Sensor Networks: A Graph Theory Perspective
7.2 FUNDAMENTAL CORRESPONDING BETWEEN WIRELESS SENSOR NETWORKS AND GRAPH THEORY
A wireless sensor network is a set of sensors deployed in a sensor field to monitor specific characteristics of the environment, to measure those characteristics, and to collect the data related to those phenomena. The sensors are small devices with limited resources: limited battery power, low memory, little computing capability, very low data rates, low bandwidth processing, variable link quality, and so on. However, despite their constraints, when the sensors are deployed in large numbers, they can provide us with a very real picture of the field being sensed. WSN can provide an area coverage that was not possible with other wired and wireless networks. They can be deployed in different environments and can be permanently attended or can be left unattended once they have been deployed in the field.
The use of the WSN potential will provide efficient and cost-effective solutions for many problems. However, it is necessary to implement mechanisms or procedures to deal with the sensor constraints. The use of clustering techniques has been proposed to help solve some of those constraints, by allowing the organization of the sensors in a hierarchical manner, grouping them into clusters and assigning a specific task to the sensor in the clusters, before moving the information to higher levels. The concept of clustering is very useful in different contexts of WSN. Clustering is a fundamental mechanism to design scalable sensor network protocols. In general terms, clustering is the classification of similar objects into different groups or subsets. The formed subsets in some sense belong together, because they share one or more similar characteristics or behaviors. Examples of such common characteristics could be: proximity according to some defined distance measure, similar behaviors, common data patterns, and so on. In the most general problem the number of clusters or groups is unknown, as are the properties that make them similar.
Clustering techniques have been proposed in wireless networks in order to achieve high energy efficiency and assure long network lifetime, for bandwidth reuse, for data gathering [1] and target tracking [2], one-to-many, many-to-one, one-to-any, or one-to-all communications, routing [3–6], and so on. Clustering is particularly useful for applications that require scalability to hundreds or thousands of nodes. Scalability in this context implies the need for load balancing, efficient resource utilization, and data aggregation [7]. Also, many routing protocols can use clustering to create a hierarchical structure and minimize the path cost when communicating with the base station. In many sensors network applications where data collection and processing can be done in situ, this hierarchical approach is a promising method for efficiently organizing the network. Also, many signal processing algorithms used for extraction of final information from the data gathered by the sensors are well-suited for local processing of data within the clusters.
Graph Theory concepts can be used to describe, analyze and represent a WSN in a very clear way. Several of these concepts refer to structures and algorithms that had been previously used to address other aspects like topology management, localization techniques and routing, not only in WSN but also in other types of wireless and wired networks [8–11]. For example, the OSPF (Open Shorted Path First) routing algorithm
164 CLUSTERING IN WIRELESS SENSOR NETWORKS: A GRAPH THEORY PERSPECTIVE
used for routing in wired LANs implements the SPF (Shorted Path First) algorithm or Dijkstra’s algorithm, which solves the single-source shortest path problem for a directed graph with nonnegative edge weights.
Graph data structures and algorithms can easily represent the network with sta-tionary wireless sensor. The construction of basic graph structures like trees, cliques or dominating sets, and the use of graph algorithms like Breadth First Search (BFS) or Depth First Search (DFS) will help in the construction of clusters to improve the communication in the WSN. However, in mobile WSN, the dynamic change of the network topology due to the sensor movement creates additional challenges when forming the clusters. In this situation it is necessary to implement additional mecha-nisms to control the changes in the cluster graph.
In this section, we give the basic concepts and definitions that provide a detailed overview of the corresponding relation between clustering in WSN and graph theory, focusing on concepts and definitions that are important for the understanding of the material to follow.
7.2.1 Wireless Sensor Networks and Graph Theory Concepts
A cluster in WNS consists of three main different elements: sensor nodes (SNs), base station (BS), and cluster heads (CH); see Figure 7.2. The SNs are the set of sensors present in the network, arranged to sense the environment and collect the data. The main task of an SN in a sensor field is to detect events, perform quick local data processing, and then transmit the data. But the greatest constraint it has is the power consumption, which usually is caused when the sensor is observing its surroundings,
SN
SN SN SN
SN SN
SN
BS
SN SN SN SN
SN
SN
CH CH
CH CH
CH CH
Figure 7.2. Elements in a clustered WSN.
and communicating (sending and receiving) data. The BS is the data processing point for the data received from the sensor nodes, and where the data are accessed by the end-user. It is generally considered fixed and at a far distance from the sensor nodes.
The CH acts as a gateway between the SNs and the BS. The function of the cluster head is to perform common functions for all the nodes in the cluster, like aggregating the data before sending it to the BS. In some way, the CH is the sink for the cluster nodes, and the BS is the sink for the cluster heads. This structure formed between the sensor nodes, the sink, and the base station can be replicated as many times as it is needed, creating the different layers of the hierarchical WSN.
The SNs and the communication links between them can be represented by an undirected graphG=(V, E), where each vertex v∈V (the set of vertices in the graph) represents a sensor node with a unique ID. An edge (u, v)∈E(the set of edges in the graph) represents a communication link if the corresponding nodesuand vare within the transmission range of each other.
The graph is formed by defining the neighborhood of each node. The neighborhood of each one of the nodes in the network and thek-neighborhood of the nodes can be defined as follows:
Definition 1. Node’s Neighborhood. The neighborhoodN(v) is the set of nodes (neighbors) that reside within the circular transmission range of nodev, which means the vertices adjacent tov. Ifvis included into the neighborhood, it is called closed neighborhood ofvand it is representedN[v].
Definition 2. k-Neighborhood.Thek-neighborhood ofv,Nk(v), is the set of nodes with distance at mostkfromv.
Nk(v)= {u|u∈V ∧d(u, v)≤k}
Following the creation of the graph, by defining the adjacent nodes in the network and the communication links between nodes, it is possible to determine which nodes are reachable from a specific node and it allows us to calculate the correspondent hop distance (see Definition 3) between any source and target nodes. It also allows us to determine thekthpower graph of a node (see Definition 4), to limit the number of nodes that will be considered among the node’s transmission radius when creating the clusters.
Definition 3. Hop Distance.The shortest path between two nodesuandvis the path with the minimum number of hops between them. The distanced(u, v) is the number of hops in the shortest path betweenuandv.
Definition 4. kth Power Graph. The kth power graph of G,Gk=(V, Ek), is the graph between the nodes inV and an edge between every pair of nodesu, v∈V, such thatd(u, v)≤kinG.
166 CLUSTERING IN WIRELESS SENSOR NETWORKS: A GRAPH THEORY PERSPECTIVE
Knowing the existent links between the nodes, the next process involved in the communication is determining the routes available to send information through the WSN. One of the most common mechanisms used for message delivery inside the network is based on a spanning trees structure, defined below, that allows the use of algorithms like Depth First Search (DFS) and Breadth First Search (BFS) to send messages along the created graph in linear time. We useT to represent the spanning tree ofGc, rooted aty. Theithlevel of the tree is the set of nodes with hop distance equal toifromy. The depth of the tree,depth(T), is the index of the farthest level in the tree.
Definition 5. Spanning Tree.A spanning tree is a connected and undirected graph, with no cycles. Each tree has withnvertices. Each pair of vertices has exactly one path connecting them, creatingn−1 edges in the tree.
Definition 6. Depth First Search (DFS).Depth First Search (DFS) is an uninformed tree search that works by expanding the first child node that appears on the graph and goes deeper each time until a goal node is found, or until it hits a node that has no children. After this the search backtracks, returning to the most recent node it hadn’t finished exploring.
Definition 7. Breadth First Search (BFS).Breadth First Search (BFS) is a graph search algorithm that begins at the root node and explores all the neighboring nodes.
After this, it explores the unexplored neighbor nodes of each of the previously found nodes, repeating this procedure until it finds the goal node.
Cluster Representation. There are many different graph concepts used for the creation of clusters in a WSN. Among them is the following definition for cluster:
Definition 8. Cluster. A cluster is any subset of nodesC⊂V.y∈V is the cluster head andGc=(C, Ec) is the cluster graph.
Ec= {(u, v)|u, v∈C∧(u, v)∈E}
IfGcis connected, then the cluster is connected.dc(u, v) is the shortest path inside the cluster, and the cluster radius is the maximal distance betweeny and any other nodev∈C.
maxv∈Cdc(y, v)
After determining the nodes’ neighbors and the distance between them, it is pos-sible to use additional graph concepts like the node’s weight to use it as one of the parameters in the definition and functionality of the clusters.
Definition 9. Graph Weight.The nodes in the network graph can have a positive weightwv. The total weight of a cluster is given by
Wsum(C)=
v∈C
wv
Dominating Sets and Covers. Another group of graph definitions which are very useful when trying to model the clusters is the concept of sets. Set definition in the context of WSN are listed below. By being able of defining these sets in the network, the nodes can calculate their real coverage and establish the stability of the communication paths between them. Generally, the clusters are defined using the vertex cover of the graph (see Definition 10) and the nodes that belong to that vertex cover are selected as the cluster heads. The remaining nodes should calculate the stability of their communication link to the neighboring CHs and join the cluster corresponding to the link with the best connection stability.
Definition 10. Vertex Cover.A vertex cover is a subsetS⊂V, such that for all edges inE, S∩e /=0, which means that every edge has at least one edge inS.
Definition 11. Independent Set.The independent setISGis the subsetS⊂V, such that there is no edge between any pair of nodes inS. The maximum independent set inGis equal to|V|, the size of the minimum vertex cover ofG.
Definition 12. Dominating Set.The dominating setDSGis the subsetS⊂V such that every node is inS or if it is inV −Sand has at least one neighbor (adjacent vertex) inS. A vertex ofSis said to dominate itself and all adjacent vertices. An edge is dominated if either of its endpoints is inS. The nondominated edges are called free edges.
Definition 13. Minimum Dominating Set.The Minimum Dominating Set (MDS) problem is the problem of finding a dominating set of minimum size. This is an NP-complete problem.
Definition 14. Independent Dominating Set. The independent dominating set IDSGis the setS⊂Vthat is both dominating and independent.
Definition 15. Connected Dominating Set.The connected dominating setCDSG is a dominating set whose induced subgraphSis connected.
Definition 16. Weakly Induced Subgraph. For any subset S⊂V, the subgraph weakly induced by S,Sw, is the graph (N[S], E∩(N[S]×S). This means thatSw contains the vertices ofS, their neighbors, and all edges with at least one endpoint in S. A vertex subsetSis aweakly connected dominating setifSis dominating and Swis connected.
168 CLUSTERING IN WIRELESS SENSOR NETWORKS: A GRAPH THEORY PERSPECTIVE