An intuitionistic fuzzy method for social network prediction problems

Abstract

Social network evolution is a major component of social network analysis. Due to ubiquitous real world uncertainty, traditional deterministic networks tend to yield analytical deviations. To lessen this bias, an intuitionistic fuzzy based method is used herein to analyze the inner-structure transformations in social networks. In addition, a conception observation community is introduced into the proposed method in order to reduce the computational load. The network density and centrality metrics are used to obtain the intuitionistic fuzzy metrics of the micro relationships. Then, the social network development trends are predicted by analyzing the changes in the metrics using a Markov chain model. The experimental results demonstrate that the proposed intuitionistic fuzzy method is advantageous to the general methods.

Keywords

Social network prediction intuitionistic fuzzy observation community Markov chain

1 Introduction

Networks have been widely used in the descriptions of complex systems. Due to the increasing number of studies regarding complex networks, a relatively complete research system has formed. In the human social environment, interactions among individuals can exhibit certain patterns or rules that reflect social structures [11].

Social network definitions are often constructed with deterministic models [1]. However, current social network analyses do not adhere to the "either-or" dualistic method due to real world uncertainty. In reality, social networks patterns are often subjective. Therefore, a lack of precise evaluation criteria can result in ambiguous results [5]. This type of scenario can be best described with a fuzzy system. Because real world time is sequential, social network predictions are essential to social network analysis [2]. Most prediction methods are based on deterministic models and probability theory. Methods based on probability theory can be used to predict results with probabilistic descriptions, but cannot be applied to existence problems since relationships are only defined as existent or non-existent. Professor Zadeh [8] proposed the concept of fuzzy mathematics in 1965. Fuzzy mathematics has since been applied to a variety of complex and uncertain systems. Atanassov [6, 7] expanded fuzzy theory in 1986 by introducing a new parameter of non-membership in order to account for the degrees of membership, non-membership, and hesitancy of information to fuzzy sets. These intuitionistic fuzzy sets can be used to more accurately and objectively study the ambiguous or uncertain information common in the real world. Intuitionistic fuzzy sets have been extensively studied [21] and applied to various areas, such as decision making [20] and forecasting [19] problems.

In this paper, an intuitionistic fuzzy system is used to perform social network predictions. By fuzzilizing the network and observing the trends of membership and non-membership in the time-series network, fuzzy language is used to obtain prediction results and eliminate defects in the network’s inner-relationship descriptions without compromising the accuracy of the results.

2 Related work

Social network analysis originated from physics adaptive network analysis, which focuses on network topology rather than individuals. The research on network relationships combines both network relationships among individuals and the structure of large-scale social systems, to study which would typically involve the use of graph theory and other quantitative analysis methods [14]. Social network evolution prediction is a major component of social network analysis. Through the analysis of a known time slice, one can predict the status of a network in the next time slice.

Social network prediction methods are usually based on probability models. However, fuzzy models can also be applied to social network prediction problems. Although they both feature uncertainty principles, fuzzy theory and probability theory are based on two different concepts. Probability is a measurement of the likelihood of occurrence of uncertain events, while fuzzy theory is a type of deterministic uncertainty. Therefore, traditional probabilistic models describe the possibility of the existence of relationships among individuals, while fuzzy models describe the extent of the relationships among individuals. Fuzzy systems have already been introduced into social network prediction problems, collectively forming a new area of research called fuzzy social networks. Nair and Sarasamma [16] further studied fuzzy relationships and proposed a definition of fuzzy social networks, yielding a fuzzy graph with meaningful results. Brunelli [10] used fuzzy logic to transform binary relationships into multivariate relationships, yielding the same positive effects. Araujo [3] developed a method of utilizing fuzzy logic to explain social relationships. Bastani [17] used a fuzzy model to perform network-featured link predictions. Recently, an ordered weighted averaging ( OWA ) operator was also applied to fuzzy network prediction [18]. Although the computational complexity of this method was high, it yielded accurate and precise prediction results. While some fuzzilized networks have been successfully developed, most adopt membership functions to measure internal relationships, resulting in insufficient information. Through the use of non-membership and hesitancy measurements, intuitionistic fuzzy systems are capable of determining the degrees of alienation and uncertainty between individuals.

In this paper, an intuitionistic fuzzy system is combined with social network to form a new community-division based network prediction method in order to better reflect the fuzziness and uncertainty associated with prediction problems. Numerous similarity measurements have been applied to intuitionistic fuzzy systems [4, 13]. In the proposed intuitionistic fuzzy network, individuals are divided into communities with fixed entities in order to investigate their inner- and inter-community relationships. Next, the intuitionistic fuzzy relationships are determined based on the degrees of membership and non-membership of the individuals. Then, the future network intuitionistic fuzzy relationships of the network are predicted using the improved intuitionistic fuzzy relationship fitted Markov chain. The results indicate that improved prediction and analysis results could be obtained by applying intuitionistic fuzzy processing to social network evolutionproblems.

3 Preliminaries

3.1 Definition of an intuitionistic fuzzy network

Let all of the nodes in a network be a set. The objective and foundation of network intuitionistic fuzzilization is the fuzzilization of the set.

Definition 1: Let U be a universe. The distance from U to the unit interval [0,1] is a map called a fuzzy set A on U . The traditional precise set represents a binary relation between a collection and a set–“belong to” or “not belong to”. If A is a classical set and the space has any element x , then the following features can be defined: $χ_{A} (x) = {\begin{matrix} 1, & x \in A \\ 0, & x \notin A \end{matrix}$ (1) where χ_A (x) is the characteristic function of set A . Values of 0 or 1 are usually selected as the characteristic degree in classical sets. However, any real number in [0,1] can be selected as the characteristic degree in fuzzy sets.

Definition 2: Let U be a universe, where x is a fuzzy subset of U . Then μ_A is a map of U in [0,1]: $μ_{A} : U \to [0, 1], x \to μ_{A} (x) \in [0, 1]$ (2)

Therefore, A is the fuzzy set on U , μ_A (x), which is the membership function of A . The membership function can be used to describe the prediction results in the fuzzy set A . Often, μ_A (x) ∈ [0, 1].

Definition 3: The definition of the intuitionistic fuzzy sets A in X can be expressed as: $A = {< x, μ_{A} (x), ν_{A} (x) > | x \in X}$ (3)

In addition, μ_A (x) : X → [0, 1], ν_A (x) : X → [0, 1] and 0 ⩽ μ_A (x) + ν_A (x) ⩽1, ∀ x ∈ X. Non-membership and hesitancy can provide more information than common fuzzy relationships.

Definition 4: The hesitancy degree π_A, also called the intuitionistic index, can be expressed as: $π_{A} (x) = 1 - μ_{A} (x) - ν_{A} (x)$ (4)

Therefore, 0 ⩽ π_A (x) ⩽1. The hesitancy degree reflects the uncertainty of the relationships in a fuzzy set. Higher degrees of hesitancy indicate greater levels of uncertainty. However, a hesitancy degree of 0 does not necessarily mean that two individuals are completely similar since their membership and non-membership degrees may differ.

3.2 Definition of an intuitionistic fuzzy network based on communitization

Consider a traditional undirected social network G=< V,E>with N nodes, V sets of individuals, and E relationships. Then an intuitionistic fuzzy social network from the traditional network G=< V,E>can be defined as $\tilde{G}$ =< V, $\tilde{E}$ >, where $\tilde{E}$ is the fuzzy relationship set among the nodes in the intuitionistic fuzzy network, and the similarity metric is often used to describe the fuzzy relationships as ${\tilde{e}}_{ij} = Similarity (v_{i}, v_{j})$ . Since the intuitionistic fuzzy network $\tilde{G}$ contains the fuzzy sets

{ < v, [μ₁ (v) ,. . . , μ_N (v)] , [ν₁ (v) ,. . . , ν_N (v)] > |v ∈ V }

The membership degree μ_N (v_i) and non-membership degree ν_N (v_i) of any individual v_i to any other individual can be used to calculate the fuzzy relationships ${\tilde{e}}_{ij}$ among individuals, as shown in Fig.1.

As shown in Fig.1, Node 1 has membership relationships with other nodes. The degree of membership is defined as 1 if a link exists between another node; otherwise, the degree of membership is defined as 0, as shown below: $v_{1} {\begin{matrix} v_{2} & v_{3} & v_{4} & v_{5} & v_{6} & v_{7} & v_{8} \\ μ & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\ ν & 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix}$ (5)

Different fuzzy relationship definitions yield different results. However, given a large number of complex social networks, calculating the membership and non-membership degrees of each node will increase the amount of required storage and computational complexity.

In order to reduce the number of calculations, the relationships among the individuals can be extended into those of the communities. That is, only the inter-community relationships are considered, while the inner-community relationships are ignored. Typically, nodes within the same community are closer [12]. Community structures reflect the regional properties of the nodes’ behaviors as well as the inter-group association properties. A network can be divided into a smaller number of communities according to these two features. By dividing the network, the computational load and storage space requirements can be lessened. When more information is needed, increasing a corresponding number of the communities can increase the accuracy of the analysis.

Assume a network G is divided into the communities {C₁, C₂,. . . , C_n}. The inter-individual fuzzy relationship can be extended into the inter-community relationship. Similarly, the degrees of membership and non-membership of the communities can be written as μ_{C
₁} (v_i) , μ_{C
₂} (v_i) ,. . . , μ_{C
_N} (v_i) and ν_{C
₁} (v_i) , ν_{C
₂} (v_i) ,. . ., ν_{C
_N} (v_i), respectively. In this paper, the relationships of each community in the network can be categorized into internal and external relationships. All of these relationships comprise the known relationships within the community. In fuzzy intuition, a larger degree of membership implies a stronger inner-community relationship, while a larger degree of non-membership implies a stronger inter-community relationship. For convenience, the network shown in Fig. 2, which was obtained from the community division of the network shown in Fig. 1, is provided as an example.

As shown in Fig. 2, nodes 1-5 are located in community 1, while nodes 6-8 are located in community 2. The relationships among the nodes within each community are considered inner-community relationships, defined as degrees of membership. Closer inner-community relationships are associated with larger degrees of self-membership within the community. This relationship become strongest with a value of 1 when the community forms a complete graph with no external relationships. Therefore, this information is considered external information rather than internal information, defined as non-membership. The degree of non-membership of a community increases as the number of inter-community connections increases.

Definition 5: Assume that the number of internal nodes is p ; the number of outer nodes related to the internal nodes is q ; and the degrees of membership, non-membership, and hesitancy of community C_i to community C_j are $μ_{i}^{j}$ , $ν_{i}^{j}$ , and $π_{i}^{j}$ , respectively. Then, the maximum number of links can be calculated as: $\sum E_{C_{in}} = \frac{1}{2} p (p - 1)$ (6)

The number of links between the internal nodes and outer nodes should be divided by 2 since the outer nodes are not included in the community. The maximum number of links can be calculated as: $\sum E_{C_{out}} = \frac{1}{2} pq$ (7)

Thus, the information contained within the structure of any community is the sum of that community’s internal and external information, calculated as: $\sum E_{C} = \frac{1}{2} p (p - 1) + \frac{1}{2} pq$ (8)

This value describes the maximum number of links within the community.

Definition 6: In this paper, the number of links the community actually contains is calculated based on network density or centrality. The network density can be defined as: $D = \frac{E}{N (N - 1)}$ (9)

The number of internal links α can be defined as: $α = \sum d (C_{in}) = D \times p (p - 1)$ (10)

This definition can be used to clearly describe the relationships between communities. In addition, the following algorithm based on network centrality is proposed: $α = p \cdot \sum d (C_{\max}) = {C_{D}}_{\max} \times p$ (11)

This definition describes the relationships between the core nodes within a community and another community, reflecting the phenomenon of imbalance. Larger values are associated with more imbalance within the community. The number of links that exists between the internal and external nodes can be written as: $β = \sum d (C_{in - out})$ (12)

This value describes the external information contained within the community.

Definition 7: The membership of any community C_n to itself can be defined as: $μ_{C_{n}} = α / \sum E_{C}$ (13)

The membership of any community C_n to any other community C_i can be defined as: $μ_{C_{i}} = \frac{1}{2} β / \sum E_{C}$ (14)

Since Community C_n does not contain outer nodes, these values should be divided by 2.

Definition 8: The degree of non-membership between any two communities can be defined as follows based on the principal that degrees of membership and non-membership should contain the same quantity of information. $ν_{C_{n}} = \sum μ_{C_{all}} - μ_{C_{n}}$ (15)

According to the above definition, the degrees of membership and non-membership between communities 1 and 2 can be calculated as follows: $C_{1} {\begin{matrix} C_{1} & C_{2} \\ μ & 0.320 & 0.040 \\ ν & 0.040 & 0.320 \\ π & 0.640 & 0.640 \end{matrix}$ $C_{2} {\begin{matrix} C_{1} & C_{2} \\ μ & 0.111 & 0.444 \\ ν & 0.444 & 0.111 \\ π & 0.444 & 0.444 \end{matrix}$

When the degree of membership of a community to itself is significantly higher than that of the community to another community, the individuals in that community are assumed to be relatively similar. When the degree of hesitancy π is not 0, there is a possibility that links exist between the nodes without links.

4 Intuitionistic fuzzy network prediction

4.1 Evolution of an intuitionistic fuzzy network

The evolution of a social network can be studied by observing changes in the degrees of membership and non-membership. The network graph provided in the last section is the original graph. The changes in membership and non-membership can be analyzed by observing the changes that occur in network diagrams. The two communities in the network shown in Fig. 3 are completely independent with no links. These types of communities often evolve from a connected state to completelyseparate.

The network fuzzy intuition can be expressed by the degrees of membership, non-membership, and hesitancy as: $C_{1} {\begin{matrix} C_{1} & C_{2} \\ μ & 0.400 & 0.000 \\ ν & 0.000 & 0.400 \\ π & 0.600 & 0.600 \end{matrix}$ $C_{2} {\begin{matrix} C_{1} & C_{2} \\ μ & 0.000 & 0.666 \\ ν & 0.666 & 0.000 \\ π & 0.333 & 0.333 \end{matrix}$

According to these calculations, compared to the original network, the degrees of membership $μ_{1}^{2}$ and $μ_{2}^{1}$ between the two communities will gradually decrease to 0, reflecting the gradual separation between the two communities. The two communities in the network shown in Fig. 4 will become increasingly intertwined without a clear dividing line. This increase in the number of links results in the gradual fusion of the two communities, resulting in the formation of a larger community.

According to calculations: $C_{1} {\begin{matrix} C_{1} & C_{2} \\ μ & 0.229 & 0.200 \\ ν & 0.200 & 0.229 \\ π & 0.571 & 0.571 \end{matrix}$ $C_{2} {\begin{matrix} C_{1} & C_{2} \\ μ & 0.333 & 0.191 \\ ν & 0.191 & 0.333 \\ π & 0.476 & 0.476 \end{matrix}$

According to these calculations, the values of self-membership $μ_{1}^{1}$ and external membership $μ_{1}^{2}$ will become increasingly similar, indicating that the nodes within the two communities will also become increasingly similar as the two communities fuse to form a new community. As shown in Fig. 5, the number of inner links within community 1 increases, indicating that the relationships within the community have become closer. In contrast, no changes occur in community 2.

According to these calculations, the degree of membership of community 1 to itself will increase $μ_{1}^{1} \to 1$ as the number of contacts within community 1 increase. In addition, the degree of hesitancy π₁ will decrease, indicating that the increased number of inter-node relationships within community 1 strengthens the community, also increasing the value of $μ_{1}^{1}$ . Due to the lack of new links between the communities, the degree of non-membership of community 1 to itself $ν_{1}^{1}$ and community 2 $μ_{1}^{2}$ will not change. In addition, the degree of membership of community 1 to itself will gradually decrease $μ_{1}^{1} \to 0$ .These results reflect the shrinking of the community and decrease in contact among individuals.

4.2 Intuitionistic fuzzy network prediction with the Markov chain

The Markov chain can be used to predict the future state of a system based on its current state. The state of the system is usually divided into ranges, where the prediction results are represented by intervals. Let x be a random variable sequences with states S₁, S₂,. . . , S_r, X_n = i, where S_i is the state at time n . Then the probability can be calculated as follows: $P (X_{n + 1} | X_{n} X_{n - 1} . . . X_{1}) = P (X_{n + 1} | X_{n})$ (16)

The relationships between the individuals in a traditional social network are certain and have two states. While the relationships in traditional networks are always certain, intuitionistic fuzzy networks do not satisfy the conditions of the traditional Markov chain model since the sum of the membership and non-membership degrees is not 1. In this paper, the number of nodes in each community did not change over time; this phenomenon was referred to as the observed-community.

In an intuitionistic fuzzy system, the non-membership degree causes the sum of all of the fuzzy states to be not equal to 1. Therefore, the hesitancy degree was introduced into the fuzzy state using the Markov chain transfer matrix. Then, the non-membership degree was calculated and the number of Markov chain steps was reduced. According to the sample data size, the network was divided into 5 communities with 5 membership degrees and 1 hesitancy degree. The state of a community classification matrix, where represents the samples at different times, can be written as: $\tilde{E} = [\begin{matrix} μ^{1} (x_{1}) & μ^{1} (x_{2}) & \dots & μ^{1} (x_{n}) \\ μ^{2} (x_{1}) & μ^{2} (x_{2}) & \dots & μ^{2} (x_{n}) \\ μ^{3} (x_{1}) & μ^{3} (x_{2}) & \dots & μ^{3} (x_{n}) \\ μ^{4} (x_{1}) & μ^{4} (x_{2}) & \dots & μ^{4} (x_{n}) \\ μ^{5} (x_{1}) & μ^{5} (x_{2}) & \dots & μ^{5} (x_{n}) \\ π (x_{1}) & π (x_{2}) & \dots & π (x_{n}) \end{matrix}]$ (17)

Next, using the Markov chain method, the probability transfer matrix P was calculated in order to predict the membership of each community in the next time slice as follows: $E_{n}^{*} = E_{n - 1} \cdot P^{T}$ (18) The predicted membership value in the fuzzy states $x_{n}^{*}$ was calculated as: $E_{n}^{*} = (μ^{1} (x_{n}^{*}), μ^{2} (x_{n}^{*}), . . ., μ^{5} (x_{n}^{*}), π (x_{n}^{*}))$ (19) Then, the real membership x_n was calculated as: $E_{n} = (μ^{1} (x_{n}), μ^{2} (x_{n}), . . ., μ^{5} (x_{n}), π (x_{n}))$ (20) According to the definition of non-membership, the status of the community for their own degree of membership can be obtained as: $E_{n}^{*} = (μ (x_{n}^{*}), ν (x_{n}^{*}), π (x_{n}^{*}))$ (21)

Finally, the changes in the network can be determined based on the changes in the degrees of membership and non-membership using the analysis method provided above.

5 Experiments

5.1 Intuitionistic fuzzy method on a small network

The Karate club network is one of the most popular small detection networks used in the field of sociological analysis. Each node in this network represents one club member, where a link between two nodes indicates that the two club members often appear together on outside occasions. Namely, two linked club members are considered to be relatively close outside of the club. The two core nodes represent the manager (Node 1) and trainer (Node 34) of the network shown in Fig. 6. In many studies, this network has been divided into two communities based on these two core nodes.

Then, the intuitionistic fuzzy method is applied to this network in order to calculate its degrees of membership and non-membership. The degrees of membership and non-membership were observed in order to analyze the variations in the relationships among the nodes. According to the definition of intuitive fuzzy methods, the degrees of membership and non-membership can be obtained using the Jaccard [15] relationship index. For example, the membership degrees of Node 1 can be represented as follows: $v_{1} {\begin{matrix} v_{2} & v_{3} & . . . & v_{33} & v_{34} \\ μ & 1.000 & 1.000 & . . . & 0.120 & 0.137 \\ ν & 0.000 & 0.000 & . . . & 0.880 & 0.863 \end{matrix}$

The variations in the relationships between Node 1 and the other nodes can be observed based on the corresponding variations in the degrees of membership. To reduce the computational load, a community-division based intuitionistic method was adopted. The entire network was partitioned into two communities using the Fast Newman community-division method. Only the changes in the communities were observed. The resulting community partition is shown inTable 1. Next, the density-based intuitionistic fuzzy method was used to calculate the degrees of membership and non-membership as: $C_{1} {\begin{matrix} C_{1} & C_{2} \\ μ & 0.182 & 0.027 \\ ν & 0.027 & 0.182 \\ π & 0.791 & 0.791 \end{matrix}$ $C_{2} {\begin{matrix} C_{1} & C_{2} \\ μ & 0.026 & 0.174 \\ ν & 0.174 & 0.026 \\ π & 0.800 & 0.800 \end{matrix}$

The Dolphin social network was also used as experimental data. Lusseauenthad studied the living habits of 62 bottlenose dolphins in New Zealand for a long period observation. According to the results, the dolphins exhibited a specific pattern of relationships. In the network, a link was placed between two nodes to indicate the existence of a relationship between two dolphins. This network shown in Fig. 7 has been divided into two communities in many studies.

The community-division of the network is shown in Table 2. As shown in this table, community 1 contained 21 nodes, while community 2 contained 41 nodes. Next, the network was fuzzilized, and the degrees of membership and non-membership were calculated. $C_{1} {\begin{matrix} C_{1} & C_{2} \\ μ & 0.147 & 0.014 \\ ν & 0.014 & 0.147 \\ π & 0.839 & 0.839 \end{matrix}$ $C_{2} {\begin{matrix} C_{1} & C_{2} \\ μ & 0.005 & 0.114 \\ ν & 0.114 & 0.005 \\ π & 0.881 & 0.881 \end{matrix}$

Membership μ reflects the closeness of the relationships among the networks. Although the Dolphin network includes a higher number of nodes and links than the Karate club network, by adopting the intuitionistic fuzzy approach, the number of memberships, non-memberships, and hesitancy can be made equal. By observing the membership, non-membership, and hesitancy degrees of each network, the similarities between their structures can be determined. The similarities among these characteristics reflect the sparse nature of social networks as well as the similarities among the inner-community properties of social networks. The two networks both yielded the result $μ_{1}^{1} ⪢ μ_{1}^{2}$ , indicating that the degree of inner-community membership was much higher than that of other relationships. In addition, since $μ_{1}^{1} ⪢ ν_{1}^{1}$ , the relationships among the nodes within each community exhibited a relatively high degree of membership, reflecting the stability of the community structures. The result $μ_{1}^{1} > μ_{2}^{2}$ illustrates that the inner community relationship of community 1 was stronger than that of community 2. In addition, the communities within the networks were not equal in size. As the size of the community increased, the number of relationships in the community decreased. Thus, in the Karate club network, the result $ν_{1}^{1} \approx ν_{2}^{2}$ indicates that the two communities were similar in size. In the Dolphin social network, the result $ν_{1}^{1} > ν_{2}^{2}$ demonstrates that community 2 was larger than community 1. Finally, the hesitancy π results indicated that the inner community relationships were relatively sparse, consistent with the inherent nature of social networks. In the next section, the time-series network membership and non-membership results will be used to analyze the inner network changes. Unlike the method presented in [9], the proposed method can be used to illustrate alienated relationships.

5.2 Intuitionistic fuzzy network prediction based on communitization

In this section, the 2001 ENRON company e-mail dataset is divided into 12 monthly time slices. Then, the network is fuzzilized and membership and non-membership predictions are performed using the time series method in order to forecast future inner network evolution trends. The ENRON dataset includes a total of 151 nodes. The Fast Newman (FN) algorithm was used to divide the nodes into communities. The January data was divided into fivecommunities.

Next, the network density based intuition fuzzy method was used to construct the division matrix for each community. Table 3 displays the division matrix of the state for the first community in the network.

The present matrix does not include degrees of non-membership. The fuzzy Markov chain was used to predict the status of the inner network community for November based on the previous 10 months of data. The following results were obtained: $E_{11}^{*} = E_{10} \cdot P^{T} = {\begin{matrix} μ^{1} & 0.102 \\ μ^{2} & 0.009 \\ μ^{3} & 0.012 \\ μ^{4} & 0.021 \\ μ^{5} & 0.013 \\ π & 0.843 \end{matrix}$ $E_{11} = {\begin{matrix} μ^{1} & 0.123 \\ μ^{2} & 0.010 \\ μ^{3} & 0.013 \\ μ^{4} & 0.021 \\ μ^{5} & 0.026 \\ π & 0.807 \end{matrix}$

The results can also be represented as degrees of membership and non-membership as follows: $E_{11}^{*} = {\begin{matrix} μ & 0.102 \\ ν & 0.055 \\ π & 0.843 \end{matrix}$ $E_{11} = {\begin{matrix} μ & 0.123 \\ ν & 0.070 \\ π & 0.807 \end{matrix}$

The prediction fuzzy state $E_{11}^{*}$ and the actual state E₁₁ can be obtained similarly via observation to demonstrate the accuracy of the prediction results. The November prediction results obtained using actual data are shown in Fig. 9.

The actual membership and non-membership values are displayed in Fig. 8 and the prediction data is displayed in Fig. 9. According to the results, the post-June predictions were fairly accurate. As the number of samples increased, the prediction data became more accurate, indicating that a sufficiently large number of samples can ensure accuracy. In addition, as shown in Figs. 8 and 9, the degree of self-non-membership increased in this period, indicating that the number of connections between the communities in the network also increased. The degree ofmembership decreased in May and June, but then gradually increased. Furthermore, the results were relatively stable, indicating that no significant changes occurred within the inner community in the later stages. The FN algorithm was used to divide the network into 7 communities in order to illustrate the relationship between the community number and prediction results. The nodes in community 1 did not change. The membership values of community 1 are shown in Table 4.

As shown in this table, when the nodes in the community remained unchanged, the community’s degree of self-membership did not change when the number of communities changed. The prediction results obtained using the Markov chain method based on the previous 10 months of data are shown below. $E_{11}^{*} = E_{10} \cdot P^{T} = {\begin{matrix} μ^{1} & 0.101 \\ μ^{2} & 0.015 \\ μ^{3} & 0.003 \\ μ^{4} & 0.010 \\ μ^{5} & 0.005 \\ μ^{6} & 0.009 \\ μ^{7} & 0.012 \\ π & 0.845 \end{matrix}$ $E_{11} = {\begin{matrix} μ^{1} & 0.123 \\ μ^{2} & 0.020 \\ μ^{3} & 0.004 \\ μ^{4} & 0.022 \\ μ^{5} & 0.002 \\ μ^{6} & 0.011 \\ μ^{7} & 0.013 \\ π & 0.805 \end{matrix}$

The membership, non-membership, and hesitancy were calculated as: $E_{11}^{*} = {\begin{matrix} μ & 0.101 \\ ν & 0.050 \\ π & 0.845 \end{matrix}$ $E_{11} = {\begin{matrix} μ & 0.123 \\ ν & 0.072 \\ π & 0.805 \end{matrix}$

The above results were compared with the network with 5 communities. When individuals in the community are decided, the prediction results of the community and self-related measures are very close. By dividing the network into more communities, the changes in the relationships among the communities can be more clearly observed. Furthermore, since the number of samples required by the Markov chain was greater than the number of states, increasing the number of communities with a fixed number of samples would not increase the accuracy of the prediction results. In addition, the OWA fuzzy method was used to perform the network predictions. The CN , Katz , Salton , RA , and PA indices were selected as the attributes. To compare our method with this method, we only focus on its membership degree. The membership degree results are shown in Figs. 10 and 11.

The prediction results obtained with the OWA method were more stable and more accurate than those obtained with the intuitionistic fuzzy method. Under the same experimental environment, the OWA method required 15.04 seconds of computation time, while the proposed method only required 0.59 seconds of computation time. Thus, the runtime and accuracy of the intuitionistic fuzzy method were superior to those of the OWA method. In addition, the membership and non-membership degree functions significantly influenced the computation time of the OWA method. Thus, appropriate membership and non-membership degree functions must be selected in the OWA method.

5.3 Intuitionistic fuzzy network predictions using different membership functions

In this section, network centrality experiments based on the intuitionistic fuzzy method are conducted on the ENRON dataset. The Markov chain method is also adopted as a prediction method in this section. $E_{11}^{*} = E_{10} \cdot P^{T} = {\begin{matrix} μ^{1} & 0.449 \\ μ^{2} & 0.010 \\ μ^{3} & 0.013 \\ μ^{4} & 0.021 \\ μ^{5} & 0.026 \\ π & 0.481 \end{matrix}$ $E_{11} = {\begin{matrix} μ^{1} & 0.481 \\ μ^{2} & 0.009 \\ μ^{3} & 0.012 \\ μ^{4} & 0.020 \\ μ^{5} & 0.012 \\ π & 0.466 \end{matrix}$

The degrees of non-membership, membership, and hesitancy are presented in Figs. 12 and 13. $E_{11}^{*} = {\begin{matrix} μ & 0.449 \\ ν & 0.069 \\ π & 0.481 \end{matrix}$ $E_{11} = {\begin{matrix} μ & 0.481 \\ ν & 0.053 \\ π & 0.466 \end{matrix}$

The predicted fuzzy state $E_{11}^{*}$ and the actual fuzzy state E₁₁ were similar, demonstrating the accuracy of the proposed method. The self-membership prediction results obtained with the centrality method were significantly higher those obtained with the density measurement method.

Unlike the density method, in the centrality method, the center of the nodes in a network reflects the aggregation within that network. The communities divided by nature property usually can exhibit internal core node status changes. As shown in the actual membership state diagram, core node 1 exhibited increased volatility and a significant increase in membership in May. Take the core node 1 in the community as an example–the core node shifted, and the degree of core nodes are also incessantly changing. This algorithm focused not only on one node, as shown in Figs. 12, 13 and Table 5. Membership can reflect the degree of aggregation within the community, so by changing the degree of membership, the membership can be analyzed via the degree of aggregation of the current time slice. As shown in Figs. 12 and 13, the degree of membership remained relatively stable rather than increasing over time. This illustrates again that, over time, the number of links between the communities in the network increased gradually. However, this did not result in any significant changes within the community.

According to the data in two experiments, the prediction of the community-based network using intuitive fuzzy method provides better performance in the time complexity and is more accurate. In addition, the proposed method yielded a greater amount of information than general fuzzy methods. Furthermore, the number of links among the communities increased over time. However, this did not result in any significant changes in the network. This result indicates that the links between nodes increased within the network core, but the cause to the finding should entail further study.

6 Conclusions and future prospects

Studies have shown that fuzzy networks can be effectively applied to social network analysis. Intuitionistic fuzzy methods possess considerable potential in social network applications in that they can be used to solve the complex problems in deterministic and weighted networks. Intuitionistic fuzzy methods reflect unobservable degrees of alientation. In addition, community-based approaches can be used to significantly reduce the computational loads associated with intuitionistic fuzzy methods. Furthermore, the degree of hesitancy must be considered in intuitionistic fuzzy methods. Moreover, inaccurate data sampling and the observation process can also affect the accuracy of the final prediction results. Universal problems, such as fuzzy network simplification and proper membership and non-membership function selection, should also be addressed in future studies.

Acknowledgments

The authors would like to thank the anonymous reviewers for their constructive comments and suggestions. This work was supported by the National Natural Science Foundation of China under grants 61272422 and 61202353.

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