A Scale-Free Topology Construction Model for Wireless Sensor Networks

A local-area and energy-efficient (LAEE) evolution model for wireless sensor networks is proposed. The process of topology evolution is divided into two phases. In the first phase, nodes are distributed randomly in a fixed region. In the second phase, according to the spatial structure of wireless sensor networks, topology evolution starts from a network of very small size which contains the sink, grows with an energy-efficient preferential attachment rule in the new node's local-area, and stops until all nodes are connected into network. Both analysis and simulation results show that the degree distribution of LAEE follows the power law. The comparison shows that this topology construction model has better tolerance against energy depletion or random failure than other nonscale-free WSN topologies.


Introduction
Wireless sensor networks (WSNs) are a kind of self-organized distributed wireless networks composed of a large quantity of energy-limited nodes. Topology construction is one of the primary challenges in WSNs for ensuring network connectivity and coverage, increasing the efficiency of media access control protocols and routing protocols, improving the routing efficiency, extending the network lifetimes, and enhancing the robustness of the network [1][2][3][4][5]. The main aim of topology construction is to build a topology to connect network nodes based on a desired topological property. A dense network topology leads to high energy consumption due to overlapped sensing areas and maintenance costs of topology, while a very sparse network topology is vulnerable to network connectivity [6].
The development of complex networks provides new ideas for topology construction of WSNs. Study on complex networks is a newly emerging subject that focuses on the networks which have nontrivial topological features [7,8]. There are many common characteristics between WSNs and typical complex networks models: networks contain a large number of nodes and have nontrivial topological features and nodes in networks connect to each other through multihop paths [9]. More importantly, typical complex network models, such as the small-world [10,11] and scale-free [12] network models, show some characteristics which are beneficial in WSNs. Small-world networks present small average path length between pairs of nodes which is beneficial to saving energy in topology construction and routing in WSNs [13]. Scalefree networks have power-law degree distributions and show an excellent robustness against random node damage [14,15]. A random attack does not significantly affect the scale-free network performance [8,16]. Therefore, it is significant to consider complex networks topology when optimising the topology in WSNs [17]. However, complex networks are a kind of relational graphs whose nodes make direct contact according to their logical relationships, while WSNs are spatial graphs in which the existence of links depends on the node's positions and radio range [18]. Thus, the complex networks theory cannot be directly used in WSNs. Some efforts have been made to tune wireless networks into heterogeneous networks with small-world [13,[19][20][21][22] or scale-free features [9,[23][24][25].
In this paper, we propose a local-area and energy-efficient (LAEE) evolution model to build a WSN with scale-free 2 International Journal of Distributed Sensor Networks topology. In this model, topology construction is divided into two phases. In the first phase, nodes are distributed randomly in a fixed region, and a node gets other node's information in its transmission range (radio range) through HELLO message. In the second phase, topology evolution starts from the sink, grows with preferential attachment rule, and stops until all nodes are added into network. Following conditions are considered when we design the evolution model: (i) links between nodes depend on the positions and transmission range. Therefore, nodes beyond transmission range cannot make direct contact. (ii) Nodes can only get local information as WSNs are distributed networks. (iii) The remaining energy of each node is considered. Nodes with more remaining energy have higher probability to be connected. (iv) In order to avoid excessive energy consumption, upper bound of degree for each node is needed.
The remainder of this paper is organized as follows: Section 2 reviews background and related works on scalefree networks and scale-free based wireless networks. In Section 3, we propose the LAEE evolution model and deduce the theoretical degree distribution. Section 4 shows simulation results based on LAEE evolution model and examines the tolerance of LAEE to random failures. Finally, we conclude in Section 5.

Traditional Topology Constructions in WSNs.
Unit disk graph (UDG) is the underlying topology model for WSNs which contains all links in transmission range. Assume that all nodes are randomly distributed in region . Each node is positioned in a particular subarea with independent probability = 2 / , where is the transmission range. The probability that a subarea has nodes is given by the binomial distribution, ( ) = ( ) (1 − ) − , where is the total number of nodes in the network. With the increase of , this probability becomes the Poisson distribution ( ) = ( ) − / !. Then the average number of neighbor nodes is close to − 1. However, UDG model has high concentration of connections that might promote excess energy consumption for periodic topology maintenance and route selection process. Therefore this is an inefficient way of topology construction.
Almost all other topology construction methods in WSNs build a reduced topology from UDG [26,27]. Based on the topology production mechanism, they can be categorized into Flat Networks and Hierarchical Networks with clustering [27].
In Flat Networks, all nodes are considered to perform the same role in topology and functionality. Typical examples include directed relative neighborhood graph (DRNG) [28], -nearest neighbor (KNN) [29], TopDisc [30], Euclidean minimum spanning tree (EMST) [31], local Euclidean minimum spanning tree (LEMST) [32], Delaunay triangulation graph (DTG) [33], and cone-based topology control algorithm (CBTC) [34]. In KNN, a node sorts all other nodes in its transmission range in Euclidean distance and then links the -nearest nodes as neighbors in the final topology. It is a scalable and parameter-free in WSNs and very easy to implement. In DRNG, a link connects nodes and V if and only if there does not exist a third node that is closer to both and V in distance. TopDisc discovers topology by sending query messages and describing the node states using three or four color system. It is a greedy approximation method based on minimum dominating set. In EMST or LEMST, each node builds its overall or local minimum spanning tree based on Euclidean distance and only keeps nodes on tree that is one hop away as its neighbors. In DTG, a triangle formed by three nodes , V, and belongs to topology if there are no other nodes in the scope of the triangle. CBTC uses an angle as a key parameter. In every cone of angle around node , there is some node that can reach.
Nodes in Hierarchical Networks with clustering are heterogeneous in functionality as cluster heads or cluster members. LEACH is a typical Hierarchical Network topology model [35] in which the network is clustered and periodic updated. The cluster heads have the responsibility to communicate directly with the sink for the whole cluster members. A node selects itself to be a cluster head with a probability related to factors such as its remaining energy and whether it has served as cluster head in the last rounds.
The WSNs topology can be indicated as graph ( , ), where the sets of and are sensor nodes and topological links, respectively. We denote the number of node's links, also the number of its neighbor nodes, as its degree. All these previous topology construction models show highly concentrated degree distribution, which means these models tend to present homogeneous graph property [36,37].

Scale-Free Evolution Models.
Barabási and Albert provide an evolution model, called BA model, to generate a scale-free network [12]. The BA model is as follows.
(2) Growth: a new node is added.
(3) Preferential attachment: the new node will connect to ( ⩽ 0 ) existing nodes according to the preferential attachment. In detail, the new node connects to an existing node according to the probability Π = / ∑ , where is the degree (i.e., numbers of topological links) of node and is in the set of all existing nodes at this moment. In BA model, the degree distribution follows the powerlaw distribution ( ) = − , where the scaling exponent is = 3. The ∑ is the sum of degrees of all existing nodes at every step; so the preferential attachment mechanism in BA model works on the global network. Therefore, the BA model is a global topology evolution model which is unable to achieve in most large-scale distributed networks.
Many other scale-free models contain growth and referential attachment features as BA models shows [38][39][40]. One of the main differences between these algorithms is that they have various referential attachment mechanisms that lead to different values of scaling exponent in degree distribution and other scale-free properties.
Li and Chen propose a local-world evolution model [38]. Similarly to BA model, the local-world model is also divided into four steps of initialization, growth, preferential attachment, and halt. But differently, in preferential attachment mechanism of local-world evolution model, the new node selects of all existing nodes randomly, refereed to as the local world of this new node. So the preferential attachment probability for a new node connecting to an existing node at time step is where Π ( ∈ local-world) = /( 0 + ). The localworld property shows that the local-wold evolution model is a distributed model. But as these nodes are selected randomly, the spatial relationships between nodes are not considered. Therefore, the local-world evolution model still cannot describe the topology evolving mechanism in wireless networks.

WSNs Topology Constructions with Scale-Free Theory.
In preferential attachment mechanisms of BA, local-world, and many other scale-free models, the new node has the tendency to connect itself to some richer node (such as with larger degree). In WSNs, the concept of richer node can be extended as a node with more resources, such as with larger degree, more remaining energy, or any other resources.
Several methods have been proposed to build WSNs with the scale-free property [9,24,25,41]. These methods take complex network characteristic such as growth, preferential attachment into account, and some of them consider the local-area feature in WSNs.
Zhang provides a model of WSNs based on scale-free network theory [24]. In this model, each node has a saturation value of degree, max , to balance energy consumption. The newly generated node has a certain probability to be damaged when it is being added to the network. The probability that the new node will be connected to an existing node as follows: where V is the distance between the new node V and the existing node , is the transmission range, and is the number of nodes which already reaches the saturation value of degree max . In (2), ( V < ) refers to the ratio of 2 to , where is the entire WSNs coverage region.
One of the main drawbacks of Zhang's model is the sum of Π which is much smaller than 1. The scaling exponent is = 1 + 2 / 2 , where is the entire coverage region and is the transmission range. Therefore, another problem is that the scaling exponent of degree distribution is much greater than 3, which is not rational in real networks.
Wang et al. propose an arbitrary weight based scale-free topology control algorithm (AWSF) [9]. All nodes in the network are coupled with a sequence of random real numbers with a power-law distribution ( ) = − , where > 1 and = ∫ max min ( ) = 1. The balance of energy consumption is not considered in this model. Therefore, there is a possibility that a node with low energy is coupled with a large weight and therefore has a large degree, which exacerbates the imbalance of energy consumption.
Zhu et al. propose an energy-aware evolution model (EAEM) of WSNs [25]. Energy is taken into account in the EAEM model. This algorithm assumes that the probability Π that a new node connects to the existing node depends on its degree and the remaining energy of that node. A function ( ) is defined to present the relationship between remaining energy and its ability to be linked. ( ) must be an increasing function, as the more energy a node has, the more probability it will be connected to the new node. Therefore the form of Π is where the local-area in the EAEM is the set of nodes locating in the new node's transmission range. The sum of all nodes' Π is less than 1 and the scaling exponent of degree distribution is 1, which is not rational in real networks.

Local-Area and Energy-Efficient Evolution Model
In this section, we propose our scale-free topology construction model for WSNs. We assume that nodes are distributed in a given region with static positions. Then connections between them are built to generate a network. Based on this fact, the process of topology construction is divided into two phases: in the first phase, nodes are distributed randomly. We define the set of scattered nodes as the nodes having no access to the network topology yet in the process of evolution, as shown in Figure 1. An arbitrary node, marked as node V, gets all other nodes' information in its transmission range via HELLO message and takes these nodes as its potential neighbor nodes. Then in the second phase, topology evolution starts from sink, grows with the preferential attachment rule, and stops until all nodes are added into the network.
The LAEE evolution model is proposed.
Step 1. Nodes are distributed randomly in region . Each node gets its potential neighbor nodes' information in its transmission range through HELLO message. All these nodes are scattered and topology has not been formed at this moment. Step 2.
(1) Topology evolution starts from sink with 0 nodes (the sink and its 0 −1 potential neighbor nodes) and 0 random links between them. (2) At every time step, add a scattered node into the network. To do that, we find the node which has the most scattered neighbor nodes and mark it as node . Choose a scattered node randomly in node 's potential neighbors as the new node, denoting as node . With this strategy, the network expands outward and fills the region as fast as possible. (3) Randomly choose nodes, which are already in the topology and are node 's potential neighbors, and link them to node . If the number of node 's potential neighbors is smaller than , all these nodes will be linked to this new node. Connect node with node in these potential neighbor nodes based on the preferential attachment: where local-area is the set of node 's potential neighbor nodes in its transmission range, and max is a predefined default upper bound of node's degree, is the number of nodes which already have the degree of max , and ( ) is the function mentioned in the EAEM model. When a node reaches the degree of max , no more links can be added to it. LAEE generates a reduced topology from UDG. In LAEE, a topology of network grows step by step via adding new nodes and connecting these nodes to network. According to the LAEE modeling algorithm, at every time step, a scattered node is added to network and then connected to the other nodes within its transmission range in network via preferential attachment rule. One of the conditions of selecting a new node as scattered node is that this node is in the transmission range of at least one node in network. Therefore, once an arbitrary node is connected to other nodes in UDG, it will be connected to at least one node via LAEE.
In (4), Π ( ∈ local-area) refers to the set of node 's neighbor nodes in its transmission range at time step ;that is, where is the number of nodes in network and = 2 / is the possibility of two nodes positioned in each other's transmission range. We assume that only few nodes reach the upper bound max , so max could be ignored here. Therefore, in local area we have where is the expected number of nodes in new node's localarea which is equal to as the expected number of nodes in transmission range mentioned in UDG model, is the expected value of ( ), and ⟨ ⟩ = 2( + 0 )/( 0 + ) is the average degree of network at time step , where 0 and 0 represent the number of nodes and links at the beginning, respectively. Then we get the varying rate of : In a very large-scale network, 0 can be ignored; then we can get As ( ) is an increasing function, we set ( ) = . Then International Journal of Distributed Sensor Networks 5 According to the initial degree of node at time step , ( ) = , we can get where = /2 . The probability that node 's degree is smaller than is Then we can obtain the probability density of the degree of a node with energy as In the above equation, ∈ ( min /2 , max /2 ), where min and max are the bounds of energy . Therefore, the distribution ( ) has a power-law form with degree exponent = (1 + 1/ ). In order to get the probability density of degree with remaining energy , we have where is the distribution of with the bounds of min and max . satisfies the equation ∫ max min = 1. Table 1 presents the parameters used in our simulation. We distribute = 1000 nodes randomly in the square region and deploy the sink at a corner marked as (0, 0). We select 0 = 10 nodes and 0 = 10 links in sink's transmission range as the initial state of our evolution model. Energy in the networks is uniformly distributed. The value of is a constant which can get from the equation ∫ max min = 1. Different values of in LAEE are considered in our simulation.

Numerical Results
The simulation and theoretical degree distributions of LAEE are presented in Figure 2. The theoretical degree distribution of LAEE model is close to the degree distribution of BA model, and the simulation result of degree distribution is close to the theoretical value when ⩾ . It is noteworthy that the degree values must be large than in BA model (each node has links at least), while there are nodes with degrees less than in LAEE simulation results. This is because in our model a node may have the number of potential neighbors less than . If this happens, this node's degree may keep in a low value. In other words, it is due to WSNs' spatial structure. Fortunately, only a small proportion of nodes have degrees  less than . The power-law degree distribution is valid for most of nodes. Fault tolerance is a key issue in WSNs. Many real applications do not require all nodes to be connected. It is appropriate to consider relaxing the connectivity requirement [42]. When a fraction of nodes are out of work, the remains may not be connected and the application of entire network may become invalid. Then, it is important to introduce the giant component, which means the largest connected component [43,44], to measure the fault tolerance of WSNs with the nodes' failure. Two types of data flows exist in WSNs, flows between any pair of nodes and between sink and other nodes. Therefore, two kinds of giant component are considered correspondingly: the normal one which contains the largest number of nodes and the one with the sink. Sometimes these two giant components are the same, whereas sometimes they are not.
We examine how the fault tolerance of WSNs can be improved by LAEE. Nodes are removed randomly to simulate the procedure of energy depletion or random failure. Typical WSNs construction models UDG, KNN, DTG, LEACH + KNN (LEACH for cluster heads election, KNN for topology construction in each cluster), and LEACH + DTG (DTG for topology construction in each cluster) are used for comparison. The degree parameters are shown in Table 2. We can see that their average degrees are close to that of LAEE with = 3. Close average degrees mean these topologies contain similar number of links. However, due to the scalefree feature, the degree distribution of LAEE is much wider than other construction models. As Figure 3 shows, with the removing of nodes randomly and gradually, the sizes of giant components decrease. UDG provides upper bounds of giant components. The LAEE presents a larger giant components than KNN, DTG, LEACH + KNN, and LEACH + DTG, though it has the minimum number of average degree. Therefore we deem that LAEE, which presents the scale-free feature in degree distribution, has better tolerance against energy depletion or random failure in WSNs.

Conclusions
Topology control is one of the primary challenges to make WSNs resource efficient. In this paper, we propose a local information and energy-efficient based topology evolution model. The process of topology evolution is divided into two phases. In the first phase, nodes are distributed randomly in a fixed region. In the second phase, topology evolution starts from sink, grows with preferential attachment rule, and stops until all nodes are added into network. The theoretical degree distribution of LAEE evolution model is approaching that of BA model. Simulation result shows that when ⩾ , the degree distribution follows the power law. The LAEE model has better tolerance against energy depletion or random failure than other nonscale-free WSNs topology with close average degrees.