How to seek the important nodes of complex networks in product research and development (R&D) team is particularly important for companies engaged in creativity and innovation. The previous literature mainly uses several single indicators to assess the node importance; this paper proposes a multiple attribute decision making model to tentatively solve these problems. Firstly, choose eight indicators as the evaluation criteria, four from centralization of complex networks: degree centrality, betweenness centrality, closeness centrality, and eigenvector centrality and four from structural holes of complex networks: effective size, efficiency, constraint, and hierarchy. Then, use fuzzy analytic hierarchy process (AHP) to obtain the weights of these indicators and use technique for order preference by similarity to an ideal solution (TOPSIS) to assess the importance degree of each node of complex networks. Finally, taking a product R&D team of a game software company as a research example, test the effectiveness, operability, and efficiency of the method we established.
All innovations begin as creative ideas or solutions, especially for product innovation; it is critical to generate creative ideas and develop them to the novel, valuable, and practical creative solutions in the frondend R&D process [
The nonhomogenous nature of complex network’s topological structure decides the large differences in the important degree of each node in the network [
Currently, there are so many methods used to assess the important nodes of complex networks [
Complex network can be represented by using figure
Here is the definitions of indicators we select.
Degree centrality is the basic parameters used to study the scalefree network topology, which can describe the direct impact by nodes in the static network; its value is the number of nodes directly connected by this node.
The degree centrality
In order to compare the centrality of different nodes, normalize the degree indicator:
Degree centrality reflects the ability of the node to directly obtain network flow content. The larger its value, the more important the node in the network.
Betweenness centrality characterizes influence of the network nodes on information flow. The betweenness centrality of node
Betweenness centrality can describe the important ability of nodes controlling the information flow between other nodes. The larger the betweenness centrality value of a node, the more information and other resources this node has and the more important the node in the network.
Closeness centrality characterizes the degree of difficulty that a node over the network has in connecting to other network nodes, and its value is defined as reciprocal of the sum of distances between the node and all other nodes.
The closeness centrality of node
As the sum of the distances between the node and all other nodes cannot be less than
Closeness centrality can be regarded as a measure of how long the information can spread from a given node to other reachable nodes in the network. The smaller the value, the smaller the time spread.
Eigenvector centrality takes into account the linear relationship between one node’s centrality indicators and the centrality indicators of other nodes around, which is the linear superposition of the centrality values of its adjacent nodes.
The eigenvector centrality of node
Structural holes can be defined as nonredundant links between two nodes in the network [
The effective size of one node is equal to the size of ego network minus the redundancy of network.
The effective size of node
The efficiency of a node is equal to the value through dividing effective size by actual size.
Constraint can be defined as the ability of the node to use structural holes in the network. The constrain of node
Hierarchy describes the degree of concentration of constraint on one node, which can be calculated by ColemanTheil disorder index:
According to the research background and definitions of each indicator, we apply the fuzzy AHP to get different indicator’s weights. In this paper, we use triangular fuzzy numbers [
We will use linguistic variables shown in Table
Definition and membership function of fuzzy scale.
Intensity of importance  Fuzzy number  Definition  Membership function 

9 

Extreme importance  (8, 9, 10) 
7 

Very strong importance  (6, 7, 8) 
5 

Strong importance  (4, 5, 6) 
3 

Moderate importance  (2, 3, 4) 
1 

Equal importance  (1, 1, 2) 
The procedure of fuzzy AHP is composed of three steps [
Construct pairwise comparison matrices among all indicators: degree centrality, betweenness centrality, closeness centrality, eigenvector centrality, effective size, efficiency, constraint, and hierarchy. And assign linguistic terms to the pairwise comparisons by asking which is more important of each two indicators, such as
Define the fuzzy geometric mean and fuzzy weights of each indicator using geometric mean technique by Buckley (1985) [
The procedure of defuzzification is to locate the best nonfuzzy performance value (BNP). We can use the following equation to calculate the BNP value of the fuzzy number:
Technique for order preference by similarity to an ideal solution (TOPSIS) is based on the concept that the chosen alternative should have the shortest distance from the positiveideal solution and the longest distance from the negativeideal solution. We will use this method to rank the node importance of complex networks, which is composed of seven steps as follows.
Suppose there are
There are some complex relationships between indicators, and the dimension of each indicators may be different, so we need to normalize the indicators. of eight indicators of this study, six indicators such as degree centrality, betweenness centrality, closeness centrality, eigenvector centrality, effective size, and efficiency belong to the benefit type indicators, while two indicators, constraint and hierarchy, belong to the cost type indicators. We use the following method to normalize the decision matrix
if the indicator belongs to benefit type and use the following method to normalize the decision matrix
if the indicator belongs to cost type, where
On the basis of normalized decision matrix, construct the weighted normalized decision matrix as the following formula:
Determine the positiveideal node and negativeideal node:
Calculate the distance of each node from the positiveideal node and negativeideal node:
Obtain the closeness coefficients:
In our study, we choose a product R&D team coming from one game software company as a research example. There are 14 members in this team, and the members always communicate with each other in the creative activities. In order to get the accurate data from the members and team, we collect data from different paths and methods, mainly including three paths: questionnaire, email and other social media records, and daily facetoface meeting records. The data collected by the latter two methods is objective; we can get it by directly collating existing records. The data collected by the questionnaire is subjective, and the data provided by one pair of investigators may be different. For example, when we asked the investigator
Statistical data of members exchange.
M1  M2  M3  M4  M5  M6  M7  M8  M9  M10  M11  M12  M13  M14  

M1  0  2  7  9  11  12  3  4  0  9  3  9  8  2 
M2  2  0  9  10  8  13  0  2  2  9  3  8  12  10 
M3  7  9  0  7  6  9  2  0  4  5  3  9  9  9 
M4  9  10  7  0  6  15  0  3  2  11  0  5  7  8 
M5  11  8  6  6  0  3  7  0  2  2  3  3  8  0 
M6  12  13  9  15  3  0  5  11  0  12  2  11  9  10 
M7  3  0  2  0  7  5  0  5  9  10  8  2  4  2 
M8  4  2  0  3  0  11  5  0  9  14  12  8  0  6 
M9  0  2  4  2  2  0  9  9  0  5  0  0  0  2 
M10  9  9  5  11  2  12  10  14  5  0  9  7  11  11 
M11  3  3  3  0  3  2  8  12  0  9  0  2  0  7 
M12  9  8  9  5  3  11  2  8  0  7  2  0  9  0 
M13  8  12  9  7  8  9  4  0  0  11  0  9  0  2 
M14  2  10  9  8  0  10  2  6  2  11  7  0  2  0 
Network structure of 14 members.
Firstly, we should get the weights of eight indicators by using the fuzzy AHP method. We construct pairwise comparison matrix as follows:
Use (
Similarly, we can obtain the other indicators’ fuzzy geometric means as follows:
The fuzzy weights of
Similarly, we can obtain the other indicators’ fuzzy weights as follows:
Use COA method to take the BNP value of the weight of “degree centrality” indicator as the example; the calculation process is as follows:
Similarly, we can obtain the other dimensions’ BNP values in the first stage’s evaluation criteria system as follows:
As the sum of eight indicators’ BNP is 1.155, we normalize them and get their weights:
Secondly, we calculate each node’s performance value in eight indicators; the results are as shown in Table
The performance values of eight indicators of 14 team members.
DC  BC  CC  EC  ES  EF  CO  HI  

M1  40.513  1.217  92.857  39.164  6.079  0.507  0.332  0.077 
M2  45.128  1.842  92.857  43.790  6.332  0.528  0.338  0.093 
M3  40.513  1.656  92.857  38.005  6.511  0.543  0.330  0.068 
M4  42.564  1.374  86.667  43.059  5.193  0.472  0.366  0.091 
M5  30.256  1.400  86.667  27.655  5.784  0.526  0.357  0.063 
M6  57.436  1.217  92.857  54.309  6.401  0.533  0.314  0.061 
M7  29.231  1.700  86.667  24.472  6.280  0.571  0.348  0.076 
M8  37.949  1.202  81.250  33.999  5.645  0.565  0.374  0.093 
M9  17.949  0.644  72.222  15.260  4.608  0.576  0.488  0.127 
M10  58.974  2.087  100.000  52.056  7.481  0.575  0.284  0.046 
M11  26.667  0.644  81.250  24.011  5.340  0.534  0.404  0.114 
M12  37.436  0.961  86.667  37.437  5.498  0.500  0.363  0.077 
M13  40.513  0.501  81.250  40.314  4.549  0.455  0.378  0.052 
M14  35.385  1.505  86.667  35.143  5.648  0.513  0.374  0.105 
DC: degree centrality, BC: betweenness centrality, CC: closeness centrality, EC: eigenvector centrality, ES: effective size, EF: efficiency, CO: constraint, and HI: hierarchy.
The normalized results of eight indicators of 14 team members.
DC  BC  CC  EC  ES  EF  CO  HI  

M1  0.687  0.583  0.929  0.721  0.813  0.880  0.855  0.597 
M2  0.765  0.883  0.929  0.806  0.846  0.917  0.840  0.495 
M3  0.687  0.793  0.929  0.700  0.870  0.943  0.861  0.676 
M4  0.722  0.658  0.867  0.793  0.694  0.819  0.776  0.505 
M5  0.513  0.671  0.867  0.509  0.773  0.913  0.796  0.730 
M6  0.974  0.583  0.929  1.000  0.856  0.925  0.904  0.754 
M7  0.496  0.815  0.867  0.451  0.839  0.991  0.816  0.605 
M8  0.643  0.576  0.813  0.626  0.755  0.981  0.759  0.495 
M9  0.304  0.309  0.722  0.281  0.616  1.000  0.582  0.362 
M10  1.000  1.000  1.000  0.959  1.000  0.998  1.000  1.000 
M11  0.452  0.309  0.813  0.442  0.714  0.927  0.703  0.404 
M12  0.635  0.460  0.867  0.689  0.735  0.868  0.782  0.597 
M13  0.687  0.240  0.813  0.742  0.608  0.790  0.751  0.885 
M14  0.600  0.721  0.867  0.647  0.755  0.891  0.759  0.438 
Thirdly, using (
The positiveideal node
The negativeideal node
Finally, use (
Use (
Similarly, we can obtain the other thirteen nodes’ closeness coefficients as shown in Table
Closeness coefficients to aspired level among different nodes.




Ranking  

M1  0.092  0.111  0.453  0.547  6 
M2  0.092  0.122  0.430  0.570  5 
M3  0.085  0.115  0.425  0.575  4 
M4  0.102  0.110  0.481  0.519  7 
M5  0.120  0.084  0.588  0.412  10 
M6  0.043  0.173  0.199  0.801  2 
M7  0.132  0.074  0.641  0.359  12 
M8  0.119  0.085  0.583  0.417  9 
M9  0.198  0.014  0.934  0.066  14 
M10  0.006  0.196  0.030  0.970  1 
M11  0.159  0.044  0.783  0.217  13 
M12  0.105  0.096  0.522  0.478  8 
M13  0.089  0.122  0.422  0.578  3 
M14  0.125  0.082  0.604  0.396  11 
From Table
Furthermore, we found the member of node 6 is the opinion leader in this product R&D team. The years he severed in this company are the longest, and he has a wealth of experience in product development and creative activities. In the creative solution formation process of the frontend innovation, when the other members generate the new ideas, they are accustomed to seek his advice, and when they meet some difficult things in work, they like to seek help from him. So the member of node 6 is the second most important node in this team, which is consistent with our analysis results.
The members of nodes 1, 2, 3, 4, and 13 have the relative strong importance degree in the complex networks of this product R&D team; they are the key members engaged in generating and developing creative ideas or creative solutions, which is the core work in the whole product innovation process. The members of nodes 5, 7, 8, 11, 12, and 13 are less important than the members previously mentioned, and the member of node 9 has the least importance degree in the team, who rarely participates in team work. Hence, in the actual management, the company leader and management can focus more on the members of nodes 1, 2, 3, 4, 6, 10, and 13, who are directly related to the team results.
Assessing the node importance of complex networks in product R&D team and seeking the member who is the most important in the team is particularly important to the companies engaged in creativity and innovation. Pervious studies have developed and established some methods in order to solve these issues, and they propose many assessment indicators. However, when we are assessing the node importance of complex network, we cannot give out the results depending on several separate variable; we need to comprehensively evaluate the node importance simultaneously taking into account a number of indicators. In this paper, we choose eight indicators from two aspects: centralization and structural holes of complex networks, such as degree centrality, betweenness centrality, closeness centrality, eigenvector centrality, effective size, efficiency, constraint, and hierarchy. We combine the fuzzy AHP and TOPSIS and establish the multiple attribute decision making model to help us seek the important nodes of complex networks. Then we took a product R&D team of a game software company as a research example; we used the model we established to seek the important members in this team and successfully find out the important members who are consistent with the fact. We confirmed the effectiveness, operability, and efficiency of the method used to seek the important nodes of complex networks.
The authors do not have any conflict of interests with the content of the paper.
This research is supported by the National Natural Science Foundation of China (Grant no. 71273076), Humanities and Social Sciences Youth Foundation of Chinese Ministry of Education (Grant no. 3YJC630166), and the Natural Science Foundation of Heilongjiang Province of China (Grant no. G201006).