According to the precise location of each unknown node obtained in the upper section, the optimal path of the mobile beacon is obtained using the optimal path selection algorithm of mobile beacon in the sensor network under non- dense distribution [18], and the detailed process is:

Chaos differential evolution method is used to obtain the optimal node energy partition of unknown nodes.

In order to describe the diversity of the direct groups of different sub-regions in the sensor networks under non-dense distribution, the variance of the fitness value of the data group and the diversity factor *mf* are set, and then there are:

$$\begin{array}{r}mf=\frac{1}{NP}\sum _{i=1}^{NP}(f({x}_{i})-\overline{f}(x){)}^{2}\end{array}$$(15)Where, *NP* represents the size of the sensor networks under non-dense distribution, *f* (*x*_{i}) is the fitness value of *i*th chromosome in similar genetic algorithm, and *̅f*(*x*) is the average fitness value.

In the iterative process, in order to avoid the algorithm entering the local optimal value too early, it is necessary to integrate the perturbation theory of chaotic time series to improve the algorithm performance. According to the chaos theory, the optimal search of unknown nodes is carried out through the Logistic chaotic time series shown in Eq. (16) [19].

$${x}_{n+1}=\mu {x}_{n}\left(1-{x}_{n}\right)$$(16)Where, *n* = 1, 2, 3, *. . .* , *x ∈* [0, 1]*μ ∈* [0, 4]. *x* is to simulate sequence elements, *n* is an iterative time step, *μ* is an adjustable parameter. Through numerical iteration, it is found that

If 0 ≤ *μ* ≤ 1, the iteration system has only *x* = 0 stable 1- periodic point.

If 0 ≤ *μ* ≤ 3, the iterative system has an unstable 1-periodic point *x* = 0, and a stable 1-periodic point *x* = 1 − 1/*μ*.

If 3 ≤ *μ* ≤ 3.449, there are two unstable 1-periodic points and *x* = 1 − 1/*u*.

$$\begin{array}{r}x=\frac{1}{2}\left(1+\mu \sqrt{(\mu +1)(\mu -3)}\right)\end{array}$$(17)$$\begin{array}{r}x=\frac{1}{2\mu}\left(1+\mu \sqrt{(\mu +1)(\mu -3)}\right)\end{array}$$(18)Eq. (17) and (18) are two stable 2-periodic points of the system: when 3.449 ≤ *μ* ≤ 3.544, the periodic point becomes unstable, and four stable 4-periodic points appear in this case.

When parameter continues to increase, *μ* > 3.544, the 4-periodic point and eight stable 8-periodic solutions. If it continues to iterate, it will find that the 8- periodic solutions fluctuate, and there are 16 stable periodic solutions, so go on, then.

Based on the characteristics of chaotic variables, the optimal search of the unknown nodes in the sensor network under non-dense distribution is carried out [20]. The chaos disturbance is integrated into the unknown node partition process, and the optimization of the unknown node partition of the sensor network is completed under the non- dense condition, and the best energy of the unknown node is obtained [21].

The optimal location of nodes in the wireless sensor network is found.

According to the above process, the optimal energy node of unknown nodes in the sensor network is obtained under non-dense distribution. By using the dynamic escape particle swarm optimization method, the optimal position of the optimal energy node in wireless sensor networks is calculated, and the network node coverage optimization is realized [22]. The dimension of wireless sensor network space is set to *k*, the number of particles in the dynamic escape particle swarm is *n*, *Y*_{k} = (*y*_{j}_{1}, *y*_{j}_{2}, *. . .* , *y*_{jk}) represents the space position of the *j*th particle. The speed of particle motion is represented by *W*_{j} = (*w*_{j}_{1}, *w*_{j}_{2}, *. . .* , *w*_{jk}). The optimal space position in the process of particle motion is represented by *Q*_{j} = (*q*_{j}_{1}, *q*_{j}_{2}, *. . .* , *q*_{jk}), and the best space position of all particles in the global search process can be calculated by the following formula.

$$\begin{array}{rl}{w}_{jk}^{l+1}=& x\left(l\right)\times {w}_{jk}^{l}\\ & +{d}_{1}{s}_{1k}\left(pbes{t}_{jk}^{l}-{y}_{jk}^{l}\right){d}_{2}{s}_{2k}\left(pbes{t}_{jk}^{l}-{y}_{jk}^{l}\right)\\ {y}_{jk}^{l+1}=& {y}_{jk}^{l}+{w}_{jk}^{l}\end{array}$$(19)Where, *d*_{1} is the weight coefficient of the running speed of the dynamic escape particle, and the *d*_{2} is the weight coefficient of the running speed of all the particles in the wireless sensor network, *s*_{1k} and *s*_{2k} are the random numbers in the range of 0~1, and ${y}_{jk}^{l}$ is the th dimensional space position parameter of the dynamic escape particle *j* during the first iteration process, and ${w}_{jk}^{l}$is the running speed of its corresponding escaping particles. $pbes{t}_{jk}^{l}$is the spatial escape position of particle *j* in the *k*th dimension, and *x* is the inertia weight coefficient. Through the operation, it can be obtained

$$x\left(l\right)=0.9{w}_{jk}^{l+1}-0.5\times l/max-step$$(20)Where, *l* is the number of iterations for network node coverage by dynamic escape particle swarm optimization, and max −*step* is the maximum number of iterations.

The following formula can be used to calculate the spatial location parameter *mbest* of the coverage area in wireless sensor network under non- dense distribution.

$$\begin{array}{rl}mbest& =x\left(l\right)\sum _{j=1}^{N}{q}_{j}/N\\ & =x\left(l\right)\left(\sum _{j=1}^{N}{q}_{j1}/N,\sum _{j=1}^{N}{q}_{j2}/N,...,\sum _{j=1}^{N}{q}_{jk}/N\right)\end{array}$$(21)Where, 1 is the spatial position parameter of dynamic escape particle *j* in th dimension. *N* is the number of nodes.

The detailed process of mobile beacon’s path selection under non- dense distribution is as follows:

Initialization. Firstly, the initialization of node energy control parameters, including the maximum number of iterations, dimensions, iterations and disturbance range, is carried out. Then the node area is initialized, and a random matrix is randomly generated as the initial population, and then an initial membership matrix is generated to select the initial optimal individual and the global optimal individual.

An evolutionary computation is carried out according to the differential evolution algorithm [23].

The threshold $\zeta $is set, the diversity factor is calculated, and *mf* and $\zeta $are made comparison analysis. If *mp* > $\zeta \phantom{\rule{thinmathspace}{0ex}},$then go to step (4), if $mp>\zeta ,$then go to step (5);

(4) The chaotic sequence is integrated into the partition to produce a random matrix *z*, which is *c* × *D*, and each component is between (0, 1), and *NP* chaotic sequence variables are obtained according to the Logistic chaotic mapping.

$$\begin{array}{r}{x}_{n+1}=4{x}_{n}(1-{x}_{n}),n=1,2,\dots ,NP\end{array}$$(22)The chaotic components that integrate into the individual perturbation variables are:

$$\mathrm{\Delta}{x}_{i}=a+\left(b-a\right){x}_{n},n=1,2,...,NP$$(23)The perturbed variables that are loaded into the population are:

$$\begin{array}{r}{x}_{n,G}={x}_{n,G}+\mathrm{\Delta}{x}_{i}\end{array}$$(24)The distance from each data to the clustering center is calculated, to adjust *U*. The population is updated through the new *U*, and the individual with the smallest fitness is the optimal individual of the present generation.

The optimized coverage of the wireless sensor network nodes is carried out by the dynamic escape particle swarm optimization (DDE-PSO) algorithm to calculate the optimal location of wireless sensor network node *mbest*.

If *G* = *mbestG*_{max}, the iteration is stopped and the global optimal individual and the optimal fitness value are the output. If it is not, then *G* = *G* + 1, and make iteration with the step (2).

The above analysis process is based on the uniform traversal characteristics of chaotic sequences and the efficient global search ability of differential evolution algorithm. Chaotic perturbation is incorporated into the node energy partition process to get the best energy node. Dynamic escape particle swarm optimization is used to calculate the optimal position of the optimal energy node in wireless sensor networks, which is the optimal beacon path.

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