## 3 A flexible Approach to a Reachable Network Topology with a Minimum Connection Degree  
### Network Model Design  
Network model is essential for this work. A graph-based network model is used to describe a certain network topology. Consider \(\bar{N}\) nodes in a network, to save node energy and to prolong network life, not all nodes works concurrently. Sleep schedule algorithm had been applied on sensor nodes to arrange nodes working in turns. Thus, sensor nodes are regarded as two states named as AwakeState and SleepState. The nodes in AwakeState are active to sense and deliver out environmental data. The nodes in SleepState are suspended In the network model, a network, denoted as _Net_, is composed of a set of active node \(N\) and a link set \(L\), _i.e._, _Net={N,L}_ in which \(N\)={\(n_{i}\), \(n_{2}\), \(n_{3}\),..., \(n_{|N|}\)}_ and \(L\)={\(l_{i}\)} where \(1\leq i,j\leq|N|\). The link, \(l_{i,j}\), stands for the connection between node \(n_{i}\) and \(n_{j}\). When node \(n_{j}\) is not the neighbor of node \(n_{i}\), there is no link to deliver data each other directly. They need a set of reply nodes, {\(n_{k}\)}, to set up a data path. Each node \(n_{i}\) has a maximum transmission power \(P_{xx}\) in watts to realize a corresponding maximum transmission distance, \(Max\_tx\), in meters. The nodes \(n_{i}\) and \(n_{j}\) deliver data directly with each other when the distance of nodes \(n_{i}\) and \(n_{j}\) is less than the Max_tx. If the distance between nodes \(n_{i}\) and \(n_{j}\) is greater than the Max_tx, nodes \(n_{i}\) and \(n_{j}\) have to indirectly communicate each other. The needed transmission power of each node is transmission_range/Max_tx in direct communication. A graph of a static WSN is shown in Figure 1.  
Each node connects to other nodes within its transmission range. In real applications, the links between two nodes cold be enabled or disabled for the reasons of energy-saving and link quality. For instance, not all nodes are in work at the same time. Due to the demand of energy-saving, sleeping schedules were applied to schedule nodes either on-duty or off-duty. Consequently, in this work, only enabled nodes are considered to construct a full reachable network. Besides, the number of connected neighboring nodes could be increased as Max-tx increasing. The number of connected neighboring nodes defines the node degree, which is varied with the transmission range of nodes. For an example, the node \(n_{i+5}\), \(n_{i+1}\), \(n_{i+2}\) and \(n_{i+6}\) are the neighbors of \(n_{i}\) and the degree of \(n_{i}\) is equal to 4. In that case, for the node \(n_{i+7}\), the packets from node \(n_{i+7}\) should be forwarded by \(n_{i+6}\). Consequently, we will develop the proposed scheme based on a three-tier network model. The operation of this hierarchical network is shown in Figure 2. The three-tier network model is composed of data sink, cluster head (router), and sensor node (node). The data sink in the first layer stores, processes and maintains data from sensor nodes. The routers in the second layer are the cluster heads to collect data from sensor nodes and forward data to data sinks. The third layer of sensor nodes is in charge of sensing environmental information and delivering data through routers as well as toward data sink. A component is a node set, in which there is at least a path between two arbitrary nodes. In a component, all nodes can deliver data each other. In a network with only a component, the network topology is full reachability. If the network topology is formed by many components, the network reachability is decreased. Hence, the transmission range is the key point for component number in a network. And the network topologies also are produced through the transmission range. To evaluate  
Figure 1: A graph of wireless sensor network. Figure 2: Three-tier network model. the connection quality of a network, three estimated values are adopted, including connectivity, reachability and percentage of k-degree node.