Design of feature selection algorithm for high-dimensional network data based on supervised discriminant projection

Data sources

The datasets used for the experiments are seven high-dimensional artificial datasets, namely Isolet, Arcene, Madelon, Gisette, Cod-RNA, Dexter and Dorothea, with sample sizes ranging from 600 to 2,000, feature numbers ranging from 5,000 to 20,000, and category numbers ranging from 2 to 26. Parts of these seven datasets have been published in the UCI machine learning knowledge base, and some are from related studies. The data format is CSV file, each row represents one sample and each column represents one feature.

Experimental setup

The experiments were done in Python 3.6 environment, mainly relying on Numpy, Scipy and Sklearn machine learning libraries. The experimental procedure was divided into three steps: (1) Min-Max normalization was applied to seven datasets, and Scaling feature values to [0,1] interval; (2) the datasets were divided into training set (70%), validation set (15%) and testing set (15%); (3) the feature selection and classification performance of the SDP algorithm proposed in this study was compared with other four algorithms. The comparison methods include information gain (IG), chi-square test (CHI), recommendation system (RS), and gradient boosting (GBDT). The evaluation metrics are chosen as F1 score, precision and recall.

Theoretical model

The theoretical basis of SDP feature selection algorithm is Fisher discriminant analysis (FDA). Due to the limited sample size of high-dimensional network data, the direct use of FDA may lead to overfitting. In this study, we propose to use sparse subspace clustering to unsupervisedly cluster the high-dimensional network data to obtain the clustering results, and then build the FDA model based on the clustering results to realize the projection to the Fisher discriminant hyperplane. SDP achieves effective feature selection by building a subset of features with low redundancy related to the category.

Algorithm steps

The main steps of the SDP feature selection algorithm are.

• 1) Formalize the sparse representation of high-dimensional network data using the Lp parametric optimization problem.

• 2) Apply sparse subspace clustering to unsupervised clustering of high-dimensional network data.

• 3) Dimensionless processing of clustering results.

• 4) Construct the FDA model based on the clustering results, and calculate the contribution of each feature to the class interval.

• 5) Combining the contribution degree and feature importance, the most important features are selected by using the sparse constraint method.

• 6) Logistic regression is used to evaluate the feature selection results and determine the optimal feature subset.

The process is iterated to select the optimal feature subset on the validation set. Finally, the performance is reported on the test set to verify the effectiveness of the proposed feature selection algorithm. The study of feature selection for high-dimensional network data has important theoretical significance and practical application value. Theoretically, feature selection is a key step in data mining and machine learning, which directly affects the effectiveness of subsequent modeling and applications. The research on feature selection for high-dimensional network data can help to understand the intrinsic correlation between data structure and features and enrich the feature selection theory. In practice, high-quality feature selection algorithms can extract the key features of high-dimensional network data and remove redundant information, thus improving efficiency and accuracy. This is important for web data mining, classification, clustering, and trend prediction.

Related work focuses on the traditional filtering, wrapping and embedding methods. These methods are not effective for feature selection of high-dimensional data sets and cannot solve the dimensional disaster problem well. Discriminant analysis-based methods can refine features by maximizing class spacing and minimizing intra-class variance, which are better adapted to high-dimensional data. The SDP algorithm proposed in this study combines discriminative projection and sparse constraints to perform feature selection of high-dimensional network data more accurately and efficiently.

The innovations of this study are: (1) combining the Lp-parametric optimization problem with SDP to realize the projection and feature selection of high-dimensional network data; (2) using unsupervised clustering of high-dimensional network data by sparse subspace clustering as the prior knowledge of SDP; (3) avoiding the overfitting problem of feature selection results by combining SDP and sparse constraints. The contributions of the study are: (1) proposing an SDP algorithm for feature selection of high-dimensional network data; (2) enriching the projection-based feature selection method; (3) providing a reference for key feature selection and mining of high-dimensional network data in practice.

In conclusion, the SDP algorithm combined with the clustering prior for discriminative projection and feature extraction can achieve accurate and efficient feature selection, which has theoretical and application values. This study enriches the theory and method in the field of feature selection and can provide inspiration for the processing and analysis of network data in the big data environment.

Sparse subspace clustering of high-dimensional network data

In order to improve the quality and efficiency of feature selection for high-dimensional network data, this article adopts the sparse subspace clustering method to cluster high-dimensional network data. The specific implementation steps are as follows:

Step 1: Based on the theory of sparse expression (Vlachos & Thomakos, 2021; Kambampati et al., 2020), use the solution of the network high-dimensional data sparse optimization problem to build a relevant matrix. The data points in the matrix may come from the same subspace. The theory should be used to find the high-dimensional data points from the same network.

Step 2: Based on the network high-dimensional data point subspace, cluster the data with the spectral clustering method to obtain accurate network high-dimensional data clustering results.

Assuming that the network high-dimensional data point is represented by $y_i\in {\cup }_{\ell =1}^n{S}_{\ell }$, it can be calculated by the following formula:

(1) $$y_i=Y{c}_i$$

The above formula, $Y$ represents a matrix composed of multiple network high-dimensional data points. Generally, high-dimensional network data points in this subspace dimension are less than the total amount of data in this space, so each $Y_{\ell }$ and $Y$ contains nontrivial zero space, so the representation of data points has complex characteristics. $c_i=\left[c_{i1},c_{i2},\cdots ,c_{iN}\right]$ represents a dataset composed of data from the same subspace. Suppose that the data point in $d_{\ell }$ dimensional subspace $S_{\ell }$ is represented by $y_i$, and the point can also be linearly described by other $d_{\ell }$ data points in $S_{\ell }$, then the point is a sparse representation of $y_i$ (Zhu, Zhao & Wu, 2021; Khandani & Mikhael, 2021).

The diagram of sparse subspace clustering is shown in Fig. 1.

Since there are countless solutions to Formula (1), the problem of sparse representation of high-dimensional data on the network can be converted into an Lp norm optimization problem, and the following formula is true:

(2) $$y_i=Y=min{\Vert c_i\Vert }_q$$when taking different values, the sparse expression of high-dimensional network data will be different. The closer $q$ is to 0, the more sparse the high-dimensional network data will be (Zheng & Li, 2021). When the value of $q$ is 0, the Lp norm optimization problem is transformed into an NP-hard problem. According to the analysis of relevant theories, the Lp norm minimization optimization problem can be used to replace the Lp norm optimization problem, so Formula (2) can be converted into the following formula:

(3) $$y_i=Y=min{\Vert c_i\Vert }_1$$

Then, the sparse optimization problem of network high-dimensional data point $y_i$ can be expressed by the following formula (Wang et al., 2022):

(4) $$y_i=min{\Vert C\Vert }_1$$

The above formula, $C=\left[c_1,c_2,\cdots ,c_N\right]\in R^{N\times N}$ represents a sparse coefficient matrix, and $i$ columns $c_i$ in the matrix correspond to the sparse representation of the data point $y_i$.

Combining Formula (4), we can get the sparse optimal representation of each high-dimensional data point on the network. The data will next be divided using the sparse coefficient matrix in order to achieve high-dimensional data clustering on the network. In this process, it is necessary to establish a weighted graph, which is expressed by the following formula:

(5) $$\varsigma =\left(\nu ,\epsilon ,W\right)$$

The above formula, $\nu$ represents $N$ vertices in the weighted graph composed of multiple high-dimensional data points of the network, $\epsilon \subseteq \nu \times \nu$ represents the edges of the weighted graph, and $W\subseteq R^{N\times N}$ represents the non-negative similarity matrix.

The sparse optimization problem of network high-dimensional data points is converted into the sparse subspace representation problem of data points (Zha, 2020) and $W$ can be constructed in the following way.

(6) $$W=\left|C\right|+{\left|C\right|}^T$$

The above formula, $T$ represents matrix transposition.

In an ideal state, there are $n$ connected components $\varsigma$, and the matrix expression corresponding to $n$ subspaces is as follows:

(7) $$W=\left[\begin{array}{ccc}{W}_1& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & W_n\end{array}\right]W_{\ell }$$

The above formula, $W_{\ell }$ represents a similar matrix located in subspace $S_{\ell }$.

Assuming that $E$ represents the estimation point set matrix and $Z$ represents the noise matrix, the sparse subspace clustering function of high-dimensional network data can be expressed by the following formula:

(8) $$D=min{\Vert C\Vert }_1+{\lambda }_e{\Vert E\Vert }_1+{\displaystyle \frac{{\lambda }_Z}{2}{\Vert Z\Vert }_F^2}$$

Dimensionality reduction of high-dimensional network data based on SDP

It is necessary to transform the original description of the data in the clustering results into data that can be compared and processed by computers in order to ensure that all network high-dimensional data formats can be unified data and enhance the effectiveness of subsequent data feature selection. This process is called dimensionless data. In general, the most commonly used dimensionless data processing methods are classified into normalization and standardization (Khaled et al., 2021; Xiong et al., 2022). The normalization formula of network high-dimensional data is as follows:

(9) $$X_i^{\mathrm{\prime }}={\displaystyle \frac{X_i-X_{min}}{X_{max}-X_{min}}}$$

The above formula, $X_i$ represents the value of the current dimension, $X_{min}$ represents the minimum value of all network high-dimensional data samples in this dimension, $X_{max}$ represents the maximum value of all network high-dimensional data samples in this dimension, and $X_i^{\mathrm{\prime }}$ represents the normalized value of this dimension.

The effect of normalization is shown in Fig. 2.

From the analysis in Fig. 2B, it can be seen that the contour lines are more rounded and smooth after normalization, and the convergence speed can be improved if the likelihood function is optimized (the red line in the figure is gradient optimization). Therefore, it is necessary to normalize.

Compared with normalization, standardization requires a much lower standard for the data distribution of the original dataset. The standardized formula for high-dimensional network data is as follows:

(10) $$X_i^{\mathrm{\prime }}={\displaystyle \frac{X_i-\mu }{\sigma }}$$

The average value of all the data in this dimension in the initial high-dimensional network data set is represented by $\mu$ in the calculation above. The following is the calculating formula:

(11) $$\mu ={\displaystyle \frac{1}{n}\sum _{i=1}^n{x}_i}$$

The above formula, $n$ represents the total amount of high-dimensional data of the original network.

$\sigma$ represents the standard deviation of all data in the original high-dimensional network data set in this dimension (Miambres, Llanos & Gento, 2020), and its calculation formula is as follows:

(12) $$\sigma =\sqrt{{\displaystyle \frac{1}{n-1}\sum _{i=1}^n{\left(x_i-\mu \right)}^2}}$$

To create a strong foundation for later network high-dimensional data selection, the dimensionless network high-dimensional data will be decreased. This study combines SDP and sparse constraint methods to implement high-dimensional network data feature selection based on the linear projection matrix and optimal transformation matrix.

In general, the objective function of dimensionality reduction of high-dimensional network data is expressed by the following formula:

(13) $$min\sum _{i,j}{\Vert y_i-y_j\Vert }^2{W}_{ij}$$

The above formula, $W_{ij}$ represents the connection weight between sample points $x_i$ and $x_j$, and $W_{ij}=W_{ji}$. $W_{ij}$ is calculated by the following formula:

(14) $$W_{ij}=\{\begin{array}{c}\exp \left(-{\Vert x_i-x_j\Vert }^2/t\right),x_i\in N\left(x_j\right)\vee x_{j}N\left(x_i\right)\hfill \\ 0\hfill \end{array}$$

The above formula, $N(\cdot )$ represents the nearest neighbor relationship function. If the projection transformation matrix (Meng, 2021; Sun et al., 2021) is brought into the linear projection matrix, the following formula is valid:

(15) $$\begin{array}{rl}& \arg \underset{V}{min}tr\left(V^{T}XL{X}^{T}V\right)\\ & s.t.tr\left(V^{T}XD{X}^{T}V\right)=1\end{array}$$

The above formula, $D$ represents a diagonal matrix and $L=D-W$ represents adjacency graph matrix.

The following formula is valid:

(16) $$D_{ii}=\sum _j{W}_{ij}$$

The constraint condition for high-dimensional data dimensionality reduction processing of the network is taken to be the orthogonalization of the linear projection matrix (Wang et al., 2020; Liu et al., 2021; Li, 2016, 2015; Lin et al., 2022), and the objective function of high-dimensional data dimensionality reduction processing of the network is constructed as follows:

(17) $$\begin{array}{rl}& max{\displaystyle \frac{tr\left\{V^T{S}_{N}VA\right\}}{tr\left\{V^T{S}_{L}VA\right\}}}\\ & s.t{V}^{T}V=I\end{array}$$

The above formula, $S_L$ and $S_N$ represent local divergence and global divergence matrix respectively, representing linear projection matrix, and $A$ represents transformation matrix.

To find an optimal transformation matrix $A=\left[a_1,a_2,\cdots ,a_r\right]$ to make the $S_N$ largest and $S_L$ smallest of the low dimensional projection space after the transformation of the discrimination vector $a$, it is necessary to establish a relevant transformation matrix processing model (Dong & Zhuang, 2011; Liu et al., 2020; Zhang et al., 2019), which is expressed by the following formula:

(18) $$J\left(A\right)=max{\displaystyle \frac{tr\left\{V^T{S}_{N}VA\right\}}{tr\left\{V^T{S}_{L}VA\right\}}}$$

Based on the above analysis, the problem of dimensionality reduction of high-dimensional network data can be transformed into a generalized eigenvalue problem (Lin et al., 2020; Huang et al., 2020; Elphick et al., 2017; Magdy et al., 2018), and the following formula is valid:

(19) $$XL{X}^{T}V=\lambda XD{X}^{T}V$$

$V$ is composed of feature vectors ${\nu }_1,{\nu }_2,...,{\nu }_d$ corresponding to the first $d$ minimum eigenvalues ${\lambda }_1,{\lambda }_2,...,{\lambda }_d$ of generalized eigen decomposition, i.e., $V=\left[{\nu }_1,{\nu }_2,...,{\nu }_d\right]$, so as to realize dimensionality reduction processing of high-dimensional network data.

The dimension reduction processing effect of high-dimensional network data is shown in Fig. 3.

Feature selection algorithm for high-dimensional data

Let $X\in R^{p\times n}$, $Y\in R^{k\times n}$, $W\in R^{p\times k}$, $h$ represent the number of features for feature selection. The high-dimensional data feature selection model based on sparse constraint is defined as follows:

(20) $$\underset{W}{min}{\Vert Y-W^{T}X\Vert }_F^2,s.t.{\Vert W\Vert }_{2,0}=h$$

If $W=AB$, $A\in R^{p\times m}$, $B\in R^{m\times k}$, Formula (20) can be converted into the following

(21) $$\underset{A,B}{min}{\Vert Y-B^T{Z}^{T}X\Vert }_F^2,s.t.Z=A,A_{2,0}=h$$

The above formula, $B$ represents the class label regression reconstruction matrix.

The following constrained optimization problems are transformed from the feature selection of high-dimensional data:

(22) $$\underset{X}{min}f\left(X\right),s.t.tr\left(h\left(X\right)\right)=0$$

The augmented Lagrangian function of Formula (22) is expressed by the following formula:

(23) $$L\left(X,\mathrm{\Lambda },\mu \right)=f\left(X\right)+tr\left({\mathrm{\Omega }}^{T}h\left(X\right)\right)+{\displaystyle \frac{\mu }{2}{\Vert h\left(X\right)\Vert }_F^2}$$

According to the augmented Lagrange multiplier method, Formula (23) can be written as follows:

(24) $$\underset{B,Z,A}{min}{\Vert Y-B^T{Z}^{T}X\Vert }_F^2+Tr\left({\mathrm{\Omega }}^T\left(Z-A\right)\right)+{\displaystyle \frac{\mu }{2}{\Vert Z-A\Vert }_F^{2}s.t.{\Vert A\Vert }_{2,0}=h}$$

Simplify the Formula (24), and the result is expressed by the following formula:

(25) $$\underset{B,Z,A}{min}{\Vert Y-B^T{Z}^{T}X\Vert }_F^2+{\displaystyle \frac{\mu }{2}{\Vert Z-A+{\displaystyle \frac{\mathrm{\Omega }}{\mu }}\Vert }_F^2-{\displaystyle \frac{1}{2\mu }{\Vert \mathrm{\Omega }\Vert }_F^{2}s.t.{\Vert A\Vert }_{2,0}=h}}$$

The alternative iterative direction multiplier method alternates and iteratively optimizes high-dimensional data features to select the model variable $B,Z,A$. These rules are obtained by minimizing Formula (25) when other variables remain unchanged. Here are the particular steps:

Step 1: Fix $A$ and $Z$, optimize variables $B$. Remove irrelevant items by fixing $A$ and $Z$, and get the optimization problem about $B$. That is, to solve the following sub-problem of Formula (25):

(26) $$\underset{B}{min}{\Vert Y-B^T{Z}^{T}X\Vert }_F^2$$

Take the derivative of the variable $B$ in Formula (26) and set it to 0, that is, the solution of Formula (26) is:

(27) $$B={\left(Z^{T}X{X}^{T}Z\right)}^{-1}{Z}^{T}X{Y}^T$$

Step 2: Fix $A$ and $B$, optimize variables $Z$. By fixing $A$ and $B$, removing the irrelevant items, and getting the optimization problem about $Z$. That is, to solve the following sub-problem of Formula (25):

(28) $$\underset{B}{min}{\Vert Y-B^T{Z}^{T}X\Vert }_F^2+{\displaystyle \frac{\mu }{2}{\Vert Z-A+{\displaystyle \frac{\mathrm{\Omega }}{\mu }}\Vert }_F^2}$$

Let $D=A-{\displaystyle \frac{\mathrm{\Omega }}{\mu }}$, take the derivative of the variable $Z$ in Formula (28) and set it to 0, then we get:

(29) $$2X{X}^{T}Z+Z\mu {\left(B{B}^T\right)}^{-1}=\left(2X{Y}^T{B}^T+\mu D\right){\left(B{B}^T\right)}^{-1}$$

Formula (29) satisfies the Silvestre equation, so that

(30) $$O=2X{X}^T$$

(31) $$P=\mu {\left(B{B}^T\right)}^{-1}$$

(32) $$Q=\left(2X{Y}^T{B}^T+\mu D\right){\left(B{B}^T\right)}^{-1}$$

Then the solution of the variable $Z$ is:

(33) $$Z=sylvester\left(O,P,Q\right)$$

Step 3: Fix $B$ and $Z$, optimize variable $A$. Fix $B$ and $Z$, remove irrelevant items, and get the optimization problem about $A$. That is, to solve the following sub-problem of Formula (25):

(34) $$\underset{A}{min}{\Vert C-A\Vert }_F^{2}s.t.{\Vert A\Vert }_{2,0}=h$$

Considering the $L2,0$ constraint term of $A$ in Formula (34), this article calculates the two norms of each row vector of $C$, sorts the row vectors in descending order according to the value of the two norms, selects the row vectors corresponding to the largest first $h$ subscript indexes of $C$, assigns the row vectors corresponding to the largest first $h$ subscript indexes to $A$, and sets the row vectors not assigned to $A$ to 0, thus obtaining the solution of $A$.

The feature selection model of dimensional data constructed in this article’s fourth stage involves updating the parameters $\mathrm{\Omega }$ and $\mu$. The specific implementation process is as follows:

(35) $$\mathrm{\Omega }=\mathrm{\Omega }+\mu \left(Z-A\right)$$

(36) $$\mu =\mu \rho$$

The SDP-based feature selection method for high-dimensional network data is demonstrated in Fig. 4:

In this work, we design a feature selection algorithm for high-dimensional network data based on SDP. The sparse representation problem of high-dimensional network data is transformed into an Lp norm optimization problem, and the sparse subspace clustering method is used to cluster high-dimensional network data. Dimensionless processing is carried out for the clustering processing results. Based on the linear projection matrix and the best transformation matrix, the dimensionless processing results are reduced by combining the SDP. The sparse constraint method is used to achieve feature selection of high-dimensional data in the network, and the relevant feature selection results are obtained.

Experimental scheme

In order to verify the effectiveness of the high-dimensional network data feature selection algorithm based on SDP designed in this article, relevant experimental tests were carried out.

(1) To ensure the authenticity and reliability of the results obtained in this experiment, the experimental parameters must be unified. Therefore, the experimental environment parameters have been set in this experiment, as shown in Table 1.

The experimental software and hardware parameters of this document are as follows: CPU: Intel Xeon E5-2640, 10 cores; memory: 64 GB; hard disk HDD 10 TB, SSD 480 GB; network card: Broadcom NetXtreme Gigabit Ethernet; Windows 10 operating system; simulation software MATLAB 7.2.

(2) Several data sets including Dermatology, newsgroups, TDT2, Tumor14, AT&T, Palm, and Mpeg7 were selected for experimental testing. The experimental data set is described in Table 2.

From Table 2 we can see that we have selected a total of seven datasets: Dermatology, newsgroups, TDT2, Tumor14, AT&T, Palm, and Mpeg7, where the data types of the seven datasets are: UCI, Text, Text, Gene, Face, Other, and Shape. The number of samples, number of characteristics, and category are shown in Table 2.

(3) Using the Tian & Zhou (2020) algorithm, the Lin et al. (2021) algorithm, the Wang & Chen (2020) algorithm, and this algorithm as experimental comparison methods, we verify the application effects of different methods by confirming the indices of different algorithms. See Table 3 for indicator statistics.

From Table 3, it can be seen that we compare the Tian & Zhou (2020) algorithm, the Lin et al. (2021) algorithm, the Wang & Chen (2020) algorithm and the present algorithm in seven aspects: Algorithm convergence, clustering effect, recall ratio, precision ratio, F1 value, accuracy of network high-dimensional data feature selection, network high-dimensional data feature selection time, where algorithm convergence are described as follows. As a result of repeated iterations, the value obtained should not increase indefinitely but rather converge to a certain number. Algorithms that do not converge cannot be used; the clustering effect is described as follows: similar samples are close to each other. The further the different samples are from each other, the better the clustering effect is.

Experimental result

The comparison outcomes of the convergence of four network feature selection strategies for high-dimensional data are shown in Fig. 5.

Analyzing the convergence comparison results of the network high-dimensional data feature selection algorithms in Fig. 5, we can see that the errors of different algorithms change with the number of iterations. Among them, the Tian & Zhou (2020) algorithm has never converged, and the error is always kept at a high level, indicating that the convergence performance of the algorithm is very poor; When the number of iterations reaches 180, the Lin et al. (2021) algorithm achieves convergence, and the error of the algorithm is low, so the convergence performance of the algorithm is poor; When the number of iterations reaches 85, the Wang & Chen (2020) algorithm achieves convergence, and the error of this algorithm is high, so it is proved that the convergence performance of this algorithm is relatively poor. Compared with the three methods, when the number of iterations reaches 24, the algorithm in this article achieves convergence, and the error at this time is always kept at a low level, proving that the algorithm's convergence performance is optimal.

Figure 6 displays the comparison findings of the clustering impacts of four network high-dimensional data feature selection algorithms.

The Tian & Zhou (2020) technique can only effectively cluster high-dimensional data of some networks, and the majority of the data are dispersed, as shown by analysis of the findings in Fig. 6. This indicates that the algorithm’s clustering performance is subpar.

The Lin et al. (2021) algorithm can process the clustering of five different types of data, including dermatology, TDT2, tumor14, AT&T, and palm, according to an analysis of the results in Fig. 7, but it cannot process the clustering of all high-dimensional data on the network, and the clustering quality is subpar.

When the clustering processing of AT&T and Palm data is attempted using the Wang & Chen (2020) technique, the results in Fig. 8 reveal that the clustering quality is poor.

The findings of Fig. 9’s analysis demonstrate that the algorithm used in this article can handle clusters of seven different types of data, demonstrating the program’s effectiveness at clustering high-dimensional network data.

Table 4 compares the outcomes of various approaches for clustering high-dimensional data in terms of recall, precision, and F1 value.

The data analysis in Table 4 shows that compared with the experimental comparison algorithm, the average value of the memory comparison results in Tian & Zhou (2020) is 65.6%, the average value of the memory comparison results in Lin et al. (2021) is 86.4%, and the recall in Wang & Chen (2020) the average of the comparison results is 75.3%, and the average of the recall comparison results in the algorithm in this article is 97.2%. Compared with the experimental comparison algorithm, the method used in this article has the highest average recall comparison result.

Analyzing the information in Table 5, we can see that the average accuracy comparison result in Tian & Zhou (2020) is 75.4%, the average accuracy comparison result in Lin et al. (2021) is 87.5%, the average accuracy comparison result in Wang & Chen (2020) is 70.5%, and the average accuracy comparison result in the algorithm of this article is 95.7% in comparison to the test comparison algorithm. The algorithm used in this article’s accuracy comparison result has the greatest average value when compared to the experimental comparison algorithm.

We can observe from the data in Tables 4–6 that the recall, precision, and F1 values of the high-dimensional network data clustering results of the four approaches exhibit a varying pattern as the number of experiments increases. The Tian & Zhou (2020) algorithm has an average recall of 65.6%, the Lin et al. (2021) algorithm has an average recall of 86.4%, the Wang & Chen (2020) algorithm has an average recall of 75.3%, and the Vlachos & Thomakos (2021) algorithm has an average recall of 97.2%, the highest of the four approaches. The Tian & Zhou (2020) algorithm’s average precision is 75.4%, the Lin et al. (2021) algorithm’s average precision is 87.5%, the Wang & Chen (2020) algorithm’s average precision is 70.5%, and the Vlachos & Thomakos (2021) algorithm’s average precision is 95.7%, the greatest of the four approaches. The Tian & Zhou (2020) algorithm’s average F1 value is 70.1%, the Lin et al. (2021) algorithm’s average F1 value is 86.9%, the Wang & Chen (2020) algorithm’s average F1 value is 72.7%, and the average F1 value of this algorithm is 96.4%, the highest among the four ways.

The correct rate of network high-dimensional data feature selection of the four algorithms is compared, and the comparison results are shown in Table 7.

The high-dimensional network data in Tian & Zhou (2020) has a maximum feature selection accuracy of 86.6%, a minimum value of 83.7%, and an average value of 85.9%. By analyzing the data in Table 7, it can be seen that the maximum, minimum, and average feature selection accuracy of the high-dimensional network data in Tian & Zhou (2020) are 88.2%, 78.2%, and 83.2%, respectively.

The four algorithms’ feature selection times for high-dimensional network data are compared, and the comparison results are displayed in Table 8.

From the analysis of the results in Table 8, it can be seen that as the number of experimental samples increases, the number of feature selections of the four algorithms for high-dimensional network data shows an increasing trend. The average selection time of the Tian & Zhou (2020) algorithm high-dimensional network data features is 182.8 milliseconds, the average selection time of the Lin et al. (2021) algorithm high-dimensional network data features is 134.7 milliseconds, and the Wang & Chen (2020) algorithm high-dimensional network data feature selection time is 192.1 milliseconds. The method in this study has the shortest selection time and the highest efficiency, and the average network selection time for high-dimensional data features is 65.1 ms, which is the lowest among the four techniques.