The studies about the convergence and complexity for random optimization-based heuristic algorithm are usually based on Markov chain, which has no aftereffect property [46,47,48,49,50,51,52]. TGSARWI DA is based on the algorithm of self-avoiding random walk and combines target guiding and intersection of paths. Since the walking process does not have after effect property, it no longer belongs to the category of Markov process. Hence, it is inappropriate to do the complexity analysis for our algorithm by common theories. This work integrates theoretical analysis and numerical simulation to discuss the algorithm performance.

In this section, simulations are conducted to verify the algorithm for the MEP problem. Firstly, the effect of the parameter *ρ* on the algorithm is evaluated. Secondly, the algorithm performance is analyzed with precision assessment, complexity analysis, and comparison with POA. Finally, the robustness of the algorithm is discussed.

In order to distinguish the application of DA to the global grid network and the sub-network created by TGSARWI, they are defined as Global DA and TGSARWI DA, respectively.

### 5.1 Effect evaluation of the target guiding factor

The effects of the target guiding factor *ρ* are discussed on the number of iterations for finding the first connected path (*T*_{
f
}), the number of iterations for finding all the required connected paths (*T*_{
l
}), and the minimum exposure *E*(*H*), respectively.

The simulation is implemented in the grid with the scale of 50 × 50, in which the length of edge is 10, the number of sensors is 50 and their directions, and coordinates are generated randomly. The initial graph created is shown in Fig. 5. For each parameter *ρ*, the algorithm is simulated 100 times and the average values of *T*_{
f
}, *T*_{
l
}, and *E*(*H*) could be obtained.

Figure 6a shows that when the parameter *ρ* increases, *T*_{
f
} and *T*_{
l
} decrease. The reason is that the influence of target guiding on the transition probability increases with the increment of*ρ*, which weakens the walker’s randomness and decreases the number of iteration. In fact, *T*_{
f
} and *T*_{
l
} decrease sharply when *ρ* belongs to the area of [0, 0.9]. However, when *ρ* > 0.9, the numbers of iterations reach stable. This means that, as *ρ* further increases, the influence on finding the connected path is little.

The effect of parameter *ρ* to minimum exposure *E*(*H*) is displayed in Fig. 6b. Roughly speaking, with the increment of *ρ*, the value of *E*(*H*) first decreases and then increases. When *ρ* is small, the randomness of the walkers is large. Thus, the sub-network formed by *n*_{
h
} connected paths is sparse (see Fig. 7a, b), leading to a large value of *E*(*H*). However, if *ρ* is too large, walkers cannot avoid moving along edges with large exposure due to the constraint of target guiding (see Fig. 7c, d). This can also make the value of *E*(*H*) large. Figure 6b shows that *E*(*H*) achieves the minimum as *ρ* = 0.9. Thus, *ρ* is set as 0.9 in simulations.

Taking *ρ* as 0.9, the MEP problem presented in Fig. 5 is addressed with TGSARWI DA algorithm. It is also solved with Global DA as a comparison. As shown in Fig. 8, although those algorithms generate different optimum paths, the values of *E*(*H*) are very close.

It is worth to note that TGSARWI DA algorithm owns good generality. Once the value of parameter *ρ* is chosen, the algorithm could solve MEP problem with various field scales and different sensor densities. In the following simulations, *ρ* is always set as 0.9.

### 5.2 Performance evaluation of TGSARWI DA

The performance of TGSARWI DA algorithm is evaluated from three aspects, i.e., precision, complexity analysis, and comparison with POA.

The minimum exposure path found by Global DA is considered as optimal path. Thus, TGSARWI DA is compared with Global DA. Considering the time efficiency, the first path found by TGSARWI is also included in the comparison. In order to overcome the randomness feature of single simulation, 30 independent repetitive simulations are performed and the average value is taken.

The numbers of walkers and connected paths which form sub-network are related with the scale of the field. As for a field with scale *m* × *n*, these values are respectively taken as \( {n}_u=\left\lfloor 30+\sqrt[4]{mn}\right\rfloor \), and \( {n}_h=\left\lfloor 30+2\times \sqrt[4]{mn}\right\rfloor \).

#### 5.2.1 Precision assessment

Since the time complexity of Global DA grows rapidly as field scale increases, simulations are initially conducted in the field whose scale changes from 10 × 10 to 150 × 150 (see Fig. 9a). In order to avoid the effect of sensor density, the density of sensors is fixed as 0.02. Namely, the sensor number is 2% of the total number of nodes in the grid network. Then, the field scale is fixed as 50 × 50, and the sensor density is adjusted from 0.04 to 0.4 (see Fig. 9b).

Figure 9a shows the trends of *E*(*H*) corresponding to the first connected path *H*_{TFP} obtained by TGSARWI, path *H*_{TDA} obtained by TGSARWI DA and path *H*_{GDA} obtained by Global DA with the increasing of scale, respectively. *E*(*H*) of *H*_{TDA} and *H*_{GDA} are almost equal, whose relative error is 4.59% and maximum value is 6.39%. In contrast, *E*(*H*) of *H*_{TFP} is much larger than that of *H*_{GDA}. Their average relative error is 85.49%. In Fig. 9, *E*(*H*) of *H*_{TDA} and *H*_{GDA} are almost the same in various sensor density fields. In fact, their average relative error is only 2.19%. However, *E*(*H*) of *H*_{TFP} is greater than *H*_{GDA}, and their average relative error is 50.41%. These results suggest that, when dealing with MEP problem with various field scales and different sensor densities, TGSARWI DA exhibits performance nearly as well as Global DA.

#### 5.2.2 Time computation complexity analysis

For a field with scale *m* × *n*, the time computation complexity of DA is *O*(*m*^{2}*n*^{2})_{,}^{Footnote 1} which means that DA is not suitable for solving MEP problem with a rather large-scale field. Here, the time computation complexity of Global DA is compared with TGSARWI DA in different grid scales and various sensor densities. The scale of the field is set from 10 × 10 to 900 × 900, and the sensor density is in the area of [0 0.4].

In this paper, the time computation complexity is measured from two aspects, i.e., first connected path found by TGSARWI and the minimum exposure path found by TGSARWI DA.

For the first connected path, the time computation complexity is mainly associated with the transition possibility \( {p}_{tj}^{(3)}\left({H}_t\right) \) of each walker at each step in the grid network, which is about *O*(*n*_{
u
} × *T*_{
f
}). Further, the time computation complexity benefit *η*_{TFP} of the first path found by TGSARWI is defined as the ratio of complexity of finding first connected path by TGSARWI and DA, which is represented as follows:

$$ {\eta}_{\mathrm{TFP}}=\frac{n_u\times {T}_f}{{\left(m\times n\right)}^2}\approx {T}_f\left(30{(mn)}^{-2}+{(mn)}^{-\frac{7}{4}}\right) $$

(13)

For the minimum exposure path, to quantitatively analyze the simplifying efficacy of TGSARWI algorithm to MEP problem, the reduction ratio *r* is defined as the norm ratio of sub-network *G*_{1} formed by *n*_{
h
} connected paths to the original grid network *G*:

$$ r=\frac{\left\Vert {G}_1\right\Vert }{\left\Vert G\right\Vert }=\frac{\left\Vert {G}_1\right\Vert }{m\times n} $$

(14)

Where ‖•‖ is the number of edges of the network. Since the time computation complexity of sub-network forming is about*O*(*n*_{
u
} × *T*_{
l
}), and the time computation complexity of minimum exposure path found by TGSARWI DA is *O*(*n*_{
u
} × *T*_{
l
} + *r*^{2} × (*m* × *n*)^{2}), the time computation complexity benefit *η*_{
TDA
} is calculated as follows:

$$ {\eta}_{\mathrm{TDA}}=\frac{n_u\times {T}_l+{r}^2\times {\left(m\times n\right)}^2}{{\left(m\times n\right)}^2}\approx {T}_l\left(30{(mn)}^{-2}+2{(mn)}^{-\frac{7}{4}}\right)+{r}^2 $$

(15)

In Fig. 10a, the iteration number *T*_{
f
} is growing linearly as the field scale increases, whose slope is about 0.9803. Compared with the time computation complexity of DA that is *O*(*m*^{2}*n*^{2}), the efficiency of first path found by TGSARWI algorithm is higher. From Eq. (13) and Fig. 10b, it is shown that the first path benefit *η*_{
TFP
} decreases significantly as the scale increases. Specifically, when the field scale is 900 × 900, *η*_{TFP} is only 8.3493 × 10^{−8}. These results suggest that TGSARWI is efficient to find out the first path for timeliness demand in large-scale field, such as real-time solution and online calculation, although the exposure performance of *H*_{TFP} is not as good as Global DA.

In Fig. 10c, although the iteration number *T*_{
l
} of TGSARWI DA increases exponentially with the change of field scale, its exponential coefficient is only about 0.0046. Meanwhile, in Fig. 10d, the complexity benefit *η*_{TDA} gradually deceases with the increasing of the scale, and finally goes close to zero. Particularly, its minimum value is 0.0042 when the field has the scale of 900 × 900, suggesting that TGSARWI DA only uses 0.42% computation time of Global DA.

Similar measures are also employed to analyze the time computation complexity of TGSARWI DA with various sensor densities. In Fig. 11a, iteration numbers of the first path found in the field for different sensor densities are visualized. The number of iterations increases linearly with the density of sensors in the areas of [0.0.312] and (0.312, 0.4], respectively. The linear relation suggests that TGSARWI is not sensitive to the change of sensor density, making it suitable to solve MEP problem in the field with high density sensors. The equations of the two fitting lines are shown in the inner figure. The slope of the straight line in the latter area is much larger than that in the former area, implying that the time to find a path increases sharply when the density of sensors exceeds the critical value 0.312. This is because if the distribution of sensors is too dense, it will be rather difficult for the walker to find an accessible path. In the area that density of sensors is reasonable (here it is [0, 0.312]), TGSARWI is expected to identify a path at a rather fast speed. For example, when the number of sensors is 780, corresponding to the density of sensor 0.312, only 41 steps are needed to find the first connected path. Figure 11b shows how the iteration number to find all the required connected paths changes with sensor density. The iteration number *T*_{
l
} is proportional to the sensor density. The curve increases gently when the sensor density is below 0.312 and grows rapidly as the density is above 0.312. The fitting function of the curve is \( y=\frac{30.2064}{0.4276-x} \). Accordingly, it is obviously shown that the complexity of finding connected path increases as the density becomes denser, while the maximum sensor density for TGSARWI algorithm to find connected sub-network is 0.4276.

### 5.3 Comparison with POA

As a representative of heuristic algorithms in solving MEP problem, physarum optimization algorithm (POA) has the ability to find the minimum exposure path in the field with various type sensors and reduce the time computation complexity [40]. However, the growth of time computation complexity is still too fast.

According to ref. [40], in the field with scales of 10 × 10, 20 × 20, 50 × 50, and 100 × 100, as well as with the sensor number of 10, 30, and 50, the minimum exposures of paths found by POA and DA are quite close. The relative errors of *E*(*H*) of paths got by POA and Global DA are 0.0240 and 0.0542 in different field scales and sensor densities. TGSARWI DA is also compared with Global DA in the same condition. The relative errors are 0.0219 and 0.0459 in different field scales and sensor densities, respectively. These results suggest that the algorithm has slightly better performance than POA.

The time computation complexity of TGSARWI DA is further compared with that of POA. Numbers of iteration for POA are taken from ref. [31]. As shown in Fig. 12, the iteration number of POA is far greater than that of TGSARWI DA, no matter what scale the field and the sensor density are. These results fully demonstrate that TGSARWI DA is more appropriate to solve MEP problem with a large-scale field and high-sensor density.

### 5.4 Robustness discussion

In this section, variation coefficient is applied to quantitatively analyze the vulnerability of TGSARWI deceased by the stochastic fluctuations of target guiding factor *ρ*, the number of walkers *n*_{
u
} and the number of connective paths *n*_{
h
} [53]. The fluctuation coefficient *φ*(*X*) of sample data *X* is defined as:

$$ \varphi (X)=\frac{\sqrt{D(X)}}{F(X)} $$

(16)

where *F*(*X*) and *D*(*X*) donate the expectation and variance of *X*, respectively. The closer the coefficient *φ*(*X*) to zero is, the smaller fluctuation the sample data *X*has.

Based on current parameters, 100 groups of normal distribution rates are generated and 50 groups of sample data are created, according to each normal distribution rate. Then, corresponding iteration number *T*_{
l
}, and the minimum exposure *E*(*H*) are calculated according to each normal distribution. Finally, values of *φ*(*ρ*), *φ*(*n*_{
u
}), *φ*(*n*_{
h
}), *φ*(*T*_{
l
}), and *φ*(*E*(*H*)) are obtained. *φ*(*T*_{
l
}) and *φ*(*E*(*H*)) are set as vertical coordinates, *φ*(*ρ*), *φ*(*n*_{
u
}), and *φ*(*n*_{
h
}) as horizontal coordinates, respectively and show their relationships in Fig. 13.

From Fig. 13a, b, when *ρ* and *n*_{
u
} fluctuate in the interval [0, 0.5], the fluctuation coefficients of the minimum exposure *E*(*H*) change little, the maximum values of *φ*(*E*(*H*)) are 0.0331 and 0.0315, respectively. They are all smaller than the corresponding fluctuation coefficients of parameters, indicating that TSAGRWI DA owns quite excellent robustness to the stochastic fluctuations of these two parameters. In Fig. 13c, *φ*(*E*(*H*)) increases quadratically with *φ*(*n*_{
h
}), but still satisfies *φ*(*E*(*H*)) < *φ*(*n*_{
h
}).The maximum value of *φ*(*E*(*H*)) is 0.0726, indicating *E*(*H*) also owns better robustness to the stochastic fluctuation of *n*_{
h
}. In summary, *E*(*H*) is not sensitive to the fluctuation of the three parameters *n*_{
h
}, *ρ*, and *n*_{
u
}.

Figure 13d–e show that *φ*(*T*_{
l
}) increases as *φ*(*ρ*), *φ*(*n*_{
u
}), and *φ*(*n*_{
h
}) grow. The values of *φ*(*T*_{
l
}) increases quadratically with the increase of *φ*(*ρ*) and *φ*(*n*_{
u
}), and the increasing rate of the former is faster than the latter. When *φ*(*ρ*) ≤ 0.1611 and *φ*(*n*_{
u
}) ≤ 0.2370, the fluctuation coefficient of *T*_{
l
} is smaller than those of *ρ* and *n*_{
u
}, suggesting that *T*_{
l
} is not sensitive to the fluctuation of *ρ* and *n*_{
u
} in these areas. *φ*(*ρ*) > 0.1611 and *φ*(*n*_{
u
}) > 0.2370, the fluctuation coefficient of *T*_{
l
} is larger than those of *ρ* and *n*_{
u
}, which indicates that *T*_{
l
} is sensitive to the fluctuation of *ρ* and *n*_{
u
} in these areas. *φ*(*T*_{
l
}) grows linearly as *φ*(*n*_{
h
}) increases, and the slope is 0.7224, which is smaller than 1, implying that *T*_{
l
} has better robustness to the fluctuation of *n*_{
h
}. In a word, the sensitiveness of iteration number *T*_{
l
} to the stochastic fluctuations of parameters could be sorted in descending order as *ρ*, *n*_{
u
}, and *n*_{
h
}.

It is worth to note that both fluctuation coefficients of *T*_{
l
} and *E*(*H*) have positive values during the fluctuations of *ρ*, *n*_{
u
}, and *n*_{
h
}, which are mainly caused by the randomness of TGSARWI.