Vol.28 No.3 JOURNAL OF ELECTRONICS (CHINA) May 2011

RESEARCH ON KEY NODES OF WIRELESS SENSOR NETWORK BASED ON COMPLEX NETWORK THEORY1

Ma Chuang Liu Hongwei Zuo Decheng Wu Zhibo Yang Xiaozong (School of Computer science and technology, Harbin Institute of Technology, Harbin 150001, China)

Abstract On the basis of complex network theory, the issues of key nodes in Wireless Sensor Networks (WSN) are discussed. A model expression of sub-network fault in WSN is given at first; subsequently, the concepts of average path length and clustering coefficient are introduced. Based on the two concepts, a novel attribute description of key nodes related to sub-networks is proposed. Moreover, in terms of node deployment density and transmission range, the concept of single-point key nodes and generalized key nodes of WSN are defined, and their decision theorems are investigated.

Key words Wireless Sensor Network (WSN); Key nodes; Fault model; Complex network theory

CLC index TP393

DOI 10.1007/s11767-011-0611-z

I. Introduction Wireless Sensor Network (WSN) research has

become a hot spot in network and communication field. Based on wireless communication technology and sensor technology, WSN is serving as an im-portant tool in monitoring and controlling appli-cations. A wide array of applications ranging from household, public health, environment to military affairs and defence make use of sensor nodes to collect data[1].

However, some problems need to be solved in WSN network applications. One of the most serious problems is the nonhomogeneity of nodes deploy-ment. In application environment, nodes often are deployed symmetrically: same types of nodes are set up in the application scenes as symmetrically as possible. But in real transmission, some nodes will be different from the others in communication because of special network topology and routing algorithm. For example, the election of cluster head nodes makes the cluster head nodes take on more

1

Manuscript received date: December 7, 2010; revised date: April 14, 2011. Supported by the National High Technology Research and Development Program of China (No. 2008AA01A201) and the National Natural Science Foundation of China (No. 60503015). Communication author: Ma Chuang, born in 1974, male, Ph.D. candidate. School of Computer Science and Tech-nology, Harbin Institute of Technology, Harbin 150001, China. Email: mc@ftcl.hit.edu.cn.

communication task. This imbalanced status of nodes in the application of WSN gives rise to key nodes in location and logic.

Complex network science can accurately and succinctly describe the WSN using mathematical language, symbols and theorems of graph theory, statistical physics and many other theoretical on the basis of modern science. However, there is not much literature of WSN based on complex network science. Moreover, key nodes issues research of WSN can not be found in literatures based on complex network science. Therefore, in this paper, on the basis of complex network science, an ab-stract description of the relation graph of WSN is presented. It incorporates the influence of the number of edges, and neglects actual physical distance between two nodes. Subsequently, aiming at the problem of fault related characteristics in wireless sensor networks, the key nodes problems using complex network theory are proposed and analyzed. Finally, based on complex network theory, characteristics of wireless nodes are discussed, and the related definitions and theorems are given.

II. Related Works Typically, a sensor networks consists of a com-

mand node or base station and sensors relay streams of data to a command node either peri-odically or upon events[2]. To describe the model system of the wireless sensor network, a simplified network topology with only one data communica-tion path is shown in Fig. 1:

MA et al. Research on Key Nodes of Wireless Sensor Network Based on Complex Network Theory 397

Fig. 1 WSN topology model

According to the WSN topology above, a ran-dom graph ( , )G V E can be used to describe its nodes and link model[3]. G is the set of nodes, and E is the set of communication links between nodes, assuming that there is one communication link at most between a pair of nodes. So, a set of wireless network nodes (point set) is expressed as below:

{ }1 2, , ..., nV v v v= (1)

And the set of communication links (edge set) is expressed as below:

( ){ }, ,i j i jE v v v V v V= ∈ ∈ (2)

If there is only one path between any two points in the graph, then G is defined as single- connected graph; if there are k paths between any two points, then G is defined as conn- ectedk graph[4].

III. Fault Model of Key Nodes The WSN is featured by large amounts of sensor

nodes[5]. Generally, the node communication dis-tance in WSN is relatively short, and in a specific network environment, a node can only communi-cate and exchange information with the nodes (called neighbor nodes or neighbors) in its com-munication range limit. If the node wants to access the nodes outside its communication range limit, the multi-hop routing approach must be used. Thus, an ordinary node plays multiple roles in: they are not only the information collectors and senders, but also the information recipients and re-transmitters. Typically, the collected data can reach the sink node (or gateway) by way of multi-hop routing[3]. Sink node is a special node with enough energy and sophisticated functions, which can communicate with the information monitoring center (i.e., Base Station) on ground via Internet, mobile commu-

nication networks, satellites, and other communi-cation networks. Additionally an air-borne base station can communicate with the sink node by wireless communications to collect the data. In order to ensure that most of the nodes in the network can establish wireless communication links with the sink node, and to ensure that the sensor nodes communication can cover the target region without any blind spot, the distribution of the nodes must be sufficiently dense. Moreover, the high density distribution of the nodes can also help to maintain the overall connectivity and function-ing of the network in case of energy exhaustion or faults in certain nodes. However, in the following six application scenarios, high density node de-ployment can not be achieved in the target area.

(1) Deployment in three-dimensional envi-ronment: when the flight deployment approach is used for the target area characterized by three- dimensional (i.e., nonplanar) terrain, the vertical drop of nodes in the target area will lead to the spatial sparseness of nodes;

(2) Real-time transmission requirement: for certain networks with a high requirement for real- time transmission, the unduly high density node deployment will cause an increase in the average hops of network transmission and consequently the overall transmission delay and hence affect efficacy of the network structure;

(3) Deployment in the enemy’s military zone: the greater the density of node deployment, the higher the possibility of being spotted and de-stroyed by the enemy;

(4) Impermissible weather conditions: when the nodes are sprayed in flight deployment, high- speed wind can make some nodes miss the targeted area, resulting in the sparseness of node deploy-ment;

(5) Network with mobile nodes: in a network with mobile nodes, the movement of nodes will change the topology of the WSN, the nodes dis-tribution in the network, which may lead to local sparseness of nodes in some areas;

(6) Special types of sensors: in networks of some special sensors, such as in a precision meas-urement sensor wireless network, high density node deployment cannot be adopted due to the high cost of the sensors.

398 JOURNAL OF ELECTRONICS (CHINA), Vol.28 No.3, May 2011

From the above analysis on node deployment density in WSNs, node sparseness is inevitable in some applications. Node sparseness is the under-lying reason that the health of certain nodes deci-sively affects the transmission effectiveness of the network.

Thus, the failure of key nodes is caused by uneven seeding or sparse deployment of nodes in WSN. Moreover, the occurrence of fault will lead to the failure of associated sub-network communica-tion. Sub-network of WSN is a special deployment region according to the difference of geographic location, the variation of distribution dimensions or artificial division aiming at the type of application and the distance from base station. Definition 1 Single-point-type key nodes A kind of bridge nodes, which are isolated from two sub- networks, serves as unique communication paths between current two sub-networks. Moreover, there is only one active node (the current node) between two sub-networks as participative communication connection node. A key node is shown in Fig. 2.

Fig. 2 The key node is isolated from sub-network in WSN

Once the fault occurs in the key nodes, the communication links will break and the transmis-sion will stop. Moreover, if the key nodes can not be repaired or maintained in specific time scope, the communication failure will be permanent failure and the message will be lost. The fault model of sub-network caused by key nodes failure can be expressed as:

( ){ }fault( , ) , , , ,i m i m i j jG V E v e v V e E v v v V= ∈ ∈ ≠ ∈

(3)

In the equation above, V and E are the nodes set and links set in failure status; faultV is the set of nodes in which fault occurs.

Obviously, key nodes are very important in WSN applications, and their failures may lead to the local or global failure in sensor networks. As a

result, they should be focused on when the fault-tolerant abilities of nodes and network are considered and designed. This can be implemented by increasing the energy of nodes or redundancy of critical nodes and components to enhance fault tolerance performance and improve network reli-ability.

IV. Probability Analysis of Key Nodes 1. Existence condition

In complex network theory, the distance ijd is defined as the sum of edges on the shortest path between node i and node .j L represents the av-erage path length of a WSN, and is defined as the average distance between all communication node pairs. It can be calculated in Eq. (4). N is the number of nodes in a certain network environ-ment[6,7].

2( 1) ij

i j

L dN N >

=− ∑ (4)

The average path length describes the spacing rate in nodes deployment. In the sub-network fault model of adjacent regions, a larger the value of L means more sparse node deployment, and the higher formation possibility of key nodes. Therefore, the L value can be used to estimate the fault gen-eration probability of adjacent regions in the sub- network. When the nodes of sub-networks are de-signed and deployed, the L value should be kept small to avoid the generation of key nodes and reduce the probability of network failures.

2. Judgement theorem

The key nodes can be described quantitatively through clustering coefficient of complex network theory[6,7]. iC represents the clustering coefficient of node i in WSN, and it is defined by the mutual connection probability of the ik nodes connected with the Node ,i as shown in Eq. (5).

( )2

, 1, 2, ,1

ii

i i

EC i N

k k= =

− (5)

In this equation, iE in the numerator is actual mutual edges of Node i ’s neighbor nodes, and the denominator is the maximum possible number of edges in current network. Theorem 1 In the network topology graph of

MA et al. Research on Key Nodes of Wireless Sensor Network Based on Complex Network Theory 399

WSN, if Node i is the only connective node be-tween two independent adjacent sub-networks and the clustering coefficient of Node i satisfies the condition 0,iC = then the Node i is a single- point-type key node between the two adjacent sub- networks. Proof As shown in Fig. 2, according to the defi-nition of the clustering coefficient, condition iC = 0 indicates that all the neighbor nodes of Node i have no communication link with each other except the links with Node .i There is no another edges across the two independent adjacent sub-networks except the edges involving Node .i So, according to the definition of single-point-type key nodes, Node i must be the key node. Q.E.D.

Average path length and clustering coefficient are two important parameters of complex network theory related directly to the key nodes of the sub- network system of WSN. Moreover, they are es-sential indicators to describe the features of small- world network[8]. Small-world network characteris-tic is an expression to describe heterogeneous WSNs based on complex network theory. Therefore, the research of key nodes is closely related to the small-world network theory. There are two research works should be emphasized: firstly, the average path length and clustering coefficient of small world network should be studied precisely; secondly, an-other attributes of small world network should be introduced to analyze more profound problems about key nodes theory. These studies need to be done in the future. Definition 2 Key path of WSN It is a commu-nication bridge path comprised of nodes from dif-ferent and disjoint sub-networks in edge set of WSN, and is the only path between the two disjoint sub- networks, as shown in Fig. 3.

Fig. 3 Key path in wireless sensor networks

Definition 3 Link-type key nodes They are a type of effective communication participation

nodes in the key path of WSN. They are the only communication link nodes in the current link be-tween two connective sub-networks except the two end points of two sub-networks. Theorem 2 The nodes isolated from sub-networks are link type key nodes only if they are on the key path. Proof It is easy to prove by Definition 3, and the steps of proof are omitted.

3. Network connectivity critical condition

(1) The critical condition of deployment density

Suppose that in a WSN of N nodes, the nodes are randomly distributed in a large enough area according to uniform distribution rule. The dis-tribution density is denoted by ,λ the transmission radius of each node is denoted by ,R and one-hop scope distance between the Node M and Node N is denoted by .MNγ Then the transmission cov-erage area of each node can be calculated as 2.Rπ Moreover, the average node number of single node transmission range can be calculated as N =

2.Rλπ The conditional probability density function should meet the Eq. (6), based on ,γ neighbour node distance between any two nodes[9,10]:

2( ) 2 / , 0g R Rγ γ γ= ≤ ≤ (6)

(2) The critical transmission range of nodes Critical Transmission Range (CTR) is the

minimum transmission range with the assumption of each node being under the same conditions, when k -connectivity is maintained in the network. In other words, when there is a Longest Link (LL) in the Minimum Spanning Tree (MST) of an e-nodn network, the LL is the critical transmission range in connectivity feature[11]. Prim-Dijkstra algorithm can be used to calculate the MST, and subse-quently the LL can be calculated based on MST. The value of LL will be the critical transmission range in certain scene. Assuming the transmission radius of each node is ,sr and the node density per unit area is ,ρ if the probability of there not being isolated nodes in network is ,p then the transmis-sion radius of each node should meets the following condition[12]:

( ) ( )( )1/21/ln 1 /n

sr p ρπ≥ − − (7)

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Definition 4 Critical connectivity probability In WSN, if critical density of the network topology meets the condition specified in Eq. (6), and the critical transmission conditions meets the condition given in Eq. (7), then this mode is called the critical connectivity. The conditional probability of this mode is defined as the critical connectivity prob-ability, denoted by .α

In other words, it is valid connectivity in time domain only if the current connection probability is greater than the critical probability of certain homogeneous network; in contrast, it is invalid if the current connection probability is less than .α Inference 1 In the network graph of wireless sensor network, an intermediate node connecting two sub-networks is the key node between the two current adjacent sub-networks, if the node clus-tering coefficient meets the condition 0 .iC α< < Proof From the definition of the critical connec-tivity probability, when 0 iC α< < is given, the network connections of related nodes is invalid within the given time domain, obviously, the con-nection number is 0. From Theorem 1, there is no other effective connectivity across two sub-net-works except the connection related to Node .i As a result, from the definition of single-point-type key node, the Node i must be the key node. Q.E.D.

V. Conclusion Problems persist in some fields of WSN research,

especially, in the energy supply, computing abilities, and maintenance abilities. As a result, a great number of faults are expected to occur in real ap-plications. In the logic topology graph and appli-cation scene, key nodes take on more important tasks in the transmission process because of the particularity of the network routing algorithm or other factors. The existence of key nodes will lead to global transmission failure when irrecoverable failures occur in the backbone key nodes, and it will become a bottleneck between high density de-ployment and low efficiency communication. Therefore, it is very important to understand the generation, identification, and configuration prob-lems so that effective fault-tolerant mechanism in key nodes field can be implemented to reduce failure probability of network.

In this letter, a fault model of the wireless

sensor sub-networks is proposed at first; secondly, the concept and definition are proposed for two types of key nodes: single-point-type and link-type. Based on complex network theory, the concept of the average path length and clustering coefficient are introduced to come up with the theorems of key nodes; thirdly, the decision theorems is proposed to separate the key nodes from wireless sensor sub- network clouds. This research is expected to be a fundamental basis of advanced researches on key nodes problem and enhancement of fault tolerance abilities of key nodes in WSN. Especially, the fu-ture work will be focused on the generalized deci-sion theorem of key nodes. Additionally, it will have an effect on research in fault, QoS, and other issues, such as reliability, fault detection, fault recovery, and fault tolerance research.

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