## 2 Related Works  
Topology control for network connectivity issues has been widely studied. Latest surveys can be found in [11, 12, 13, 14]. One of the fundamental problems in network is how to make the network connected or \(k-connected\). For instance, if two networks are \(k-connected\), the resultant network formed by joining them together is also a \(k-connected\) network if there are \(k-vertex\) disjoint edges connecting them [15]. The connectivity problem can be formalized as a linear programming (LP) problem [16, 17, 18]. Study [18] considers the special issue that some nodes' batteries are renewable. It presents the algorithm to optimize the power assignment by employing LP model. The connectivity problem can also be formalized as a Steiner tree or minimal spanning tree problem, e.g., [2, 3, 19, 20, 21, 22, 23]. Many works focus on study of the node degree \(k\). One of the most interesting problems is the value of \(k\) to make the whole network connected. If all nodes in a network are uniformly deployed at random, there are some asymptotical results showing the condition when the entire network is connected with high probability as \(n\rightarrow+\infty\), where \(n\) is the number of nodes in a network. There is a generic form: if \(k\geq c_{1}\log n\), the network is connected with high probability; if \(k<c_{2}\log n\), the network is not connected with high probability. For instance, if \(k\geq 0.5193\log n\), the network is connected with high probability [24]. If \(k<0.074\log n\), the network is disconnected with probability one; if \(k\geq 5.1774\log n\), the network is connected with probability one [25]. Study [26] finds that if \(k\leq 0.3043\log n\) the network is not connected with high probability; if \(k\geq 0.5139\log n\), then the network is connected with high probability. There is no evidence showing that \(c_{1}=c_{2}\). It is worth mentioning that once the network is connected with high probability, it is also immediately \(k-connected\) with high probability [24, 27]. Previous results are asymptotical in nature, so they may not be able to be applied to real world scenarios easily. From a more practical perspective, it is necessary to propose topology control algorithms/protocols to make the network connected with high probability, while conserving as much energy as possible. However, the optimal solutions are usually not available. The problems, such as minimizing power consumption while maintaining a \(k-connected\) network [28]; minimizing number of links to obtain a \(2-connected\) network [29]; minimizing the number of node placement for \(k-connected\)[30]; minimizing the number of relay nodes for \(2-connected\) network [31, 32], are NP-hard or NP-complete. Therefore, heuristic algorithms/protocols are proposed, such as [33], SPAN [34], CCP [35], PEAS [36], k-Neighbor [10], FGSS [2], FLSS [2], LTRT [3] and FTC, which is shown in this paper. Some of them are centralized algorithms, such as FGSS [2]. FGSS has proved that the maximum transmission radius among all nodes in the network is minimized. In WSNs, because the number of the nodes deployed in the field is very large, processing capability is relatively low, and power in each sensor nodes is very limited, a centralized algorithm is not very practical. Hence, localized and lightweight algorithms/protocols are expected for WSNs. Ref [33] proposes a greedy algorithm: it starts from a complete graph, and then reduces nodes to reach desirable connectivity, or starts from an empty graph and later connect edges until connectivity is reached, then it cuts off unnecessary nodes but maintains the same connectivity. In this paper, our proposed FTC is a localized approach based on fuzzy-logic theory and does not depend on neighbor location information. The results show that FTC is able to achieve comparable performances to FLSS and better than LTRT, but FLSS and LTRT both utilize the neighbor location information. Control theory has been applied to WSNs.