## 1 Introduction  
In Wireless Sensor Networks (WSNs), topology plays a vital role in minimizing various constraints, such as limited energy, latency, computational resource crisis, and quality of communication. For example, energy consumption is proportional to the number of packets sent or received. The receiving cost depends on packet size, while the transmission energy depends on the distance between the nodes. As topology inherently defines the type of routing paths, indicates whether to use broadcast or unicast, determines the sizes and types of packets and other overheads, choosing the right topology helps to reduce the amount of communication needed for a particular problem and thus save energy. An efficient topology, which ensures that neighbours are at a minimal distance, reduces the probability of message being lost between sensors. A topology can also reduce the radio interference, thus reducing the waiting time for sensors to transmit data. Moreover, topology facilitates data aggregation, which greatly reduces the amount of processing cycles and energy, thus giving a longer lifetime for the network.  
In addition, topology inherently defines the size of a group, how to manage new members in a group, or how to deal with members who have left the group. With the awareness of the underlying network topology, more efficient routing or broadcasting schemes can be achieved. Furthermore, the network topology in WSNs can be changed by varying the nodes' transmitting ranges and also by adjusting the wake/sleep schedule of the nodes. Therefore, more energy can be saved if the network topology is maintained in an optimal manner.  
All the discussions and examples above infer that logical topology plays a vital role for wireless networks, especially for resource-constraint WSNs. Moreover, logical topology facilitates WSNs in many ways to overcome different constraints such as minimizing energy consumption, maximizing lifetime, reducing interference, making networks scalable, _etc_. Thus, it is very crucial to compare different topologies and to choose the best topology while designing protocols or algorithms for WSNs. To do so, in this paper, we compare various topologies of wireless sensor networks using various performance metrics.  
This paper primarily consists of two parts. In the first part, we provide detailed descriptions of existing topologies, which are identified from different protocols for WSNs. To design various protocols for WSNs, different underlying logical topologies have been used. Each topology has its own advantages and disadvantages under a specific working environment. Because of this, to compare and evaluate the effectiveness of each topology, a set the performance evaluation metrics is required. In doing so, this paper also focuses on two major issues, namely (i) the system model of the WSN, which would be used throughout the paper, and (ii) the list of performance metrics to evaluate existing topologies.  
The second part of this paper presents a comparative discussion of performance of different existing topologies of WSNs. In doing so, various performance metrics are first identified, and then detailed comparisons among identified topologies are provided for each of these metrics. The paper ends with a table, which shows the comparison summary of the performance evaluation of different topologies.