Measuring node importance in complex networks has great theoretical and practical significance for network stability and robustness. A variety of network centrality criteria have been presented to address this problem, but each of them focuses only on certain aspects and results in loss of information. Therefore, this paper proposes a relatively comprehensive and effective method to evaluate node importance in complex networks using a multicriteria decision-making method. This method not only takes into account degree centrality, closeness centrality, and betweenness centrality, but also uses an entropy weighting method to calculate the weight of each criterion, which can overcome the influence of the subjective factor. To illustrate the effectiveness and feasibility of the proposed method, four experiments were conducted to rank node importance on four real networks. The experimental results showed that the proposed method can rank node importance more comprehensively and accurately than a single centrality criterion.
In recent years, the study of complex network theory has received sustained attention in various academic fields, such as aviation networks [
The term “important nodes” refers to certain nodes that can affect network structure and function to a greater extent than other nodes in the network [
Current methods of node importance analysis include social network methods and systems science methods [
Although existing centrality criteria have been widely used, they do have some shortcomings and deficiencies. Degree centrality (DC) [
Degree centrality, closeness centrality, and betweenness centrality of each node in the Kite network.
DC | CC | BC | |||
---|---|---|---|---|---|
node | value | node | value | node | value |
Diane | 0.6667 | Fernando | 0.6429 | Heather | 0.0160 |
Fernando | 0.5556 | Garth | 0.6429 | Fernando | 0.0099 |
Garth | 0.5556 | Diane | 0.6000 | Ike | 0.0093 |
Andre | 0.4444 | Heather | 0.6000 | Garth | 0.0080 |
Beverly | 0.4444 | Andre | 0.5294 | Diane | 0.0012 |
Carol | 0.3333 | Beverly | 0.5294 | Andre | 0.0010 |
Ed | 0.3333 | Carol | 0.5000 | Beverly | 0.0010 |
Heather | 0.3333 | Ed | 0.5000 | Jane | 0.0000 |
Ike | 0.2222 | Ike | 0.4286 | Ed | 0.0000 |
Jane | 0.1111 | Jane | 0.3103 | Carol | 0.0000 |
Kite network.
To overcome the shortcomings of node importance ranking using a single criterion, a range of criteria must be considered that affect node importance from different perspectives. Therefore, this paper proposes a multicriteria decision-making (MCDM) method to rank node importance effectively. MCDM has been widely used in many fields [
Therefore, the TOPSIS method and the entropy weighting method were combined to propose a novel method called EW-TOPSIS. The proposed method is based on degree centrality, closeness centrality, and betweenness centrality and computes node importance through integrated computation of these criteria. Because the measurement of node importance takes into account multiple factors that impact node importance without a one-sided emphasis on any single factor, the measurement is more accurate compared to when using a single criterion. Moreover, compared with other multicriteria decision-making methods, the proposed method uses an entropy weighting method to calculate the weight of each criterion, overcoming the disadvantage of the original TOPSIS using equal weights. This research solves the problem of partial and inaccurate ranking of node importance. It provides a beneficial supplement to network node importance measurement, enriches existing research on complex networks, and has great academic value. To demonstrate the effectiveness of the proposed method, four real networks (Zachary’s karate club, the dolphin social network, American college football, and jazz musicians) were used as experimental data. The susceptible-infected (SI) model [
The primary contributions of this paper can be summarized as follows:
(i) A novel method of node importance ranking in complex networks based on multicriteria decision making is proposed. It comprehensively combines the advantages of various criteria from different perspectives and makes the measurement more accurate and universal.
(ii) An entropy weighting method is proposed to calculate the weight of each criterion. It can overcome the influence of subjective factors and obtain an objective result.
(iii) Four experiments on four real networks have been conducted, and the experimental results show that the proposed method has superior performance in identifying important nodes in complex networks.
The rest of this paper is organized as follows. Section
An undirected network can be denoted as
The degree of node
The closeness centrality of node
The betweenness of node
The betweenness centrality of node
The above criteria can measure node importance, but they are a one-sided way to rank node importance with a single criterion. Therefore, to conduct a more comprehensive and objective ranking of node importance, a novel method is proposed here, integrating the above criteria based on the entropy weighting method and TOPSIS.
TOPSIS is a common method to solve multicriteria decision-making problems. Original TOPSIS has been used to identify important nodes [
If the set of nodes in a network is
For cost criteria, the standardization process can be expressed as
The normalized decision matrix can be denoted as
The entropy weighting method, which is used to calculate the weight of each criterion, determines the weight according to the variability of the criterion. The information entropy of the
When
Then the weighting coefficient of the jth criterion can be calculated as
The positive ideal solution
The closeness degree to the ideal solution can be calculated as
Combined with the above theoretical analysis, the specific steps of node importance ranking in complex networks can be given as in Algorithm
This section describes the use of four actual networks to verify the feasibility and effectiveness of the proposed method. (i) Zachary’s karate club [
In this experiment, the EW-TOPSIS method is used to identify the top 10 nodes based on the four actual networks, and the three centrality criteria DC, CC, and BC are also used for comparison. Table
Top 10 nodes ranked by DC, CC, BC and EW-TOPSIS.
Karate club | Dolphin | ||||||||
---|---|---|---|---|---|---|---|---|---|
Rank | DC | CC | BC | EW-TOPSIS | Rank | DC | CC | BC | EW-TOPSIS |
1 | 34 | 1 | 1 | 1 | 1 | 15 | 37 | 37 | 37 |
2 | 1 | 3 | 3 | 34 | 2 | 46 | 41 | 2 | 2 |
3 | 33 | 34 | 34 | 3 | 3 | 38 | 38 | 18 | 46 |
4 | 3 | 32 | 33 | 33 | 4 | 34 | 46 | 46 | 18 |
5 | 2 | 33 | 32 | 32 | 5 | 52 | 21 | 38 | 38 |
6 | 32 | 9 | 6 | 2 | 6 | 18 | 2 | 41 | 41 |
7 | 4 | 14 | 2 | 6 | 7 | 21 | 15 | 8 | 8 |
8 | 9 | 20 | 28 | 9 | 8 | 30 | 29 | 52 | 52 |
9 | 14 | 2 | 24 | 4 | 9 | 58 | 34 | 30 | 15 |
10 | 24 | 4 | 9 | 14 | 10 | 2 | 8 | 58 | 30 |
| |||||||||
Football | Jazz musicians | ||||||||
Rank | DC | CC | BC | EW-TOPSIS | Rank | DC | CC | BC | EW-TOPSIS |
| |||||||||
1 | 1 | 59 | 1 | 1 | 1 | 136 | 136 | 136 | 136 |
2 | 2 | 81 | 4 | 4 | 2 | 60 | 60 | 60 | 60 |
3 | 3 | 89 | 21 | 21 | 3 | 132 | 168 | 153 | 153 |
4 | 4 | 107 | 22 | 59 | 4 | 168 | 70 | 5 | 5 |
5 | 6 | 7 | 39 | 17 | 5 | 70 | 83 | 149 | 149 |
6 | 7 | 1 | 59 | 22 | 6 | 99 | 132 | 189 | 189 |
7 | 8 | 16 | 17 | 39 | 7 | 108 | 122 | 167 | 167 |
8 | 16 | 17 | 2 | 7 | 8 | 83 | 194 | 96 | 96 |
9 | 54 | 25 | 7 | 2 | 9 | 158 | 174 | 115 | 83 |
10 | 68 | 93 | 83 | 83 | 10 | 7 | 158 | 83 | 70 |
According to Table
By comparing the top 10 nodes using four methods based on four actual networks, it is clear that the ranking results of DC, CC, BC, and the EW-TOPSIS method are different. Different centrality criteria measure node importance from different perspectives. The EW-TOPSIS method comprehensively considers multiple criteria, and the ranking results are more scientific and reasonable.
Different nodes may have the same ranking, which makes it impossible to rank nodes with the same ranking accurately. For a node importance ranking method, the higher the frequency of the same ranking, the worse the performance of the ranking method. Therefore, the frequency of nodes with the same ranking can be used as an indicator to measure the performance of the method. The frequency of nodes with same ranking was compared using the four methods, with the results shown in Figure
Maximum frequency comparison of the four methods.
DC | CC | BC | EW-TOPSIS | |
---|---|---|---|---|
Karate club | 32.35% | 17.65% | 55.88% | 5.88% |
Dolphin | 14.52% | 4.84% | 25.81% | 3.23% |
Football | 57.39% | 6.96% | 0.87% | 0.87% |
Jazz musicians | 4.55% | 3.03% | 45.96% | 1.52% |
Frequency of nodes with the same ranking using the four methods.
According to Figure
In this experiment, the SI model is used to examine the infection ability of the top 10 nodes. The importance of nodes can be regarded as equivalent to infection ability; that is to say, the higher the importance of a node is, the stronger its infection ability will be. Therefore, the average infection ability of nodes can be used as an indicator to evaluate the effectiveness of a ranking method. In the SI model, every node has a susceptible state and an infected state; infected nodes infect susceptible nodes with a certain probability, and nodes cannot be recovered once infected. The infection source node
Average infection ability of the top 10 nodes between the EW-TOPSIS method and the centrality criteria.
In Figure
In addition, the spread time to reach a state of 90 percent infected nodes by the EW-TOPSIS method is 11 intervals, but by BC, it is 14 intervals. Clearly, the EW-TOPSIS method has better performance than DC and BC for both the average number of infected nodes and the spread velocity, and the performance of the EW-TOPSIS method is similar to that of CC.
In short, the proposed method has almost the same performance as DC and is slightly better than CC. In the case of BC, it is obvious that the proposed method performs better. Hence, the experimental results illustrate the effectiveness of the proposed method.
To compare further the ranking performance of the four methods, the average infection abilities of a single node at the same infection rate are compared, with the results shown in Figure
: Average infection ability of a single node.
For the karate club network, from Figure
The four experiments described above indicate that the proposed method has better performance than a single centrality criterion. Node importance is ranked with different methods, and the top 10 nodes of the four real networks are obtained. Based on the ranking results, the frequency of nodes with the same ranking is analyzed, and it is discovered that the proposed method has the lowest frequency; that is to say, the proposed method is more effective from this perspective. In addition, an indicator called the average infection ability is defined to describe the infection ability of the top 10 nodes, and the average infection ability of the top 10 nodes is obtained with the SI model. The proposed method performed better in terms of both infection scale and spread velocity. The infection abilities of a single node were also compared, and the results demonstrate the superiority of the proposed method.
Many criteria are used to rank node importance in complex networks, which considers only one aspect of networks. To solve this problem, a multicriteria decision-making method has been proposed here to perform a comprehensive evaluation of node importance. In this study, an entropy weighting method was used to obtain criterion weights that can overcome the subjective effect existing in other methods [
This paper proposes a novel method of node importance ranking based on an entropy weighting method and TOPSIS. The proposed method takes multiple centrality criteria as its decision criteria and uses an entropy weighting method to obtain the corresponding weight of each criterion, thus overcoming the effect of subjective factors. Multiple criteria were chosen from different perspectives on complex networks, and the advantages of each criterion were combined to obtain more objective and reasonable ranking results. To verify the effectiveness of the proposed method, four experiments based on four actual networks were conducted, and the SI model was used to simulate the spread ability of the top 10 nodes. The experimental results show that the proposed method has superior performance.
In this paper, the proposed method is applicable to undirected and unweighted networks, but a complex network may be directed and weighted in real life, and therefore a future research object of the authors is directed and weighted networks. Furthermore, experiment 2 showed that there are still some nodes with the same ranking; in such a case, how should their ranking be determined? In addition, some researchers have found that node importance is involved in dissemination mechanisms in addition to network topology. Therefore, future research will focus on a combination method of dynamic characteristics and network structure to measure node importance.
The simulation data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
The work described in this paper is supported by the National Natural Science Foundation of China (Grant No. 61472443). We thank International Science Editing (