission across a distance is proportional to . Since ranges from 2.5 to 6 in most real environments [4], especially over longer distances, a single transmission over distance often consumes more energy than two transmissions each over distance . This motivates the goal of most topology control algorithms to choose multiple smaller hops in place of a single longer hop with the intent to reduce overall energy consumption. While device electronics can sometimes be such that choosing smaller hops—especially at smaller distances—does not always guarantee lower energy consumption, there is another good reason to choose smaller hops: reducing interference in all communications. Therefore, a general goal of a topology control algorithm is to achieve lower transmission powers for all the sensor nodes in order to reduce both energy consumption and interference [7]. In other words, the topology illustrated in Fig. 1(c) is more desirable.
In this paper, we employ game theory to facilitate such topology control that allows cooperation between nodes as illustrated in Fig. 1. Our approach is through developing an ordinal potential game [8, 3, 9] into which our problem can be mapped, so that all nodes pursue a localized strategy that can be expressed through a single global function, or the global potential function. Our approach also allows an adaptive strategy so that a node does not end up with the same power level through its entire lifetime. This is significant to extending the network lifetime because it is almost always the case that different nodes consume energy at different rates. Our approach to allowing adaptation is through incorporating the energy remaining on the nodes in the neighborhood into the decisions made by each node. Since this remaining energy changes over the life of a network, our topology control algorithm adaptively adjusts the power levels at each node. This constantly keeps shifting energy consumption from nodes with less energy reserves to those with more energy reserves, thus extending the life of the network.
Ia Problem Statement
Given a wireless sensor network, let graph represent its topology at time , where represents a node within the network, and represents the fact that node is within node ’s communication radius and can hear from directly at time . Assume is a connected graph at time . Topology control algorithms have traditionally emphasized preserving connectivity as a constraint while pursuing the goal of reduced energy consumption at each node.
However, depending on the type of application for which an ad hoc sensor network is deployed, it is possible that a network is functional even if a certain subset of nodes runs out of energy [10, 11, 12]. The functional lifetime of a network, therefore, depends on the application in use and consequently, there is some debate on how best to define the functional lifetime of a network. In an ad hoc sensor network with a nonhierarchical topological organization, one may assume an applicationdependent parameter, , to define the functional lifetime as the length of time the network topology possesses at least one connected component with or more nodes (where is the total number of nodes in the network). When , the functional lifetime of the network is the length of time is a connected graph, which is until any node is disconnected or runs out of energy.
We find that a definition of functional lifetime using is a more versatile one for two reasons: firstly, on any application, there may be some crucial nodes which, when they die, can disable the functionality of the network; secondly, a definition based on the case can form the foundation of greedy algorithms designed to extend functional lifetime for .
In this paper, therefore, we consider the functional life of the network to have ended when one of the following two cases occurs:

Case 1: A node reduces its current transmission power in order to save energy, but becomes unable to reach certain nodes and, consequently, loses connection from part of the network.

Case 2: A node runs out of energy, thus getting disconnected from the rest of the network.
If Case 1 happens, the communication links whose removal caused the network to become disconnected can be restored back into the network to restore the functional life of the network. On the other hand, if Case 2 happens, the network’s functional lifetime cannot be extended in any way. Therefore, to improve the lifetime of the network, (i) Case 1 should be avoided by always ensuring connectivity in the assignment of power levels to the nodes, and (ii) Case 2 should be pushed as far into the future as possible by reducing the rate of energy consumption at the node that is estimated to have the smallest remaining lifetime. The problem can now be defined as one of periodically reassigning the power at which each node makes its transmissions so that the first occurrence of either Case 1 or Case 2 is pushed as far ahead in time as possible.
IB Contributions and Organization
Section II reviews the related work on approaches that have been employed to increase a wireless sensor network’s lifetime through topology control. Section III analyzes the rationale behind the approach used in this paper and presents a few definitions and lays out the foundational concepts for the gametheoretic approach used in this paper. Section IV proves the existence of a Nash equilibrium for the ordinal potential game used to map our problem. Our proof is based on showing that the difference in individual payoffs for each node from unilaterally changing its strategy and the difference in values of the global potential function have the same sign.
The pseudocode for the Cooperative Topology Control with Adaptation (CTCA) algorithm is presented in Section V. A simulationbased evaluation of its performance and a comparative analysis with other topology control algorithms are described in Section VI. Our results show that the CTCA algorithm extends the life of a network by more than 50% compared to the best previouslyknown algorithm. This section also compares the topology delivered by the distributed CTCA algorithm to the optimal solution obtained using a centralized algorithm. We also study the dependence of the performance of CTCA in relation to the optimal on the communication ranges of nodes and on the node density.
Section VII concludes the paper.
Ii Related Work
The task of extending the life of a wireless sensor network can be tackled through multiple complementary ways involving routing protocols, medium access strategies or any of several other protocols that facilitate network operations. In this section, we will discuss only the approaches most related to this paper; that is, approaches based on changing the topology of the network by individual nodes changing the power levels at which they make their transmissions while preserving network connectivity.
Traditional topology control algorithms such as Small MinimumEnergy Communication Network (SMECN)[13], Minimum Spanning Tree (MST) [14], DRNG[1], DLSS[1] and STC[2] usually start the topology control process with each node transmitting at its maximum transmission power to discover all of its neighbors. Local neighborhood and powerlevel information is next exchanged between neighbors. The minimum transmission power of each node such that the graph is still connected is later computed at each node without further communication between nodes. The Weighted Dynamic Topology Control (WDTC) [15] algorithm improves upon the work of MST, and considers the remaining energy of each node in addition to the energy cost of communication across each pair of nodes. The algorithm, however, forces bidirectional communication between each pair of nodes and, in addition, requires periodic communication by each node at its maximum possible power level. Other related algorithms seek to offer a robust topology where the graph can stand multiple channel failures; for example, a connected graph is sought in [16, 17] and a twotiered network in [18].
Other topology control algorithms may require communication between nodes throughout the topology control process. One typical example is the work described in [19], which is based on a selfish game on network connectivity to help reduce the transmission power on each node. By offering a utility function which indicates a high profit if the node’s transmission power is small and a low profit if the node’s transmission power is large, each node selfishly reduces its transmission power to maximize its profit. On the other hand, if the node has reduced its transmission power to such an extent that the graph becomes disconnected, the profit of each node becomes 0. This algorithm was later improved in [3], where the requirement of global information (to establish connectivity) is eliminated and a distributed topology control algorithm is proposed. Among the first works on using game theory in topology control problems is [20] which gives tight bounds on worstcase Nash equilibria for a game in which the network is required to preserve connectivity. However, this study only considers selfish nodes which try to minimize their energy consumption without considering potential sacrifices nodes can make (by expending more energy) to extend a network’s lifetime.
Another class of topology control algorithms is represented by the work reported in [21], where the authors provide a decentralized static completeinformation game for power scheduling, considering both frame success rate and connectivity. Yet other approaches to increasing the lifetime of a wireless sensor network include grouping nodes into clusters to create a communication hierarchy in which nodes in a cluster communicate only with their cluster head and only cluster heads are allowed to communicate with other cluster heads or the sink node [22, 23, 24, 25, 26, 27]. In the work of [28], the authors tried to assign sensor nodes with different initial energy levels so that sensor nodes with high traffic load will be assigned more energy than those with smaller loads. By doing so, with the same amount of overall energy, the network’s lifetime may be extended.
If the network’s lifetime is measured in terms of how many transmissions can be made before the sensor nodes run out of energy, then maximizing the network’s lifetime can be interpreted as maximizing the throughput of the network. In the work of [29], the authors studied the relationship between throughput of the network and its corresponding lifetime under an SINR model. But they focus on a specific network setting where sensor nodes’ neighbors and the communication links are predetermined and the topology of the network remains constant throughout the network’s lifetime.
A survey of topology control algorithms can be found in [30, 31] and a survey of the applications of game theory in wireless sensor networks can be found in [32, 33].
The CTCA algorithm proposed in this paper is the first to use a gametheoretic approach that also adapts to changes in the remaining energy levels of nodes and which allows cooperative behavior amongst nodes. As will be discussed in the following sections, these features allow it to extend the life of a network by more than 50% compared to the best previouslyknown algorithm.
Iii Definitions and Preliminaries
In this section, we define terms and concepts that will enable us to specify the localized goals that each node should pursue in order to achieve the global goal of increased lifetime for the network.
Let denote the amount of energy remaining at node at time . Let denote the power at which node makes its transmissions at time . As an estimate of the additional length of time before a node runs out of energy, we define the estimated lifetime of node at time , denoted by , as the ratio between the amount of remaining energy on the node at time and the power at which it makes its transmissions at time . That is, . Note that the estimated lifetime may or may not accurately capture the actual remaining lifetime of a node (because its transmission powers may change later or its energy reserves may deplete slower/faster than estimated.) When the context is clear, for brevity, we refer to the estimated lifetime as just the lifetime.
In a system in which the rate of energy consumption is largely balanced across the nodes (which is the goal of this paper as a means to improve network lifetime), the node with the smallest estimated lifetime is likely the one that determines the network’s lifetime. We consider the estimated lifetime of a network as the estimated lifetime of the node with the smallest estimated lifetime. If is the node with the smallest estimated lifetime within the network, then it may be possible to improve the network’s lifetime by improving ’s estimated lifetime. Fig. 1 shows an example where node is able to reduce its transmission power with help from node , thus increasing its estimated lifetime and likely the lifetime of the network.
We further illustrate the definitions in this section using the topology shown in Fig. 2. Suppose at time , the topology of the network is as shown in Fig. 2(a). Denote by the minimum transmission power at which nodes and have to transmit to reach each other. We refer to the set of transmission powers that a node may switch to at time as its available transmission powers at time . Then, according to the topology given by Fig. 2(a), node ’s available transmission powers are: , and , while its current transmission power is (note that there is no need for node to transmit at any power level other than the ones in this available set).
Let denote a mapping of nodes in the network to power levels. For example, in Fig. 2(a), the mapping implemented is given by . Since node has the potential to transmit at power while still keeping the graph connected, we refer to power as node ’s potential transmission power under this nodetopower mapping .
In general, the potential transmission power of a node is the smallest available transmission power that the node can use such that the graph is still connected while the power levels at all other nodes remain the same. Let denote the potential transmission power of node under the nodetopower mapping . Note that a node’s potential transmission power is no greater than its current transmission power provided that the network is currently connected. That is, if is a connected graph and is the nodetopower mapping implemented at time .
Transmitting at the potential transmission power as defined above can increase the lifetime of a node beyond its estimated lifetime and, consequently, of the network. To estimate the best lifetime a node can achieve without changing the transmission powers of other nodes and without disconnecting the network, we define the potential lifetime of a node as the ratio between the node’s current remaining energy and its potential transmission power. Let denote the potential lifetime of node at time under the nodetopower mapping . Let denote its potential transmission power under the nodetopower mapping . Then, .
If , then . Therefore, to increase its estimated lifetime, a node should always try to change its current transmission power to its potential transmission power, if they are not the same. Figs. 2(a) and 2(b) illustrate such a process for node , where it changes its current transmission power from in Fig. 2(a) to its potential transmission power as illustrated in Fig. 2(b).
In Fig. 2(a), suppose node is the node that has the smallest estimated lifetime within the network. Then, ’s estimated lifetime has to be improved in order to improve the network’s lifetime. In Fig. 2(b), node reduces its transmission power but this does not improve the potential lifetime of node . This implies that ’s estimated lifetime cannot be improved by node reducing its transmission power. On the other hand, if node chooses to transmit at a higher power level, , as illustrated in Fig. 2(c), then node ’s potential transmission power reduces to . Now, node can reduce its transmission power to the new potential transmission power as illustrated in Fig. 2(d), thus improving the network’s estimated lifetime.
Let denote the set of nodes can reach at time ; i.e., . Let denote the set of nodes that can reach node at time , i.e, . Then for any , we have . We refer to the nodes in the set as the reachable neighbors of node and the nodes in the set as the reverselink neighbors of . For example, in Fig. 2(a), ’s reachable neighbors are , and while has only one reachable neighbor, . Also, is a reverselink neighbor of , and . Let and let .
In Fig. 2(a), note that ’s potential transmission power will not be reduced unless either or has increased its transmission power to be able to reach . In general, ’s reachable neighbors are the only nodes who can help reduce its potential transmission power; and only the nodes who are ’s reverselink neighbors may benefit from ’s increase in its transmission power level.
A question worth answering at this point is about what might be the relationship between improving the network’s lifetime and improving node ’s estimated and potential lifetime. As we have stated previously, the network’s estimated lifetime is dependent upon the node with the smallest estimated lifetime. If we compare the power mapping illustrated in Fig. 2(a) and Fig. 2(d), we can see that node not only has sacrificed its chance to improve its lifetime but also has sacrificed its own estimated lifetime (increases its current transmission power from to ) in order to help improve node ’s potential lifetime which is the smallest in Fig. 2(a). When the topology is as illustrated in Fig. 2(d), suppose node is the node with the smallest estimated lifetime. If its estimated lifetime in the topology illustrated in Fig. 2(d) is less than that of node ’s in the topology illustrated in Fig. 2(a), the network’s estimated lifetime in fact reduces rather than increases. In such a case, node should not choose to increase its transmission power to help improve ’s lifetime, but should instead focus on improving its own estimated lifetime.
For any node within the graph, its estimated lifetime will not improve unless it switches to its potential transmission power. That is to say, no node can help node with its estimated lifetime except for node itself; but as illustrated in Fig. 2, another node may help with ’s potential lifetime. In our example, node helps node improve its potential lifetime which eventually allows to increase its estimated lifetime by changing its transmission power to the potential transmission power.
The above discussion leads to the following primary and secondary goals for each node in order to improve the network’s lifetime while also conserving energy as much as possible:

Primary goal: Let denote the node with the smallest potential lifetime amongst the reverselink neighbors of node . Let denote the potential lifetime of node . The primary goal of node should be to increase the potential lifetime of node above while making sure that its own estimated lifetime does not reduce below .

Secondary goal: The secondary goal of node should be to increase its own estimated lifetime.
The secondary goal is achieved once a node adopts its potential transmission power as its current transmission power (in this situation, its potential lifetime then becomes its estimated lifetime). Note that, if the primary and the secondary goals conflict, a node should always choose to meet the primary goal. This leads us into the design of the cooperative game that each node can play with its reachable neighbors and its reverselink neighbors.
One may argue that since the ultimate goal is to extend functional lifetime (the primary goal), the secondary goal should be unnecessary. However, there are three reasons why we include the secondary goal in our design of the algorithm.
Firstly, for any realistic algorithm, it is not possible to precisely predict the network lifetime since it is not possible to predict with precision the future traffic load experienced by any given sensor node. As a result, an algorithm can only seek to maximize the estimated functional lifetime of a node and not the actual functional lifetime. Therefore, the secondary goal helps each sensor node save energy as much as possible so that when the estimated functional lifetime deviates from the actual functional lifetime, the node that actually determines the network’s functional lifetime (the one that dies earliest) does not waste any energy by transmitting at a power larger than what was necessary to stay connected. Secondly, as discussed in Section IA, we define the functional lifetime assuming . However, if the functional lifetime for some application is defined assuming that the network function survives until nodes are disconnected, a good heuristic for extending the functional lifetime emerges if we implement the secondary goal in conjunction with the primary goal assuming on the surviving largest connected component every time a node dies. Thirdly, reducing the transmission power of a sensor node may help reduce the interference among transmissions, reducing retransmissions and also helping extend the functional lifetime of the network.
In the above discussion, without loss of generality, we assume that there is only one node with the smallest potential lifetime amongst reverselink neighbors of . If there is a tie with two nodes having identical potential lifetimes, one can always break the tie in the algorithm using a consistently applied second criterion such as the node id.
Iv The Ordinal Potential Game
In this section, we present the notation and the utility function governing the ordinal potential game into which we map the problem of extending the lifetime of the network.
Iva Notation
Table IVA presents a glossary of terms used in this section. In this paper, for brevity and clarity, we sometimes omit the time index whenever the corresponding instant of time is clear from the context.
Notation  Definition 

The set of nodes within the network. .  
Graph representing the network topology at time .  
or  Node ’s transmission power at time . 
A mapping, , of the nodes in the network to transmission power levels. For example, the mapping actually implemented at time is  
A mapping of all the nodes except to transmission power levels.  
The minimum transmission power needed for node ’s transmissions to reach node . We assume .  
or  The set of nodes reachable by ’s transmissions at time . . This set of reachable neighbors is also called the reachable neighborhood of . 
or  The set including and its reachable neighbors. . 
or  The set of nodes at time which can directly reach with their transmissions. . This set of reverselink neighbors of is also called reverselink neighborhood of . 
or  The set including and its reverselink neighbors. . 
A mapping, , of and its reachable neighbors to transmission power levels.  
A mapping of and its reachable neighbors except to transmission power levels.  
’s potential transmission power when its and its reachable neighbors’ power levels are given by the mapping .  
’s possible transmission power choices.  
or  ’s remaining energy at time . 
’s estimated lifetime (remaining) at time if set to transmit at power .  
’s potential lifetime at time when its and its reachable neighbors’ transmission powers are given by the mapping .  
or  The node with the smallest potential lifetime among ’s reverselink neighbors at time . . 
Suppose that at time , each node is transmitting at its maximum transmission power . Then, the set of nodes that includes ’s reachable neighbors is . Let denote the size of the set . Therefore, we define the available transmission powers for node as where, for any , there exists a node such that is the minimum transmission power required for to reach . Note that a node does not need to transmit at power levels other than the ones needed to reach other nodes within its maximum range.
Let denote a mapping of the nodes in the network to transmission power levels. The mapping implemented at time is , where is the power at which node is set to make transmissions at time . We write where is a mapping of node to a certain power level and is a mapping of all other nodes in the network to power levels.
Let denote a mapping, , of and its reachable neighbors to power levels. We write where is a mapping of node to a certain power level and is a mapping of all other nodes in to power levels.
Let denote the estimated lifetime of node at time if set to transmit at power level . Per the definition of estimated lifetime, . If , then .
Denote by , or for brevity, the node in with the smallest potential lifetime. We define the potential lifetime of ’s reverselink neighborhood at time as the potential lifetime of node at time , i.e., .
IvB The utility function
In the following, we present and justify the utility function governing the ordinal potential game upon which our topology control algorithm is based.
As stated in Section III, the primary goal for each sensor node is to improve the potential lifetime of its reverselink neighborhood without also causing a reduction in the network’s estimated lifetime. While prioritizing the primary goal, the secondary goal of the sensor node is to improve its own estimated lifetime. Both of these goals are captured in the utility function presented in this section.
Let denote a power level that is available to node at time . Define the primary utility function (corresponding to the primary goal described in Section III) for node with power level at time as:
(1) 
This is the minimum of the estimated lifetime of node at power level and the potential lifetime of the node whose value is the minimum amongst its reverselink neighbors. Maximizing this is the primary goal as explained in Section III.
Define the secondary utility function for node with power level at time as:
(2) 
This is the estimated lifetime of node when transmitting at power level at time . Maximizing this is the secondary goal as also explained in Section III.
Bearing in mind the two goals, primary and secondary, for each node, we define the utility function for node with power level at time as:
(3) 
where and are defined in the following paragraphs.
The term in Eqn. (IVB) is a binary function indicating whether node , when set to transmit at power , is connected to every node . More specifically,
If node has lost connectivity with a certain node by transmitting at power at time , i.e., , then, the network’s connectivity is lost and by Case 1 in Section IA, the network’s life has ended. This should be reflected in node ’s own utility function, and thus, when . Note that checking for the existence of a path to every is a localized function and does not require global or centralized knowledge.
The term in Eqn. (IVB) is a binary function indicating whether the node’s own estimated lifetime should be considered when calculating its utility at power level . In the following, we will describe the conditions under which takes on the values of either or .
As discussed in Section III, improving its own lifetime is only the secondary goal for every sensor node. When the primary and the secondary goals of a node are in conflict, the secondary goal of improving its own lifetime should yield to the primary goal. Therefore, for that leads to this situation, indicating that the secondary goal of node yields to the primary goal. On the other hand, for the power level at which node is able to achieve its primary goal without conflict with the secondary goal, should take on the value of 1 and node should now focus on its secondary goal as well. In cases where node is not able to achieve the primary goal at whichever power it is transmitting, it should then focus on improving its own estimated lifetime and therefore, the function then takes on the value of 1 for every selected. The following paragraphs present a formal definition of function .
Suppose at power level , node is able to help node reduce its potential transmission power, and its own estimated lifetime at power is larger than its reverselink neighborhood’s previous potential lifetime. Under such a circumstance, we refer to power level as a preferred power level of node . Note that, for a nodetopower mapping , there may exist several preferred power levels for node . We denote the set of preferred power levels for node under the nodetopower mapping as . For any power level , node ’s reverselink neighborhood’s potential lifetime is extended by node transmitting at power and node ’s lifetime at power level exceeds its previous reverselink neighborhood’s potential lifetime. Therefore, at a power level , the primary goal for node is met and node should focus on optimizing for its own estimated lifetime (the secondary goal) through the utility function. Therefore, we can conclude that in such a case.
If there exists no preferred power level at which can transmit to increase its primary utility function (indicated by ), then also node should focus on improving its own estimated lifetime through the utility function. In this case, ’s lifetime should be still be relevant in the local utility function and so, for any value of .
If neither of the above two cases is valid, then transmitting at power level causes a conflict between node ’s primary and secondary goals. In this case, node should focus on its primary goal only and therefore, .
Based on the above reasoning, is defined as:
IvC The ordinal potential game
We are now ready to describe the strategic game as having the following three components:

Player set : where is the number of nodes in the network.

Action set : is the space of all action vectors, where each component represents the set of available power levels at which may transmit.

Utility function set : For each player , utility function as given by Eqn. (IVB) which models the node’s preferences for its available power level choices. The vector of these utility functions is .
Theorem IV.1
The game is an ordinal potential game and its ordinal potential function is given by
(4) 
where is the binary connectivity function indicating whether the graph is connected with nodetopower mapping , i.e,

We will prove this by applying the definition of an ordinal potential game and proving that, at any time instant , the difference in individual utilities for each node from unilaterally changing its strategy and the difference in values of the global potential function have the same sign [9, 34]. Denote the mapping of nodes to power levels as follows: when is transmitting at power level as and when is transmitting at power level as . First, for the difference in an individual node’s utilities, we have:
Omitting for brevity, we can rewrite this equation as:
Note that, with ’s power level being either or , the power levels for the rest of the nodes within the network remain the same. Since at any time instant , are the only nodes whose potential lifetime may be affected by ’s power level, we can thus conclude that for node , .
Now, the difference in the values of the global potential function, , is:
Since this equation holds for any value of , we can omit to simplify the notation.
where is the smallest potential lifetime amongst node and its reverselink neighborhood when the nodetopower mapping is . Recall that and therefore:
(5) is similarly defined. and are also defined similarly as and , respectively. Since nodes within are the only nodes whose potential lifetime may be influenced by node ’s change in its transmission power, we can therefore conclude that .
Without loss of generality, we assume that , indicating that if , then . We can also conclude that . According to the definition of , if , then . The possible cases of and are (omitting for brevity):

Case 1:

Case 2:

Case 3:
In Cases 1 and 2, the network is not connected with or or both. In these cases, it is easy to prove that and have the same sign. We consider Case 3 in detail in the following.
In Case 3, the local graph within ’s range is connected whether ’s power level is or . Since all other nodes except ’s power levels remain the same at time , we can conclude that . This leads us to two situations: in one, , i.e, the full graph is not connected because of some node located outside of ’s range, and in the other, , i.e, the full graph is connected. In the case the graph is not connected, and, therefore, . Thus, we can conclude that and have the same sign. In the following, we now focus on the situation in which the full graph is connected.
The Case 3 situation in which the graph is connected, i.e., , can be further categorized into four subcases:

Subcase (3a):
, and . 
Subcase (3b):
, and . 
Subcase (3c):
, and . 
Subcase (3d):
, and .
Case (3a): In this case, whether ’s power level is or , the node with the smallest potential lifetime lies either within ’s reverselink neighborhood or is node itself. Since and , we can conclude that:
(6) (7) Since a node’s potential transmission power is no larger than its current transmission power, we can conclude that , and . Also, is at least one power level larger than and, therefore, . We conclude:
(8) Now, there are four subsubcases based on the values of and :

Case (3ai):

Case (3aii):

Case (3aiii): , and

Case (3aiv): , and
In the following, we consider each of the above subsubcases.
Case (3ai): According to the definition of , either both power levels and can help improve node ’s reverselink neighborhood’s potential lifetime or neither of them can. Therefore, we have . Note that, . Now, we can rewrite Eqn. (6) as
As for , since , and , we can rewrite Eqn. (7) as:
Therefore, we have and . Thus, as for Case (3ai), and share the same sign.
Case (3aii): Since , it indicates that node ’s reverselink neighborhood’s potential lifetime cannot be extended when node is transmitting at either power level or . Thus, we can conclude that . Following a similar line of deduction as in Case (3ai), we can conclude that .
As for , also following the same line of deduction as in Case (3ai), we have:
