A LSGDM method based on social network and IVIFN’s geometric characteristics for evaluating the collaborative innovation problem 1

2.1 Definitions about fuzzy sets

In this research, the first question discussed is determining a tool to describe the decision-making information. One priority selection is introducing the fuzzy numbers. Definitions about IFS and IVIFS are given below. which is better to reflecting DMs opinions.

Definition 1. [26] An intuitionistic fuzzy set (IFS) R = { < x , μ R x , ν R x > | x ∈ R } is defined on a fixed set R, satisfying μ _R  (x) : X → [0, 1] , x ∈ X → μ _R  (x) ∈ [0, 1], ν _R  (x) : X → [0, 1] , x ∈ X → ν _R  (x) ∈ [0, 1], where μ _R  (x) and ν _R  (x) stand for the membership degree and non-membership degree, respectively, with μ _R  (x) + ν _R  (x) ≤1, ∀x ∈ R.

Furthermore, let μ A ¯ ( x ) = [ μ A ¯ L ( x ) , μ A ¯ R ( x ) ] , ν A ¯ ( x ) = [ ν A ¯ L ( x ) , ν A ¯ R ( x ) ] . Then the function π A ¯ ( x ) = [ π A ¯ L ( x ) , π A ¯ R ( x ) ] stands for the uncertainty degree of x to X, satisfying π A ¯ L ( x ) = 1 - μ A ¯ R ( x ) - ν A ¯ R ( x ) and π A ¯ R ( x ) = 1 - μ A ¯ L ( x ) - ν A ¯ L ( x ) . 〈 [ μ A ¯ L ( x ) , μ A ¯ R ( x ) ] , [ ν A ¯ L ( x ) , ν A ¯ R ( x ) ] 〉 is called interval valued intuitionistic fuzzy number (IVIFN) for simple. The IVIFN will degenerate to intuitionistic fuzzy number (IFN) when μ A ¯ L ( x ) = μ A ¯ R ( x ) and ν A ¯ L ( x ) , ν A ¯ R ( x ) , whose precise definition is as below. For the reason that, IFN has better comparability, in Section 3 constructs a transform function to connect the IVIFN and IFN.

Definition 2. [27] An interval-valued intuitionistic fuzzy set (IVIFS) A = { 〈 x , μ A ¯ ( x ) , ν A ¯ ( x ) 〉 | x ∈ X } is defined on a fixed set X, satisfying μ A ¯ ( x ) : X → L [ 0 , 1 ] , ∀ x ∈ X ; ν A ¯ ( x ) : X → L [ 0 , 1 ] , ∀ x ∈ X ; supμ _A  (x) + supν _A  (x) ≤1, ∀x ∈ X, where μ A ¯ ( x ) and ν A ¯ ( x ) stand for the membership degree and non-membership degree, respectively.

2.2 Definitions about social net constructing

In this section, some basic definitions about social analysis are given. Suppose T decision makers D = {DM₁, DM₂, . . . , DM _T }, who will be regarded as T nodes, belongs to set V = {v₁, v₂, . . . , v _T }. Edge e (d _p , d _q ) exists when decision maker d _p and d _q has a certain connection, and all connections compose a set E = {e _pq , p, q ∈ {1, 2, . . . , T}}. The value of e (d _p , d _q ) is calculated by Equation (1). The number of edges adjacent to node v _t , t ∈ {1, 2, . . . , T} is called its degree, written as d _t . This paper considers simple graph, that is to say, the direction and weight of the edges are not distinguished. Then the network is constructed by the edges and nodes.

e pq = { 1 if d p and d q has a connection ; 0 otherwise .(1)

Centrality is usually used to describe the position of a node. One of the most direct concepts is degree centrality. Suppose a network with T nodes, whose biggest degree is T - 1. For convenience of comparison, the centrality index is normalized.

Definition 3. [44] The degree centrality of node t is defined by Equation (1) whose degree is d _t . DC t = d t T - 1(2)

Eigenvector centrality’s basic idea is about the importance of a node. It depends on both number of its neighboring nodes and importance of its neighboring nodes.

Definition 4. [44] Let EC _t stands for the importance degree of node t, then EC t = c ∑ j = 1 N a tj x j(3) where c is a constant number, A = (a _tj ) is the adjacent matrix, j ∈ {1, . . . , N}.

Denote x = [x₁, x₂, . . . , x _T ]^′, then Equation (3) could be written as below. x = c Ax(4) where x means the eigenvector of matrix A and eigenvalue c^-1. That is why call it Eigenvector-centrality.

3.1 Aggregating algorithm constructed with network analysis

An important definition in this section is Modularity, which is used to deal with the aggregating problem, defined by Equation (5):

Δ Q = [ W c + s i , in 2 W - ( S c + s i 2 W ) 2 ] - [ W c 2 W - ( S c 2 W ) 2 - ( S i 2 W ) 2 ](5) where s_i,in is the edge numbers of node i and the other nodes in community C. Aggregating results will be calculated by fast greedy method based on the modularity gain degree. The fast greedy method’s core idea is proposed by Newman [32].

A principle of network analysis in the LSGDM problem is that let each node be a separate community initially. In order to maximize modularity of the whole community, calculate the biggest local contribution of each node community. The basic algorithm process contains four steps.

Algorithm 1

Step 1. Let each nodes to be an original community.

Step 2. Based on the modularity, some neighbors are determined to be merged, after one iteration.

Step 3. Each community is regarded as one new node. And calculate its degree and connect information, that is the basic for next iteration.

Step 4. Repeat step 3 until the modularity of the community no longer increases.

In order to prevent the dense inside and sparse connection to the outside, determine the subgroups’ weight vector based on two aspects. One aspect is considering the nodes degree and eigenvector centralities; the other aspect is the influence of a subgroup to the others. In a subgroup, one node has two kinds of edges, an inside one and an outside one. The inside one connects nodes belong to the same subgroup. And the outside one connects two kinds of nodes, where one belongs to the subgroup and the other one belongs to the outside. A subgroup may get high weight, for the nodes contained in this subgroup with high degree. But the connections of this subgroup and the outside nodes may be not as dense as the inside. Depending on the aggregating method, all T nodes will be separated to S parts, each subgroup has Q _s nodes, s ∈ {1, 2, . . . , S}. The main progresses for determining the subgroups’ weight are as follows.

Algorithm 2

Step 1. Calculate each node’s degree centrality (DC) and eigenvector centrality (EC). Combine the two centralities to get a node weight λ_node,t for node t, by Equation (6). The normalized weight is calculated by Equation (7).

λ node , t = DC t · EC i(6)

λ node , t ¯ = λ node , t ∑ t = 1 T λ node , t(7)

Based on all nodes weight, the network’s average centrality λ _net is computed by Equation (8) and the subgroup’s average centrality λ comm s is computed by Equation (9):

λ net = 1 T ∑ t = 1 T λ node , t(8)

λ comm s = 1 S t ∑ t = 1 S t λ node , t(9)

A subgroup’s inside weigh λ_in,s is determined by the distance of this subgroup’s average centrality and the network’s average centrality, computed by Equation (10).

λ in , s = 1 λ comm s - λ net(10)

And normalize λ_in,t by Equation (11), obtaining λ in , s ¯ .

λ in , s ¯ = λ in , s ∑ s = 1 Q s λ in , s(11)

Step 2. Suppose there are Q _s nodes in subgroup C _s . Calculating the number of edges for all Q _s nodes, which connects the inside nodes and outside nodes, denoted as λ_out,s. Normalize λ_out,s and get λ out , s ¯ , a outside weight λ_out,s, by Equation (12).

λ out , s ¯ = λ out , s ∑ s = 1 S λ out , s(12)

Step 3. Combine the inside weight and outside weight of the network together, by Equation (13), obtaining subgroup’s weight λ. α is moderator variables which can adjust the importance degree of λ in , s ¯ and λ out , s ¯ .

λ = α λ in , s ¯ + ( 1 - α ) λ out , s ¯(13)

4.1 A new score function based on IVIFNs

This paper proposes distance measure from the perspective of geometric center of gravity, a novel score function defined after it, considering the IVIFN’s geometry characteristics. As IVIFN could be understood as an extension of IFN, give a transform function to connect the IFN and IVIFN. Let α = 〈 [μ^-, μ⁺] , [ν^-, ν⁺] 〉 be an IVIFN. l _PQ is a line defined by x + y = 1. P and Q are the intersection of l _PQ and coordinate axis. Coordinates of A, B, C and D are A = (μ^-, ν^-), B = (μ⁺, ν^-), C = (μ⁺, ν⁺) and D = (μ^-, ν⁺), respectively. Lengths of L _AB , L _CD , L _AD and L _BC are L _AB  = L _CD  = μ⁺ - μ^- and L _AD  = L _BC  = ν⁺ - ν^-. Obviously, the quadrilateral determined by points A, B, C and D is a rectangle. So, its barycentric coordinate is G = ( μ - + ( μ + - μ - ) 2 , ν - + ( ν + - ν - ) 2 ) , which could be regard as the most representative point of the decision maker’s preference. For the reason that different rectangles may obtain the same barycenter, some other parts contained in triangle determined by points O, P and Q.

Name the region determined by point A, B, C and D “possible region”. Name the region determined by point C, E and F “impossible region”. And name the region determined by point O, N, C and T “possible fuzzy region”. There regions are shown in Figure 1. Next part introduces a ratio which measures the uncertainty degree of the decision making results.

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Fig. 1

The barycenter of IVIFN shown in rectangular coordinate system.

Definition 5. Let α = 〈 [μ^-, μ⁺] , [ν^-, ν⁺] 〉 be an IVIFN, written as α = 〈 [a, b] , [c, d] 〉 for simple. Define R = ( b - a ) × ( d - c ) + 1 2 × ( 1 - b - d ) 2 b × d + 1 2 × ( 1 - b - d ) 2(14) IVIFN’s Certainty Ratio.

Definition 6. An INIFN α = 〈 [a, b] , [c, d] 〉 can be transformed to an IFN-from β ˜ = ( u , v ) based on the following Equation (15),

β = ( u , v ) = ( a + ( b - a ) / 2 × R , c + ( d - c ) / 2 × R )(15) where R is the INIFN’s Certainty Ratio.

Next, a theorem is given to prove the novel "IFN" obtained by Definition 4.1 satisfies the formal IFN’s conditions.

Theorem 7. Let α = 〈 [a, b] , [c, d] 〉 be an INIFN. The number β ˜ = ( u , v ) obtained by Equation (15) is an IFN.

Prove

Let α = 〈 [a, b] , [c, d] 〉 be an INIFN. From the Theorem 1,

R = ( b - a ) × ( d - c ) + 1 2 × ( 1 - b - d ) 2 b × d + 1 2 × ( 1 - b - d ) 2 ,

β ˜ = ( u , v ) = ( a + ( b - a ) / 2 × R , c + ( d - c ) / 2 × R ) .

Based on the definition of INIFN, cases are classified by value of b + d.

i) b + d = 1.

Because b + d = 1, 1 - b - d = 0 and a ≤ b, c ≤ d, a, c ∈ [0, 1], R = ( b - a ) × ( d - c ) + 0 b × d + 0 = ( b - a ) × ( d - c ) b × d , obtain 0 ≤ R ≤ 1. In addition, R = 0 if a = b or c = d; and R=1 if a = 0 and c = 0.

ii) b + d < 1.

Because b + d = 1, 1 - b - d = 0 and a ≤ b, c ≤ d, a, c ∈ [0, 1], Then

(b - a) × (d - c) ≤ b × d;

( b - a ) × ( d - c ) + 1 2 × ( 1 - b - d ) 2 ≤ b × d + 1 2 × ( 1 - b - d ) 2 .

obtaining 0 ≤ R ≤ 1. In addition, R=1 if a = 0 and c = 0.

iii) b + d = 0. The decision result is reduced to a number 0. This condition will not be considered in the following procedure. That is to say, all evaluating results are positive real number.

In all, 0 ≤ R ≤ 1. Next will discuss the ranges of u + v.

Firstly,

u = a + b - a 2 × R ≤ a + b - a 2 = a + b 2 ≤ 1 ;

v = c + d - c 2 × R ≤ c + d - c 2 = c + c 2 ≤ 1 ;

then u ∈ [0, 1] and v ∈ [0, 1].

Secondly, calculate the value of u + v.

u + v = ( a + c ) + R 2 ( b + d - a - c ) ≤ ( a + c ) + 1 2 ( 1 - a - c ) = 1 + a + c 2 ≤ 1 + b + d 2 ≤ 1

Finally, conclusion of Theorem 1 could be proved.

Depending on Definition 4.1 and Definition 4.1, introduce a new score function for INIFNs. And any different INIFNs could be compare by their score values.

In addition, π _R  (x) =1 - μ _R  (x) - ν _R  (x) is defined as the uncertainty degree of x to R. If π _R  (x) =0, R is reduced to a fuzzy set which is proposed by Zadeh [29].

For the reason that, two different IFNs cannot be compared directly. In order to measure the difference degree between two different IFNs, some distance measures have been proposed. Researches about this comparison problem are mainly related to calculate the distance of one certain fuzzy numbers.

Recently, You and Yang [47] give a distance function h ¯ ( α , β ) to measure any two IFNs’ difference degree, which is more effective for classifying IFNs, shown as below.

h ¯ ( α , β ) = | μ A ( x i ) - μ B ( x i ) | + | ν A ( x i ) - ν B ( x i ) | + | π A ( x i ) - π B ( x i ) | + | S A ( x i ) - S B ( x i ) |(16)

Definition 8. Let α be an INIFN, written as α = 〈 [a, b] , [c, d] 〉 without confusion. Its new score function based on gravity center is given by Equation (17).

f ( score ) = ( a + ( b - a ) / 2 × R ) + ( c + ( d - c ) / 2 × R )(17) where R is the INIFN’s Certainty Ratio.

In order to show the effectiveness of the novel Score function, give the following Example 1 from [40], which also be used by references [15] and [38].

Example

The weights of Criterion 1, Criterion 2, Criterion 3 are determined after the original papers as "0.10, 0.65, 0.25", and the alternatives’ decision results are shown in Table 1. Comparison results from different score functions are shown in Figure 2, with methods from [49], [40], [15] and [38] and novel score function proposed by this paper. From Figure 2, it is obvious to see that, the novel score function is able to compare the four alternatives. However, the comparison results are different with each other. Then, in order to show the effectiveness of Certainty Ratio, next will do more research.

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Fig. 2

Comparison results of different Score functions.

In order to determine the criteria’s weights and the decision makers weights, studying the characteristics of IVIFNs made by decision makers, when dealing with the multi-criteria group decision making problem. It is easy to understand that the criterion’s weight will be higher when the decision result is more accurate based on it. And the criterion’s weight will be higher when its deviation degree is higher. Depending this idea, the function H (β) = μ + ν means accurate degree for IFN β = (μ, ν). Let C = {C₁, C₂, …, C _n } be a set with n criteria, j ∈ N, N = {1,2, …, n}, X = {X₁, X₂, …, X _m } be a set with m alternatives with i ∈ M, M = {1, 2, …, m}. Decision results will be give in the following form as Equation (18). a _ij stands for decisions of alternative X _i made by decision maker considering criterion C _j . For any criteria C _j , its accurate degree is determined by Equation (19).

A ij = [ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a n 2 ⋯ a mn ](18)

f ( w j ) = ∑ i = 1 m w j ( μ ij + ν ij )(19)

Model 1 is constructed to ensure the maximum accurate degree.

Model 1 { max f 1 ( w ) = ∑ i = 1 m ∑ j = 1 n w j ( μ ij + ν ij ) s . t . ∑ j = 1 n w j = 1 0 ≤ w j ≤ 1 .(20)

And the deviation degree is obtained by Equation (21), based on the distance equation (16).

f ( w j ) = ∑ i = 1 m ∑ 1 ≤ i < k ≤ m w j d ( a ij , a kj )(21)

Model 2 is constructed to ensure the maximum deviation degree.

Model 2 { maxf 2 ( w ) = ∑ i = 1 m ∑ j = 1 n ∑ 1 ≤ i < k ≤ m w j d ( a ij , a kj ) s . t . ∑ j = 1 n w j = 1 0 ≤ w j ≤ 1 .(22)

Taking the accurate degree and deviation degree into consideration simultaneously, the following Model 3 is given to calculate the criteria’s weights, where β standing for the weight of accurate degree determined by actual requirement.

Model 3 { max f 3 ( w ) = β f 1 ( w ) + ( 1 - β ) f 2 ( w ) s . t . ∑ j = 1 n w j = 1 0 ≤ w j ≤ 1 .(23)

Next, the criteria’s weights of Model 3 is calculated by constructing a lagrangian function L (w, λ).

L ( w , λ ) = f ( w ) + λ ( ∑ j = 1 n w j - 1 )(24)

Take the derivative of w _j and λ simultaneously.

{ ∂ L ∂ w j = ∑ i = 1 m ( β ( μ ij + ν ij ) ) + ∑ 1 ≤ i < k ≤ m ( 1 - β ) d ( a ij , a kj ) ) + λ = 0 ; ∂ L ∂ λ = ∑ j = 1 n w j - 1 = 0 .(25)

The criteria’s weight is obtained by Equation (26).

w j = ∑ i = 1 m ( β ( μ ij + ν ij ) + ∑ 1 ≤ i < k ≤ m ( 1 - β ) d ( a ij , a kj ) ) ∑ j = 1 n ∑ i = 1 m ( β ( μ ij + ν ij ) + ∑ 1 ≤ i < k ≤ m ( 1 - β ) d ( a ij , a kj ) ) .(26)

Decision maker’s weight is determined by his/her position and evaluation quality. The position aspect could be obtained by the node’s weight from Equation (26). The evaluation quality is represented by the deviation degree of decision result, which is calculated based on Equation (21). The weight of decision maker is computed by Equation (21), introducing the different value ρ.

w DM t = ρ λ node , i ¯ + ( 1 - ρ ) ∑ i = 1 m ∑ 1 ≤ i < k ≤ m w j d ( a ij , a kj )(27)

Depending on the criteria’s weight and decision makers’ weights, each community’s weight is calculated by Equation (28), where a ij tl standing for the evaluation of decision maker DM _l from community t with alternative Alter _i under criterion C _j . Let B t = ( b i t ) m × 1 be the community’s representing matrix. Then the alternative A _i could obtain a weighted decision result by Equation (29) where b i s is the definition of an IFN.

b i s = ∑ j = 1 n ∑ l = 1 S t w DM l ( a ij sl )(28)

b i = ∑ s = 1 S λ s b i s(29)

Let b _i  = (μ _i , ν _i ), satisfying the definition of an IFN. The alternatives’ will de ranked by values calculated by Equation (30).

a i = μ i - ν i(30)

6.1 An example about coordinated development for a collaborative innovation problem

In China, innovation is treated as the first driving force leading development. The Chinese government recognizes the importance of innovation from a strategic perspective, believing that innovation supports the construction of a modern economic system. Under the current economic development situation, the competition trend of technological innovation among countries, regions, industries and enterprises is particularly obvious. Due to the constraints of resources, technology, and knowledge, it is difficult to innovate independently by relying on only one region, one industry or one enterprise. Therefore, the innovation goals will be easier to achieve, through integrating resources between regions, industries and enterprises. During the developing process, the strengths will be maximized and weaknesses will be avoided, with discovering advantages and disadvantages and and avoid.

There are three regions named B, T and H for simple. Their economic development levels from high to low are Region-B, Region-T and Region-H. Because independent development of the three regions has encountered a certain of bottlenecks. It is significant to implement coordinated development, that is treating three regions as a whole. The B-T-H region has intensive intellectual resources, strong technological complementarity, a solid manufacturing base, and rapid development of high-tech industries. This combine-region is a core demonstration zone for China to participate in international competition and promote collaborative innovation. It is also a strategically positioned part of China’s three major economic zones. The coordinated development of the B-T-H region is necessary to realize the complementary advantages of the B-T-H region, promoting the development of a bigger economic zone and leading the development of the northern hinterland. It is a major national strategy. Systematically study on the participants’ degree of collaboration, will encourage the further gathering of innovative talents in Region-B, Region-T and Region-H, sharing innovative elements and interconnect technology market. Then, transfer and transformation of scientific and technological achievements will also be improved, increasing the success rate of innovation activities and the quality of transformation. In addition, promote the sustainable development of the economy. However, the economic development levels and industrial structures of Region-B, Region-T and Region-H are different. Although government proposed a policy for the coordinated development of B-T-H. This case hope to show that whether it is suitable to adjust development structure slightly. Depending on the reality, it is hard to develop the three regions in perfect synchronization. Propose four combinations as alternatives: 1. develop Region-B, Region-T and Region-H as Region-B-T-H simultaneously; 2. prioritize the implementation of coordinated development of region-B and region-T as Region-B-T simultaneously, then Region-H; 3. prioritize the implementation of coordinated development of region-B and region-H as Region-B-H simultaneously, then Region-T; 4. prioritize the implementation of coordinated development of region-T and region-H as Region-T-H simultaneously, then Region-B.

In order to analysis which development structure is better, 30 experts are invited to evaluate the criteria, as shown in Figure 4. Build an evaluation standard system, after discussing with the invited experts and our project team staff.

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Fig. 4

The criteria considered by decision makers.

The decision makers’ origin social network is shown in Figure 5.

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Fig. 5

The original decision makers network.

The original decision results of DM₁ is given by Table 2. And the remaining 29 are given in Appendix A.1, Table A1-A29.

Applying the proposed aggregating method of Section 4, the decision makers are parted to 4 subgroups, shown in Figure 6. Let α = 0.5, meaning the network’s inside weight and outside weight are equal important. Let β = 0.5, meaning the accurate degree and deviation degree with equal importance degree. Let ρ = 0.5, meaning the decision makers’s relationships and evaluation results have the same importance degree. The 4 alternatives’ evaluation results are (0.3430, 0.0867), (0.3739, 0.0657), (0.3221, 0.1097), (0.2897, 0.1356). And the final ranking values are 0.2564, 0.3082, 0.2124, 0.1541, meaning Alter2 ≻ Alter1 ≻ Alter3 ≻ Alter4. From the ranking results, the second alternative is the best one. That is to say, developing Region-B and Region-T first, then Region-H, has the highest creative degree. This result could be explained by the three regions’ economic development levels, where Region-B and Region-T are better than region-H. At the same time, Region-B’s innovation output can be better undertaken by Region-T, the two regions technology industries will have a better expected performance from the perspective of innovation collaboration. Region-H is a traditional agricultural major province. Prioritize the development of B and H, from the perspective of recent economic benefits, which has advantages over other programs. However, from the perspective of long-term economic and social benefits, the second option of simultaneously considering the three regions is more competitive. Because Region-B and Region-T have a smaller area. Many basic needs rely on external support. Especially, relying on the Region-H. Region-B is about to encounter various "big city diseases". In China, the coordinated development of the three places is currently mainly option. At present, certain phased results have been achieved, but the synergy effect needs to be improved.

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Fig. 6

The aggregating result of the proposed method.

Apply the LSGDM method proposed by T. Wu et al [45], who also considering the social relations of decision makers, obtain 3 subgroups, shown in Figure 7.

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Fig. 7

The aggregating result of Lovain method.

Apply the LSGDM method proposed by You and Yang [47], who only discussed the decision results and did not considered the social relations. Figure 8 shows the numbers of aggregating result will be stable after a large number of evolutions. Determine the threshold value 0.76, after trying different threshold values from 0.75, 0.76, 0.77, 0.78, 0.79. The aggregating results will be too large when the threshold value is bigger than 0.79. From Figure 9, find that the aggregating result is 5, except for a single one. The cluster numbers, mean-consistency and std-consistency from evolution number 7501 to 8000 are shown in Figure 10.

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Fig. 8

The cluster numbers with threshold value γ = 0.76.

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Fig. 9

The aggregating result with threshold value γ = 0.76.

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Fig. 10

The consistency degree and cluster results with threshold value γ = 0.76.

Ranking results by different methods are shown in Figure 11. The overall rankings of the three method are the same with each other, that shows the validity and rationality of the proposed method in this paper. Although the specific results obtained by each method are different. You’s [47] method did not take the relations of decision makers into account, only considering evaluation results. Wu’s [45] method weighted the decision makers and subgroups by their connections, without discussing their decision information’s quality. In this paper, the proposed method considers the network of decision makers and the evaluations simultaneously.

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Fig. 11

Comparisons of different LSGDM methods.

Advantages of the proposed method.

a. The model constructed takes into account the social relations between decision makers, during dealing with the large scale group decision making problem. Subgroups are detected by every decision maker’s position in the network. Each subgroup’s weight is calculated by its importance degree, which is determined by the nodes’ position and their edges contained in it.

b. Decision maker’s weight is considered by both of the position in whole network and effectiveness of decision information. The importance of eigenvector centrality and degree centrality is considered at the same time, for calculating the subgroups’ weights.

c. Utilize modularity to realize clustering of network nodes, depending on the nodes importance degree of the whole network.

d. Decision maker’s evaluating results are represented by the interval valued intuitionistic fuzzy number. This paper constructs a transform function, considering the barycenter of IVIFN, which is displayed by using rectangular coordinate system.