## 2 Related Work
For instance, in [41], energy, concentration and centrality are combined to optimize the CH selection in order to extend the network lifetime efficiently. Energy and local distance are used in cluster head election mechanism using fuzzy logic (CHEF) [42] to compute the probability of being selected as CHs. The clustering algorithms proposed in [43, 44] are similar to CHEF. The distance of the cluster centroid, the residual energy of nodes and network flow are selected to compute the selection probability of the CH. multi-weight based clustering (MWBC) [44] is a clustering algorithm based on multiple factors, such as the degree, current energy, transmission power, link quality and relative position of nodes when selecting CHs, aiming at maximizing the network lifetime. Simulation results demonstrate that these algorithms can make a good selection of CHs, and the network lifetime, to varying degrees, can be extended. However, in these algorithms, the weights of different factors can only be determined by experience instead of the precise calculations, which leads to the instability of these algorithms and, thereby, affects the performance of the entire network. Generally, some algorithms mentioned above, such as [24, 26, 27, 28], can take full advantage of the high energy of nodes. Selecting some high-energy nodes to take on the CH role ensures that the energy consumption of these CHs is balanced. However, there often exist the following cases. If the cluster size is not well-controlled, on the one hand, the energy consumption of some high-energy nodes may be also higher; after a round, the residual energy of these high-energy nodes is less than of these low-energy nodes, and on this occasion, the advantages of these nodes are not obvious. On the other hand, if the energy consumption of some high-energy nodes is lower, in this case, there will be much energy left while some other nodes have been dead, which cause the the network energy to be wasted. That is, in these cases, these algorithms cannot guarantee the local energy consumption optimization, let alone the global energy consumption optimization. Thus, to avoid these situations, we propose the concept of the local energy consumption ratio and then introduce it into the cluster head election phase. Some other algorithms, such as [41, 42, 43, 44], also overlook this influencing factor in the choice of CHs. These hybrid algorithms consider the local energy consumption ratio as a parameter for nodes to compete for CHs, which is also helpful. If CHs send their data to the BS directly, energy consumption increases if CHs are far away from the BS. Accordingly, for communication among CHs, searching for different communication routing paths among CHs to alleviate the loads of the CHs is also studied. Applying the multi-hop communication method, each CH can find out the appropriate forwarding node, based on their distance to the BS, to relay data. The inter-cluster communication tree construction in energy-aware distributed unequal clustering (EADUC) [36] is based on the residual energy and relaying cost of forwarding nodes, while in EADC (energy aware distributed clustering) [37], the routing algorithm increases the forwarding tasks of the nodes in scarcely covered areas by forcing CHs to choose nodes with higher energy and fewer member nodes as their next hops, which can help EADC achieve load balance among CHs in non-uniform networks. For more results related to clustering protocols, one can refer to [5]. In this paper, to further reduce and balance the energy consumption of nodes, we construct a simple and energy-efficient inter-cluster communication tree based on the local energy consumption ratio of nodes in a distributed way.