## 1 Introduction  
Wireless sensor networks (WSNs) have been widely used in recent years. The properties of WSNs--which contain hundreds or thousands of nodes--are limited by energy, bandwidth, computing capability, etc. [1; 2; 3; 4]. Interference is an important factor of WSNs, affecting not only the data transmission efficiency, but also the energy consumption. High interference means low communication quality and high energy consumption. In multihop WSNs, due to the characteristics of wireless communication, the network performance is affected by interference more seriously than that of the wired network. On one hand, the node which is receiving data packets will be affected by the nodes with transmission ranges that cover this node; on the other hand, the nodes which are sending data packets will interfere with the nodes that fall into their transmission ranges [5; 6]. In addition, high interference causes serious propagation delays and packet retransmissions, which consumes a large amount of energy. At worst, the serious interference can make the network fail to work. Many algorithms have been proposed to reduce the network interference. One remarkable approach is topology control. The aims of topology control are: improving the network connection and reducing the energy consumption [7; 8]. Many remarkable topology control algorithms have been proposed, such as [5; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20]. However, based on the conclusions of [5; 6], reducing the transmission range cannot guarantee low network interference, i.e., a small transmission range does not mean small network interference. This property is important. It is necessary to take this property into account when designing the interference-optimal topology control algorithm. However, the related issues shown below have not been analyzed accurately so far:  
1. What is the probability that the network interference is reduced when adjusting the transmission range? 2. What are the properties of this probability, such as how this probability varies when the network topology or the transmission range changes? 3. If the network topology is fixed, which transmission range has the highest probability of interference-optimality? 4. What is the relationship between the interference-optimal and the energy-efficient, i.e., the probability that the interference-optimality and energy-efficiency can be achieved at the same time? Therefore, these issues will be investigated and analyzed in detail in this paper. Motivated by these issues, in this paper, we firstly analyze the probability of interference-optimality by adjusting the transmission ranges. We prove that the probabilities in homogeneous networks and heterogeneous networks are \(p_{\text{hom}}\) and \(p_{\text{het}}\), respectively. With an increase in node density, this probability increases, too. Moreover, if the transmission range is large enough, when the transmission range reduces, the interference decreases with probability 1. This transmission range is large in sparse network and small in a density network. Secondly, we calculate the probability of interference-optimality under a specific transmission range and find out the optimal transmission range which has the maximum probability of interference-optimality. In homogeneous networks, this probability is \(P_{\text{hom}}\) and the optimal transmission range is \(0.85r\). In heterogeneous networks, this probability will be \(P_{\text{hels}}\) and the optimal transmission range is \(0.84r\). Finally, we investigate the relationship between energy-efficiency and interference-optimality when adjusting the transmission range. We prove that if the network is energy-efficient, the network is also interference-optimal both in the homogeneous and heterogeneous WSNs.