## 2 Related Works  
During the past few decades, clustering techniques have been extensively studied to improve the performance of WSNs. In this section, we will review the existing clustering schemes which are most related to our work. One of the most famous clustering protocols named Low Energy Adaptive Clustering Hierarchy (LEACH) is proposed in [32]. Sensor nodes in the network are grouped into several clusters, and each cluster consists of one CH and some cluster members. As the CH has responsibilities of aggregating the data from its member nodes and forwarding the aggregated data to BS, it consumes energy much faster than the member nodes. To balance the energy consumption among sensor nodes, a random rotation mechanism of CH is designed to insure that each sensor node acts as CH once during a deliberately designed time interval. However, this random CH re-selection mechanism may result in uneven distribution of CHs. Furthermore, CHs in LEACH adopt single-hop communication mode to transmit data to BS. Then the energy consumption among sensor nodes cannot be well balanced since CHs further from BS have longer data transmission distance than these closer to BS. In addition, nodes usually have limited communication abilities in real scenario. Thus, it is better to employ multi-hop communication method for inter-cluster data transmitting when designing clustering protocols for WSNs. Hybrid Energy-Efficient Distributed clustering (HEED) is another famous clustering protocol presented in [33]. The CHs are periodically selected according to a hybrid of node's residual energy and the intra-cluster communication cost. This CH re-selection mechanism effectively reduces the communication cost of the network and can avoid the case that more than one CH occurs in a close proximity. Furthermore, the multi-hop communication model is adopted in HEED so that CHs in the network cooperatively transmit data to BS by a multi-hop way. However, since the CHs closer to BS have heavier relay traffic and consume energy much faster than those farther from BS, the energy hole problem may occur which is not considered in HEED. To solve the energy hole problem, unequal clustering structures have been adopted to balance the energy consumption among CHs. In [31], [PERSON] _et al._ present an Unequal Cluster-based Routing (UCR) protocol to group the network into several clusters with unequal sizes. The CHs closer to BS have smaller cluster sizes than those farther from BS, thus they can burden less intra-cluster traffic and preserve more energy for inter-cluster data forwarding. However, the size of cluster is simply designed to decrease with the decrease of the distance between CH and BS, which needs to be analyzed and optimized. [PERSON] _et al._ in [34] theoretically analyze the energy consumption with different cluster radius in clustering network and then give the expression of cluster radius which can maximize the network lifetime. In addition, by employing an unequal cluster radius and an alternate operation scheme between work and dormancy, a simple clustering protocol, which can mitigate the energy hole problem, is proposed. However, the network scalability is not considered in this protocol. An Energy-Balancing unequal Clustering Approach for Gradient-based routing (EBCAG) protocol is proposed in [35]. Each node maintains a gradient value which is defined as the minimum hop count to BS. In addition, based on this gradient, a ring-based network model is constructed. CHs in a ring transmit data to the heads in the neighbor ring closer to BS. By balancing the energy consumption among the CHs in different rings and minimizing the total energy consumption of the network, the size of clusters is determined in each ring. EBCAG can adapt to the expansion of the network, but how to calculate the optimal number of rings is not concerned. A protocol named Constructing Optimal Clustering Architecture (COCA) is proposed in [36]. It divides the sensor field into a number of equal-size square units.