Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 165136, 7 pages

http://dx.doi.org/10.1155/2015/165136

## A Topology Evolution Model Based on Revised PageRank Algorithm and Node Importance for Wireless Sensor Networks

^{1}School of Mathematics and Statistics, Xidian University, Xi’an 710071, China^{2}School of Computer Science and Technology, Xidian University, Xi’an 710071, China^{3}Guangxi Key Laboratory of Trusted Software, Guilin University of Electronic Technology, Guilin 541004, China^{4}College of Computer and Information Engineering, Zhejiang Gongshang University, Hangzhou 310018, China

Received 25 November 2014; Revised 14 April 2015; Accepted 15 April 2015

Academic Editor: Elmetwally Elabbasy

Copyright © 2015 Xiaogang Qi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Wireless sensor network (WSN) is a classical self-organizing communication network, and its topology evolution currently becomes one of the attractive issues in this research field. Accordingly, the problem is divided into two subproblems: one is to design a new preferential attachment method and the other is to analyze the dynamics of the network topology evolution. To solve the first subproblem, a revised PageRank algorithm, called Con-rank, is proposed to evaluate the node importance upon the existing node contraction, and then a novel preferential attachment is designed based on the node importance calculated by the proposed Con-rank algorithm. To solve the second one, we firstly analyze the network topology evolution dynamics in a theoretical way and then simulate the evolution process. Theoretical analysis proves that the network topology evolution of our model agrees with power-law distribution, and simulation results are well consistent with our conclusions obtained from the theoretical analysis and simultaneously show that our topology evolution model is superior to the classic BA model in the average path length and the clustering coefficient, and the network topology is more robust and can tolerate the random attacks.

#### 1. Introduction

Wireless sensor network (WSN) consists of spatially distributed autonomous sensors to monitor physical or environmental conditions, such as temperature, sound, vibration, pressure, motion, or pollutants, and to cooperatively transmit their data through the network to a sink node. Today WSNs are more and more widely used in variety of industrial and consumer applications, such as industrial process monitoring and control, machine health monitoring, environment and habitat monitoring, health care applications, home automation, and traffic control [1].

In military and warfare applications, WSNs are deployed in hostile monitoring environment, and the sensor node is of limited energy support. Energy exhaustion and natural damage of some sensor nodes often result in the failure of the whole networks. As a vital technique, the construction and control of the topology play an important role in conquering these problems in WSNs. The main purpose of construction and control of topology is to achieve a higher communication quality, energy utilization efficiency, and strong robustness topology.

Complex network model becomes increasingly popular in the complex communication networks topology control and performance optimization. Complex network is the abstract description of the complex systems which emphasize characteristics of topology [2]. A complex system contains a mass of units which are considered as nodes, and the forces among units are considered as edges [3]. To achieve better performance and more robust topology against the attacks in the network, people need to better understand the complex network theory and the possible application scenarios in these fields [4]. With the development of WSN, small-world model [5] and scale-free model [4] are widely used to optimize the WSN’s topology.

As one classical complex network model, small-world model has the shorter average path length and the larger clustering coefficient and also a few of short cuts, so this kind of network can be easily destroyed under the targeted attacks [6]. As another classical complex network model, scale-free model has the power-law node degree distribution characteristic and the better robustness against random attacks, because its physical topology is tightly associated with the evolution process, which depends on the mechanism of its growth and preferential attachment, and lots of researchers have contributed their works for improving the performance of the network topology [6, 7]. In this paper, our purpose is to propose a new topology control method which is based on the scale-free model, and its evolution process includes two aspects as follows.(1)Growth: the scale of the network is expanding.(2)Preferential attachment: the newly joined node is more inclined to join those nodes with higher degree. Namely, the rich become richer.

It is generally known that the preferential attachment is the most important one because it provides the principal rules about how a new node connects to the previous network topology and which node should be the potential candidate node to be connected. To achieve better performance of the network, we aim to revise the preferential attachment method and to consider more details for selecting the candidate node from the existing topology and connecting to the new node. PageRank algorithm is a method for node ranking in a network and also for evaluating the node’s importance in some application scenarios, so some revised PageRank algorithms were proposed to evaluate the node importance and select the important candidate nodes, and also some detailed rules were considered in the proposed algorithms [8–10]. However, we can see that these preferential attachment rules only concern more node’s degree.

In some challenging applications, there are lots of nodes with same degree, which is not sufficient to make a rational decision. So some of the background information, such as location and flow, was taken into account to design the PageRank-based algorithms. Dorogovtsev showed that main properties of scale-free evolving networks may be described in frames of a simple continuous approach and also obtained the scaling relations for networks with nonlinear, accelerating growth and described the temporal evolution of arising distributions [11]. Unlike the well-studied models of growing networks, where the dominant dynamics consists of insertions like nodes and connections, and rewiring of existing links, Sarshar studied the scale-free evolving networks application in ad hoc network, where one also has to contend with rapid and random deletions of existing nodes (and the associated links); the dynamics discovered in the paper can be used to craft protocols for designing highly dynamic peer-to-peer networks and also to account for the power-law exponents observed in existing popular services [12]. However, the above algorithms missed some inner relationships, and the transform possibility matrix holds the same value for every link, which will be revised in our model.

In this paper, we introduce node degree, centrality, and betweenness to revise the PageRank algorithm to evaluate the node importance of the WSNs, and a dynamic transmission matrix is built up; these will be helpful to form the new preferential attachment rules to select a suitable candidate node to be connected to the newest node. Thus, the robustness and survivability of WSNs are enhanced to some extent.

The rest of the paper is organized as follows. Section 2 introduces basic definitions of complex networks model and the limitations of the existing PageRank Algorithm. Con-rank node importance evaluation algorithm is proposed in Section 3; Section 4 proposes the new rule for topology evolution based on Con-rank algorithm; finally, the simulation results and conclusions will be shown in Section 5.

#### 2. Complex Networks Model and PageRank Algorithm

In this section, we formally describe the complex networks and the PageRank algorithm in detail, which is the basis of our node importance evaluation algorithm and topology evolution.

##### 2.1. Complex Networks Model

Complex networks can be modeled by graph , where is a nondirection connected graph with nodes and edges. is the set of nodes and represents the set of edges. is adjacent matrix of with rows and columns, and the detailed definition of in is shown as follows:

At the same time, some terms to be used to evaluate the node importance are listed as follows.(i)*Degree *. It is the number of edges connected to node .(ii)*Shortest Distance *. It is the number of hops along the shortest path between node and node .(iii)*Centrality *. It is the reciprocal of the total cumulative sum of the distance from all other nodes to node ; namely, .(iv)*Betweenness *. It is a ratio of the number of the shortest paths through the node to the total number of the shortest paths which include all the node pairs of a network.(v)*Largest Effective Component*. It is the effective subgraph with the largest size in a network.

##### 2.2. PageRank Algorithm

According to PageRank algorithm, a weight of each web page can be computed through iteration, which depends on weights of the pages connecting to this page. Here is the PageRank equation:where indicates the set of pages destined for page , and surfers connect to another page with probability , and follow the links in current page with probability .

#### 3. Con-Rank Node Importance Evaluation Algorithm for WSNs

It is known that PageRank algorithm is proposed for node ranking and importance evaluation in Internet, and its limitation is that it only depends on hyperlink relationship. We have improved the original PageRank algorithm and proposed a so-called “Con-rank algorithm,” which is based on node similarity and the characteristics of WSN. First, a transformation probability matrix is built up, namely, the probability of each node transforming data to its neighbors, in which the more similar the nodes are, the higher probability they will have. Second, the normalized centrality is used to revise the uniform distributed parameter in PageRank algorithm.

##### 3.1. Basis of Node Contraction

Here, we use node contraction algorithm proposed in [13] to calculate contraction value and is further used to compute the node importance. Because the node position and the degree are taken into account at the same time, the node importance in the network can be obtained.

*Definition 1. *Node contraction: there are nodes, which are directly connected with node , and these nodes are all replaced by a new node . One extreme example is that it becomes one-node network when the central node of a star network is contracted.

*Definition 2. *The cohesion degree of is that , where and is the shortest distance from node to node , when , . Obviously , and gets the maximal value 1 when there is only one node in the network. So network cohesion degree is decided by connection ability of nodes in network, and one assumes that the average shortest distance between a couple of nodes is . In addition, the number of nodes can also influence network cohesion degree.

*Definition 3. *The contraction value of node is , where indicates the graph after contraction of node .

Hence, according to Definitions 1 and 2, we get where is the same as that mentioned in Definition 2 and is the same as that mentioned in Definition 1.

##### 3.2. Node Contraction Algorithm

Detailed node contraction procedure is listed in Algorithm 1, in which the contraction value of node can be obtained.