2.1 LMST algorithm
LMST algorithm [8,9,10] is a typical control algorithm grounded on neighborhood graph. Such type of graph control algorithm sets the topology graph G formed when all the nodes transmit using the maximum transmit power. According to a certain neighbor judgment condition q, the neighboring graph G’ of the graph is obtained. Finally, each node in G’ determines the transmit power by its nearest communication node. The main algorithms include RNG, MST, LMST, and so on. The LMST algorithm [11,12,13,14] is to send Hello packets to any node in the plane with the maximum statement radius d_{max}, accumulate all the info from closer nodes, and starts greedy algorithm to construct the local minimum spanning tree which takes itself as the root node.
The communication power of the node is set, and the communication radius is adjusted to reach all the onehop neighbors on its LMST to attain the goal of evaluating the total energy consumption by local minimum communication. The system is in fact a distributed approximate optimal implementation of the global minimum spanning tree algorithm. By adjusting the locally independent minimum spanning tree, the local minimum spanning tree algorithm can recognize the double connectivity of the topology, and it successfully lessen the transmit power of maintaining the comprehensive global connection and improve the collision of the MAClayer. Furthermore, this algorithm constructs the local accumulated informationbased topology to minimize the number of interactive packets and the time delay required. This local topology repair can be carried out in a definite direction to the node movement problem. Compared with other neighbor graph methods (such as UDG), it reduces energy consumption, contains fewer links, reduces interference in the network, and improves performance provided in Fig. 1.
Although the LMST type algorithm constructs a network topology efficiently by locally minimum total energy consumption, ignoring the problem of energy balance. The middle node bears too much load in the network, which consumes a lot of energy and easily leads to poor network connectivity.
2.2 Network model
There are some assumptions included in this article such as (1) every node has uniform maximum transmit power capability, that includes a section having a radius of d_{max}, and can modify that radius by regulating the transmit power. (2) Each node has the same preliminary energy, and the position info can be acquired through localization algorithm. (3) A relationship between energy utilization and communication gap in wireless communication is [15, 16]:
$$ \mathit{\mathsf{E}}={\mathit{\mathsf{kd}}}^{\mathit{\mathsf{n}}}+\mathit{\mathsf{C}} $$
(1)
Where E = consumed energy, k = a constant associated with the working system, d = communication gap, and C = working system constant and its value is taken as “0.” Here, n has a value of 3. That is, energy consumption for communication is related to the third power of communication space.
Furthermore, LMST algorithm based on energy balance considers the remaining energy of the communication node while taking into account the node communication energy consumption. Therefore, the algorithm link weight [17, 18] is
$$ \mathit{\mathsf{R}}={\mathit{\mathsf{k}}}_{\mathsf{1}}\mathit{\mathsf{d}}\left(\mathit{\mathsf{r}},\mathit{\mathsf{t}}\right)+{\mathit{\mathsf{k}}}_{\mathsf{2}}\left(\mathsf{2}{\mathit{\mathsf{E}}}_{\mathsf{0}}/\left({\mathit{\mathsf{E}}}_{\mathit{\mathsf{r}}}+{\mathit{\mathsf{E}}}_{\mathit{\mathsf{t}}}\right)\right) $$
(2)
Where k_{1} + k_{2} = 1, and d (r, t) = space between the transferring node and accepting nodes, E_{0} = the node’s initial energy, E_{t} = remaining energy of the transferring node. E_{r} = remaining energy of the accepting node; k_{1} and k_{2} are variable parameters. However, if the remaining energy of two nodes after transferring or accepting is high, the weight of link parameter (R) becomes lower. While if the remaining energy of nodes is lower, the weight (R) value becomes relatively high and the intermediate node is less likely to be responsible for the task of data forwarding, which saves its own energy and prolongs the life of the node. The whole network gradually reaches the state of energy balance during operation.
2.3 Description of network topology control algorithm based on energy balance
This algorithm takes balancing energy consumption as the goal. When the topology is constructed, the energy utilization and the remaining energy of the nodes for communication are considered, and the effect of energy balance is achieved [19, 20].

(1)
Collect node information. The node “u” periodically transmits a Hello packet to other node in the valid range at the d_{max}, and the subsequent acceptor node feeds back the response packet to the sender node u, so that the node u can acquire the response of all the nodes in its visible neighboring area NVu(G).

(2)
Constructing topology graph. The node u works on the basis of the greedy algorithm to obtain the LMST in perspective of the visible neighbor node information and calculates the link weight coefficient R according to the formula (2). With the increase of service time, the gradual increase of weight between the intermediate node and other neighbor nodes has been observed, and there is a greater probability in topology construction from intermediate node to leaf node, which is no longer responsible for data forwarding and saves energy.

(3)
Determine the transmit power. Based on the determined structure of the LMST, the node u determines its own transmit power and regulates the communication radius d_{max} to grasp all onehop neighbors.

(4)
Bidirectional processing. The node u directs a check packet for whole onehop neighbors in the tree and waits for receiving the response packet from the neighbor node. After the timelag, the received response packet defines a link between the node u and neighbor node is bidirectional. If not, the edge is deleted.

(5)
Broadcast topology. The node u broadcasts its own reasonable neighbor node, and the advised node stores information into the data construction Info_Root Node, marking the considered node as the reasonable neighbor node of the node u and receiving the information packet from the u. After accepting a data packet by each node, it requires to examine the Info_Root Node to determine the receiving of the data packet. If it cannot be received, it is discarded.

(6)
Reconstruct the topology. Due to energy consumed by the node or may be dead node, every network node needs to operate the algorithm periodically, to perform topology reconstruction based on the existing visible information by the neighbor node, to obtain a new LMST, and to determine the transmit power.