The Impact of Clustering on the Average Path Length in Wireless Sensor Networks

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3 size. For example, the dominators can choose its one-hop neighbors that can cover the largest number of two-hop neighbors to become dominators. The advantage of is that it can build a CDS quickly using minimum energy resources and low communication overhead. III. ASSUMPTIONS AND IMPLEMENTATION In this section, the description on the assumptions and the detailed implementation of the study are presented. A. Network Model An undirected graph G = (V, E) is used to represent the WSN, where V is a set of sensor nodes in the network, called vertices and E is a set of a communication link between a pair of sensor nodes, called edges usually denoted as (u, v) E. Two vertices u and v are neighbors if () they are within their maximum transmission range R max and () the communication links between them are symmetrical. We assume all sensor nodes have same hardware capabilities. The sink node on the other hand has more powerful communication and processing features. B. Average Path Length Average path length is one of important measures of clustering given that the network topology generated is often used for routing. A path P in a graph G is a sequence of vertices connected by edges. To find the average shortest path length in the graph G = (V, E), the greedy method based on classical Dijktra s algorithm is applied on the network. The main objective is to find the shortest path from each node i in the graph to a sink node j in which the edge weight is the distance between the vertices. It is assumed that the graph G is connected if the shortest path length between these two nodes is a finite number and it is disconnected if there is no connecting edge between the nodes. In this paper, Dijkstra s technique was adopted to obtain the shortest path length measure. A total of topologies were used to obtain the shortest path length measure. These topologies were referred to the original topologies. The clustering algorithms also utilize these topologies to form the backbone or clusters. Recall that the backbone topology is derived from the original topology in which the number of nodes remain the same except that redundant links in the backbone topology might be removed. Thus, it is possible for the backbone topology to have longer path between nodes. In order to investigate the performance of the average path length in the original topologies against the topologies of the backbone, we compute the average shortest length on both (i) original topologies and (ii) topologies constructed by the clustering algorithms (known as backbones). IV. PERFORMANCE OF ALGORITHMS algorithm has the following performance: Theorem. has O(n) message complexity and O( C + ) time complexity, where n is defined as the overall number of nodes, is the maximum node degree while C is the number of selected connectors. Proof. has O(n) message complexity because each node u exchanges exactly one message when building the CDS. The time complexity of the algorithm refers to the amount of time required for computing the connector set. takes the advantage of MPR property [] when determining the connector, where the time complexity of the algorithm is O( C + ). algorithm has the following performance: Theorem. has O(n) message complexity and O(n) time complexity, where n is defined as the total number of nodes. Proof. Each node sends at most one message during the first and second phase of the algorithm. Hence, its message complexity is is O(n). The time complexity of is bounded by the MIS construction in which node requires at most O(n) time complexity. algorithm has the following performance: Theorem. has O(m) message complexity and O( ) time complexity, where m is the number of edges and is the maximum nodal degree. Proof. The message complexity of is contributed by the number of messages sent to each edge, in this case are two messages. Whereas the time complexity refers to the O( ) time required for constructing and pruning the CDS. V. SIMULATIONS AND DISCUSSIONS In this section, we present our simulation results. We evaluated the presence of the average path length with respect to network density (or average node degree) and network sizes to investigate the scalability of the algorithm. We also compare the performance of all algorithms to study how well they preserve the average path length. The discrete event simulator OMNeT++ (version.) [] was used for simulating various algorithms. In order to have more realistic model, MiXiM [] which is an extension of OMNeT++ framework was used to provide models of radio wave propagation and MAC protocols. Realistic topologies that were extended from a small network calibrated using the testbed in our department were created. We generated different types of topologies to represent various network sizes and densities. These topologies are categorized into three densities; network (node degree ), network (node degree ) and network (node degree ). To obtain various network sizes we varied the number of nodes N in each density from to with an interval of. To achieve a % confidence interval, the simulations were repeated with runs. We assumed that the nodes were deployed in a D dimensional space and their transmission range varied.

5 (a) Comparison of average path length against network densities for. (a) Average path length versus network size for network. (b) Comparison of average path length against network densities for. (b) Average path length versus network size for network. (c) Comparison of average path length against network densities for. (c) Average path length versus network size for network. Fig.. Average path length versus various network densities. Fig.. Average path length comparison against network size.

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