## 1 Introduction  
Topology control has been proposed as a technique to address many problems in networks by adding or deleting nodes/links according to certain algorithms/protocols, with the aim of obtaining expected network properties. In Wireless Sensor Networks (WSNs), the topology control is usually achieved by means of changing the communication range (equivalently, transmission power), scheduling sensor nodes to active/sleep mode, placing sensor nodes in specific positions, _etc_. Since energy availability is one of the most precious resources for sensor nodes, one of the goals of the topology control in WSNs is usually to reduce energy consumption, while at the same time preserving other fundamental properties, such as network connectivity, reliability, fault-tolerant, coverage, _etc_. The node degree means the number of one-hop neighbors a node has. If all nodes in a network are still connected after removal of any \(k-1\) nodes, the network is called \(k-connected\) network. A network that is \(k-connected\) indicates that the node degree of each node is at least \(k\), but the network is not necessary \(k-connected\) if the node degree for each node is at least \(k\). The reasons considering the node degree as the goal of the topology control are the following. Firstly, the node degree suggests fault-tolerance capability of network. If a node has a higher node degree, it implies that there are more alternative paths to route the data when one or more neighbors fail. Secondly, if the node degree is the order of \(\Theta(\log n)\), where \(n\) is the total number of nodes in a network, the whole network is connected with high probability [1]. This result helps to design a localized algorithm, because each node in the network only has to take care of its own node degree. Thirdly, the node degree has an impact on the signal inference. Nodes experience lower contention when accessing a wireless channel if the node degree is lower. As a result, a tradeoff between network connectivity and link quality has to be taken into account. In order to obtain the desired node degree, many algorithms are proposed to adjust sensor nodes' communication range to control the desired number of nodes in neighbor list, e.g., [2,3]. Because collecting global information is a time-consuming and energy-consuming process in large and distributed WSNs, these algorithms are usually localized, so they only rely on local and its one-hop neighbors' information. However, they also usually depend on calculating its neighbors' locations, e.g., through sensor nodes equipped with GPS, which is not always available in sensor nodes. In this paper, we propose a localized fuzzy-logic approach, without the assumption that location information is needed, to adaptively control the communication range of each node in order to achieve the desired node degree. It is named Fuzzy-logic Topology Control, FTC for short. Fuzzy-logic is a very powerful control algorithm. Fuzzy-logic control has been proved to be a very successful control approach to many complex nonlinear systems [4,5]. It is relatively simple to convert knowledge of domain experts to control algorithms. Usually, the design of fuzzy-logic controllers starts with constructing the membership functions for linguistic input/output and if-then rules, such as [6,7]. Unfortunately, it is not easy to tune the parameters of membership functions and if-then rules for fuzzy-logic controllers to achieve desired performances when the system is very complicated or dynamic. For instance, if the number of nodes in a network increases or decreases because of node failures or adding new nodes, the membership functions may have to be re-designed, which would be a tedious work. Alternatively, we can also start from a training data set obtained from already acquired knowledge, such as the node degree distribution [8,9]. The fuzzy-logic controller can be constructed from the training data set, leaving thecumbersome process of tuning fuzzy controller parameters to learning algorithms.