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Abstract

Aimed at the limited energy supply and imperfect topological tolerance for underground wireless sensor networks, one energy-efficient random-walk scale-free topology model is proposed in this article, and a power network topology structure with adjustable rate index gets generated. At first, the network is divided into several clusters, and the cluster heads are selected with the use of random-walk strategy. During the growth of the network, with the introduction of preferred connection for scale-free network, together with considering both the node’s residual energy and the distance among nodes, nodes with larger residual energy present higher connectivity probability, so that the energy balance of the network gets realized. Simulation results show that the communication among the cluster heads selected by the proposed random-walk scale-free topology model presents not only the power-law characteristics of scale-free networks but also has better stability, higher fault tolerance, and it can still balance the energy consumption for nodes and the network and therefore can prolong the lifetime of the network.

Keywords

Underground wireless sensor networks topology evolution random walk scale-free network energy efficient

Introduction

The safety production of coal mine has always been a priority among all the priorities in coal mine work system, where the coal mine safety production monitoring system has always played an important role. First, in traditional safety monitoring systems, the installation and deployment of monitoring equipment usually adopt the mode of wire communication, which is complicated in wiring with fixed monitoring positions and locations and single network topology. It also demands for large amount of hand labor and is hard to satisfy dynamic changes in underground mine channels. Second, the traditional safety production monitoring control systems preserve low degree of intelligence and are not able to automatically adjust network structure according to the actual conditions, which means that it is apt to form the dead monitoring zone, resulting that the underground environmental information in coal mines cannot be mastered by the monitoring center located above the ground. Besides, in relatively harsh underground coal mine conditions, if the communication wires get damaged, the maintenance cost will be high and the production tasks may even get delayed. Therefore, traditional safety production monitoring system, which is based on wire communications, is hard to satisfy the demand for dynamic and all-around underground monitoring.

Wireless sensor networks (WSNs) are distributed networks composed of large low-cost, low-power, multi-functional sensor nodes that are small in size and communicate untethered in short distances. They have broad applications in surveillance and monitoring of the environment, collaborative processing of information, gathering scientific data from spatially distributed sources for environmental modeling and protection, hazardous gases detection, and so on.¹ There is a special application scenario for WSNs, the underground supervision with long-chain-type topology,² such as the underground utility tunnels supervision³ and the underground coal mine supervision.⁴ For this kind of WSNs, the wireless communication environments are supposed to be closed or semi-closed, and the communication channels are supposed to be the fading ones.^5–7 Taking the underground coal mine supervision system as an example, the communication environment in coal mine is becoming more and more complex since the underground pressure, the high-level water pressure, and the coal mine gases are the main threats for the coal mine safety mining.⁸

Topology control is the basis of WSNs’ network construction and communication, which directly affects the performance of network’s routing protocol, the time synchronization protocol, the localization protocol, and so on. The goal of topology control is to control the topology of the graph representing the communication links between network nodes with the purpose of maintaining some global graph property, such as the connectivity, while reducing the energy consumption and/or interference that are strictly related to the nodes within the transmitting range.⁹ Study on complex networks is a newly emerging subject that focuses on the networks which have non-trivial topological features. The complex networks are maps of multiple nodes connected by edges, and many things in our daily life can be described by the complex networks,¹⁰ such as the Internet,¹¹ the power grids,¹² and the World Wide Web.¹³ From the study by Jiang,¹⁴ if the node degree of nodes in the network obeys the power rate distribution, that is, $p (k) ~ C k^{- γ}$ , then the network is called scale-free network. Figure 1 shows the simulation diagram for a scale-free network with 100 nodes within the area of $100 \times 100 m^{2}$ , where the connection edges among network nodes are not uniform, some nodes with smaller degree, such as node 54, while some others with a larger degree, such as node 74.

Figure 1.

Communication topology for a scale-free network.

It is mentioned in the study by Jiang et al.¹⁵ that the typical complex network models, such as the small-world and scale-free network models, show some characteristics that are beneficial in WSNs. It is the common characteristics between WSNs and typical complex networks models that motivated us to optimize the topology in underground WSNs with the use of complex networks theory.

In recent years, energy-saving topology control algorithm of network and nodes has become a hot issue.¹⁶ Low-energy adaptive clustering hierarchy (LEACH) is well known because it is simple and efficient,¹⁷ the nodes using LEACH are divided into clusters, and the advantage of LEACH is that each node has the equal probability to be a cluster head, which makes the energy dissipation of each node be relatively balanced.¹⁸ To measure the communication cost in a cluster, Younis and Fahmy¹⁹ put forward the hybrid energy-efficient distributed (HEED) clustering approach and select the cluster head based on the residual energy.

It should be pointed that in some special cases, especially for the scenario in underground coal mines, the energy restriction of nodes is still a major problem since in many contexts it is hard to recharge the batteries or scavenge energy from the environment. Therefore, in order to ensure the stability of the network, it is an important task to design the topology with low-energy consumption and long life cycle.²⁰ Motivated by Xiangning and Yulin¹⁸ and Younis and Fahmy,¹⁹ considering the energy restrictions of nodes in underground WSNs, with the introduction of random walking mechanism, in this article, we put forward an energy-efficient random-walk scale-free topology model (RSTM), where the cluster heads are selected with the use of HEED algorithm, and the probability of a new node to the next node is determined by the residual energy and the distance between the two.

The organization of this article is as follows: section “Energy consumption model” introduces the energy consumption model for nodes in underground WSNs; the evolution of RSTM and the degree distribution features are analyzed in sections “The evolution of RSTM” and “Degree distribution analysis,” respectively; and the simulation results and the conclusion remarks are given in sections “Simulations and analysis” and “Conclusion,” respectively.

Energy consumption model

According to Liu et al.,²¹ the lifetime of the WSNs is determined by the survival time of the first failed node, and the lifetime of network is closely related to the network energy consumption. In this section, through the establishment of model of node’s energy consumption, we try to find the relationship among the network energy consumption, the network residual energy, and the network degree and then apply this relationship to the priority connection selection in the scale-free network point of view.

In the WSNs, the energy of a node is mainly consumed by the data sending and data receiving. For node i, the energy needed to send l bits data is shown as

$E_{iT} = E_{elec} \times l \times k_{i} + ε \times l \times d^{σ}$ (1)

while the energy needed to receive l bits data is

$E_{iR} = E_{elec} \times l \times k_{i}$ (2)

where l is the length of data to be received or sent, $k_{i}$ is the degree of node i, d is the distance between the sender and receiver, and $E_{elec}$ is the energy consumed for sending and receiving per unit bit data. It can be seen from equations (1) and (2) that the energy consumption to send per bit data is greater than to receive per bit data, and the extra energy is caused by the amplification of sending data done by the amplifier. Moreover, the energy consumption of the amplifier is determined by the parameter $ε$ , the distance d between nodes, and the path loss index $σ$ . The values of $ε$ and $σ$ further depended on the distance d, when d is less than the distance threshold $d_{0}$ defined as

$d_{0} = \sqrt{\frac{ε_{fs}}{ε_{amp}}}$ (3)

then $σ = 2$ and $ε = ε_{fs}$ ; otherwise if $d < d_{0}$ , then $σ = 4$ and $ε = ε_{amp}$ . Where $ε_{fs}$ is the power amplification coefficient for the free space attenuation channel model, and $ε_{amp}$ is the power amplification coefficient for the multi-path attenuation channel model.

For any node i in the WSNs, the energy consumption is the sum of the energy consumption for sending data and receiving data. Therefore, the average energy consumption for node i is

$\begin{matrix} E_{i} (t) = 2 E_{elec} \times l \times k_{i} + ε \times l \times d^{σ} \\ = {\begin{matrix} 2 E_{elec} \times l \times k_{i} + ε_{fs} \times l \times d^{2}, d < d_{0} \\ 2 E_{elec} \times l \times k_{i} + ε_{amp} \times l \times d^{4}, d ⩾ d_{0} \end{matrix} \end{matrix}$ (4)

Then, the energy consumption of the network is

$E_{itotal} = \sum_{i = 1}^{M} (2 E_{elec} \times l \times k_{i} + ε \times l \times d^{σ})$ (5)

where M is the number of nodes in the network.

From equation (5), when $E_{elec}$ and l are all known, the value of $E_{itotal}$ is only related to $k_{i}$ and d. Thus, the energy consumption model of node i can be defined as the ratio of the energy consumption, denoted as $E_{i} (t)$ , to its residual energy $E_{i}$ . Therefore, the energy consumption model for node i can be of the following form

$E_{s} (i) = \frac{E_{i} (t)}{E_{i}}$ (6)

Then, the energy consumption model for the network can be

$E_{s} = \frac{E_{itotal}}{{\bar{E}}_{s}}$ (7)

where ${\bar{E}}_{s}$ is the average residual energy of the network and is given as ${\bar{E}}_{s} = \sum_{i = 1}^{M} E_{i} / M$ .

In this article, we use $1 / E_{s}$ to evaluate the network lifetime $τ_{N}$ , which is shown as

$τ_{N} = \frac{1}{E_{s}} = \frac{{\bar{E}}_{s}}{E_{itotal}} = \frac{\sum_{i = 1}^{M} \frac{E_{i}}{M}}{\sum_{i = 1}^{M} 2 E_{elec} \times l \times k_{i} + ε \times l \times d^{σ}}$ (8)

From equation (8), we can see that when we construct the network topology, the nodes’ residual energy and the distance between nodes together with the node’s degree determine the lifetime of the network. The greater the distance between the nodes $d_{i}$ , the larger the node’s degree $k_{i}$ and the higher the node’s energy consumption $E_{i}$ , thus the less the average remaining energy ${\bar{E}}_{s}$ , the shorter the lifetime $τ_{N}$ .

The evolution of RSTM

As the scale of underground WSNs are relatively large and the topology changes dynamically due to new nodes joining and old nodes dying, one layered network structure should be used for the WSNs. In this article, nodes in the WSNs are divided into multiple clusters according to the geographical positions or the density of nodes. Each cluster is composed of one cluster head node and many cluster members. The heads of lower clusters are the members of higher clusters. Cluster head is responsible for data forwarding, data fusion, and routing selection, while members in clusters are only responsible for data collection and communication with their cluster heads.

In this article, we adopt the cluster head generation algorithm in HEED protocol proposed in the works of Xiangning and Yulin¹⁸ and Younis and Fahmy,¹⁹ and to make the cluster heads uniformly distributed in the WSNs, and we mainly consider evolutionary process of topology between WSNs’ clusters in layered structure and generative process of our network model, and schematic diagram of nodes distribution and cluster heads of a WSN with 100 nodes in a $100 m \times 100 m$ area is shown in Figure 2, where the communication range R is set as 30 m, the green circle indicates the sink node in the underground WSNs, red circles and black circles indicate cluster heads and the nodes, respectively; moreover, dotted blue lines indicate the communication edges among cluster heads and other nodes.

Figure 2.

Distribution of cluster head nodes in HEED protocol.

The evolution of RSTM is shown as follows:

Cluster distribution: with the use of HEED approach, all the nodes in the WSNs are classified into clusters and then the degree distributions of nodes are derived.

Initial network: it is formed by the base stations, the adjacent $m_{0}$ initial nodes, and the $e_{0}$ lines, where the base stations and the initial nodes are connected in pairs and all nodes are distributed uniformly in the distribution area.

Starting node: it is randomly chosen among the cluster head nodes during the random-walk process.

Node growth: for each time step, one new cluster head is added to the network and gets connected with m existing nodes, where $m ⩽ m_{0}$ .

Priority connection: during each walk, the connectivity probability between the adjacent node i of the cluster head node j and the random-walk node is dependent on its residual energy $E_{i}$ and the distance $d_{i}$ between these two nodes

$Π (i) = Π (E, d) = \frac{E_{i}}{\sum_{k_{j}} E_{k_{j}}} + \frac{d_{i}}{\sum_{k_{j}} d_{k_{j}}}$ (8)

and node i will get marked after the connection, and such procedure is repeated until there are $m ⩽ m_{0}$ different nodes marked.

If the walk node gets repeatedly connected with one certain marked cluster head, then a new round of random walk will be started from this cluster head.

The new cluster head will become connected with m marked cluster head and the process will get repeated until all nodes in WSNs are mutually connected.

Degree distribution analysis

Suppose there are N cluster heads in the WSNs to be considered. The probability for cluster head j chosen as the starting point is $P (j) = 1 / N$ , and the probability of this cluster head j walking to its adjacent cluster head i is with the following form

$P (i) = P (E, d) = \frac{p (i / j) p (j)}{p (j / i)}$ (9)

where $P (i / j)$ is the probability of random-walk node j to connect with its adjacent cluster head i after j has been taken as the starting point. Hence, we have

$P_{E} (i / j) = Π (E) = \frac{E_{i}}{\sum_{k_{j}} E_{k_{j}}}$ (10)

and

$P_{d} (i / j) = Π (d) = \frac{d_{i}}{\sum_{k_{j}} d_{k_{j}}}$ (11)

In equations (10) and (11), $E_{i}$ refers to current residual energy of node i, $d_{i}$ refers to the distance between node i and j, $k_{j}$ refers to number of adjacent nodes of cluster head j, and

$\sum_{k_{j}} E_{k_{j}} = k_{j} < E_{k_{j}} >, \sum_{k_{j}} d_{k_{j}} = k_{j} < d_{k_{j}} >$ (11)

where $< E_{k_{j}} >$ is the mean value of residual energy of all adjacent nodes of node j; $< d_{k_{j}} >$ is the mean value of distances between node j and its adjacent nodes.

Similarly, $P (i / j)$ can be expressed as

$P_{E} (\frac{j}{i}) = \frac{E_{j}}{\sum_{k_{i}} E_{k_{i}}}, P_{d} (\frac{j}{i}) = \frac{d_{j}}{\sum_{k_{i}} d_{k_{i}}}$ (11)

where

$\sum_{k_{i}} E_{k_{i}} = k_{i} < E_{k_{i}} >, \sum_{k_{i}} d_{k_{i}} = k_{i} < d_{k_{i}} >$ (11)

Summing up equations (9)–(11), we can get

$\begin{matrix} P (E) = \frac{P_{E} (\frac{i}{j}) P (j)}{P_{E} (\frac{j}{i})} = \frac{\frac{E_{i}}{\sum_{k_{j}} E_{k_{j}}} \times \frac{1}{N}}{\frac{E_{j}}{\sum_{k_{i}} E_{k_{i}}}} = \frac{\frac{E_{i}}{k_{j} < E_{k_{j}} >} \times \frac{1}{N}}{\frac{E_{j}}{k_{i} < E_{k_{i}} >}} \\ = \frac{E_{i} k_{i} < E_{k_{i}} >}{E_{j} k_{j} < E_{k_{j}} >} \times \frac{1}{N} = \frac{E_{i} < E_{k_{i}} >}{E_{j} < E_{k_{j}} >} \times \frac{k_{i}}{N k_{j}} \end{matrix}$ (12)

and

$\begin{matrix} P (d) = \frac{P_{d} (i / j) P (j)}{P_{d} (j / i)} = \frac{\frac{d_{i}}{\sum_{k_{j}} d_{k_{j}}} \times \frac{1}{N}}{\frac{d_{j}}{\sum_{k_{i}} d_{k_{i}}}} = \frac{\frac{d_{i}}{k_{j} < d_{k_{j}} >} \times \frac{1}{N}}{\frac{d_{j}}{k_{i} < d_{k_{i}} >}} \\ = \frac{d_{i} k_{i} < d_{k_{i}} >}{d_{j} k_{j} < d_{k_{j}} >} \times \frac{1}{N} = \frac{d_{i} < d_{k_{i}} >}{d_{j} < d_{k_{j}} >} \times \frac{k_{i}}{N k_{j}} \end{matrix}$ (13)

where $N k_{j} = N < k_{i} > = \sum_{j \in Λ} k_{j}$ .

Denoting $f (E) = E_{i} < E_{k_{i}} > / E_{j} < E_{k_{j}} >$ , $f (d) = d_{i} < d_{k_{i}} > / d_{j} < d_{k_{j}} >$ , $f (E, d) = f (E) + f (d)$ , and $P (E, d) = P (E) + P (d)$ , it can be drawn

$P (i) = P (E, d) = f (E, d) \times \frac{k_{i}}{\sum_{j \in Λ} k_{j}}$ (14)

Suppose that the new node n enters WSNs at time t, $R_{n}$ is denoted as the communication radius of the new node n; $R_{0}$ and $R_{t}$ as the network communication radius at initial time $t = 0$ and time t, respectively, then the improved prioritized growth process of the network can be shown as Figure 3.

Figure 3.

The join of new nodes to the WSNs.

From what we have stated above, based on the continuum theory,²² we can know that if node $k_{i}$ changes continuously, the rate of its change can be of the following form

$\frac{\partial k_{i}}{\partial t} = mP (i) = mf (E, d) \times \frac{k_{i}}{\sum_{j \in Λ} k_{j}}$ (15)

Since all nodes in the network are uniformly distributed in the monitoring zone, the probability of new node n being adjacent to the cluster head is similar to the ratio of the communication range of node n to the area of the whole network, that is, $R_{n}^{2} / 2 R_{t}^{2}$ , then, it can be calculated that

$\begin{matrix} P (i) = f (E, d) \times \frac{k_{i}}{\sum_{j \in Λ} k_{j}} \\ = f (E, d) \times \frac{R_{n}^{2}}{2 R_{t}^{2}} \frac{k_{i}}{N_{t} \frac{R_{n}^{2}}{2 R_{t}^{2}} < k >_{t}} \end{matrix}$ (16)

where $N_{i}$ is the number of node i in the WSNs at time t, $R_{n}^{2} / 2 R_{t}^{2}$ is the size of adjacent node set $Λ$ of new node n at time t, and $< k >_{t}$ is the degree of uniformity of nodes in the whole network at time t.

Motivated by Chen et al.,²³ by virtue of the improved prioritized growth theory, the topology of a network with $m_{0} + t$ nodes and mt edges can be generated for every period t. Therefore, we can derive $N_{t} = m_{0} + t$ , $< k_{t} > = 2 mt / m_{0} + t$ . Equation (16) can be simplified into

$P (i) = f (E, d) \times \frac{k_{i}}{N_{t} < k >_{t}} = f (E, d) \times \frac{k_{i}}{2 mt}$ (17)

By substituting equation (16) into equation (15), one can get

$\frac{\partial k_{i}}{\partial t} = f (E, d) \times \frac{k_{i}}{2 t}$ (18)

and

$k_{i} = C t^{\frac{1}{2} f (E, d)}$ (19)

where C in equation (19) is an arbitrarily chosen constant.

By setting $t_{i}$ as the time when a new node i enters the network, together with the initial condition set as $k_{i} (t_{i}) = m$ , the differential equation (15) can get solved, and the expression of $k_{i} (t)$ has the following form

$k_{i} (t) = m {(\frac{t}{t_{i}})}^{\frac{1}{2} f (E, d)}$ (20)

Therefore, $p (k_{i} (t) < k)$ can be expressed by

$p (k_{i} (t) < k) = p (\frac{m^{\frac{2}{f (E, d)}} t}{k^{\frac{2}{f (E, d)}}} < t_{i})$ (21)

During the improved prioritized growth process, for each time interval, there is one new cluster head added to the network, which means that the entering of new cluster heads to the network is totally uncertain and arbitrary. Hence, $t_{i}$ is uniformly distributed in the range of $(0, t)$ , and the probability density function can be expressed by $p (t_{i})$ , shown as

$p (t_{i}) = \frac{1}{m_{0} + t}$ (22)

Combining equations (21) and (22), one can get

$p (k_{i} (t) < k) = 1 - p (t_{i} < \frac{m^{\frac{2}{f (E, d)}} t}{k^{\frac{2}{f (E, d)}}}) = 1 - \frac{t}{m_{0} + t} {(\frac{m}{k})}^{\frac{2}{f (E, d)}}$ (23)

Then, the degree distribution of the network is

$p (k) = \frac{\partial p (k_{i} (t) < k)}{\partial k} = \frac{t}{m_{0} + t} \frac{2}{f (E, d)} \frac{m^{\frac{2}{f (E, d)}}}{k^{1 + \frac{2}{f (E, d)}}}$ (24)

then, as $t \to \infty$ , equation (24) can be simplified into

$\begin{matrix} p (k) = \frac{\partial p (k_{i} (t) < k)}{\partial k} \\ = \frac{2}{f (E, d)} \frac{m^{\frac{2}{f (E, d)}}}{k^{1 + \frac{2}{f (E, d)}}} \propto k^{- (1 + \frac{2}{f (E, d)})} \end{matrix}$ (25)

From equation (25), we can know that the exponent of RSTM model is $1 + 2 / f (E, d)$ , and thus, the distribution of network’s topology degree follows the power-law characteristic, which means that the proposed topology model is a scale-free network.

Simulations and analysis

In this section, we will carry out the simulations on the degree distribution, the node’s degree, and the residual energy. All the figures are drawn based on the mean of 100 runs. At each run, the simulation parameters are set as shown in Table 1.

Table 1.

Simulation parameters.

Parameter	Value
Number of nodes, N	100
Number of initial nodes, $m_{0}$	5
Newly added edges, m	3
Size of network A, $(m^{2})$	100 × 100
Transmission range, $R_{\max} (m)$	50
Initial energy, $e (J)$	2
Data fusion energy consumption, $E_{elec} (nJ \cdot bi t^{- 1})$	50
Power consumption of amplifier, $ε_{amp} (pJ \cdot bi t^{- 1} \cdot m^{2})$	100
Number of packet per unittime, $l (bit)$	100

Simulation on the degree distribution

As a key characteristic of the network topology, the degree distribution relates directly to the safety and the stability of network. In section “The evolution of RSTM” of this article, the topology degree distribution formation in RSTM model has been analyzed using the continuum theory. When $f (E, d) = 1$ , the degree distribution of the proposed RSTM model is similar to that of the Barabasi–Albert (BA) model proposed in the study by Barabasi and Albert.²⁴

In Figure 4, the straight line refers to the theoretical values of degree distribution, and the scattering points indicate the actual values. It can be seen from Figure 4 that the degree distribution of the proposed RSTM model follows the power-law characteristics with the power exponent being 3, and the bigger the degree of nodes in the network, the smaller the probability $p (k)$ , which is well suited with the characteristics of the scale-free network. Therefore, the proposed RSTM model has good fault tolerance to the random failure of nodes in the WSNs.

Figure 4.

Network average degree distribution when $f (E, d) = 1$ .

Figure 5 is the topology degree distribution of RSTM model when $f (E, d) = 1$ and $f (E, d) = 0.5$ . According to theoretical analysis stated above, we know that when $f (E, d) < 1$ , the degree distribution among clusters in the WSNs follows the power-law distribution with the exponent being $1 + 2 / f (E, d) > 3$ . From Figure 5, we can see that the sparsity of the network will get decreased when the value of $f (E, d)$ declines.

Figure 5.

Network average degree distribution when $f (E, d) = 1$ and $f (E, d) = 0.5$ .

Figure 6 shows the theoretical network degree distributions when $f (E, d) = 1$ , $f (E, d) = 0.5$ , $f (E, d) = 2$ , and $f (E, d) = 4$ . From Figure 6, we can know that the WSNs’ topology degree distribution $p (k)$ is affected by the value of $f (E, d)$ . $p (k)$ increases as $f (E, d)$ increases, and the sparsity of the WSNs topology gets influenced; moreover, the larger the $f (E, d)$ , the larger the probability of nodes being dominated by nodes with larger degrees, and thus the more irregular the network topology will be, the value of $f (E, d)$ should be adjusted to realize the adjustment of network degree distribution so as to reach the demands of network characteristics.

Figure 6.

Theoretical network degree distributions when $f (E, d) = 1$ , $f (E, d) = 0.5$ , $f (E, d) = 2$ , and $f (E, d) = 4$ , respectively.

The evolution of network’s average degree and average residual energy

Figure 7 shows the relationships between the node degree k and the residual energy of nodes, constructed by the BA model, the RSTM model when $f (E, d) = 0.5$ , and the RSTM model when $f (E, d) = 4$ , respectively. It can be seen from Figure 7 that the model constructed by RSTM model $f (E, d) = 0.5$ and $f (E, d) = 4$ has more node degree, meanwhile the corresponding residual energy is larger, while the BA model does not preserve this characteristic. This is due to the fact that for the BA model it evolutes the topology only considering the node degrees, but for the RSTM model, it considers not only the node degree but also the residual energy and makes those nodes with more residual energy the large probability to be connected with other nodes. The proposed RSTM model can avoid this situation: node with large degree fails fast due to the running out of energy, which means that the proposed RSTM model can balance the energy consumption and thus can prolong the network lifetime.

Figure 7.

Theoretical degree distribution.

Conclusion

Aimed at the application of WSNs to the underground coal mines, taking into account the limited energy supply for nodes, in this article, we have proposed one energy-efficient topology control method RSTM based on the complex networks theory and the random-walk strategy. During the topology initial phase, the network is classified into several clusters, and the cluster heads are selected with the use of the random-walk strategy. When it comes to evolutionary stage, for any new node to join the network, the probability to connect it with other existing node is jointly determined by the existing node’s residual energy and the distance between them, so as to avoid the failure of node due to the run out of energy. Simulation results show that proposed RSTM model conforms to the basic characteristics of scale-free networks, and it not only has strong tolerance to the random failure but also can prolong the network lifetime by balancing the node’s residual energy.

As a preliminary study, this article just describes the general process of RSTM model and has not analyzed it systematically, which is the main task for our future works.

Footnotes

The authors would like to thank the reviewers for their insightful comments that have greatly improved the current presentation of the paper.

Handling Editor: Pengpeng Chen

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: This work was supported by National Natural Science Foundation (NNSF) of China under Grant nos 51274011,51874205,and 61472003;Jiangsu Natural Science Research Major Projects in Colleges and Universities under Grant no. 17KJA520005;and Suzhou University of Science and Technology Talents Research Funding Project under Grant nos 331711602 and 491811104.

ORCID iD

Zhenping Chen

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