Potential social media influencers discrimination for concept marketing in online brand community

Abstract

Identifying potential social media influencers (SMIs) accurately can achieve a long-time and effective concept marketing at a lower cost, and then promote the development of the corporate brand in online communities. However, potential SMIs discrimination often faces the problem of insufficient available information of the long-term evolution of the network, and the existing discriminant methods based on link analysis fail to obtain more accurate results. To fill this gap, a consensus smart discriminant algorithm (CSDA) is proposed to identify the potential SMIs with the aid of attention concentration (AC) between users in a closed triadic structure. CSDA enriches and expands the users’ AC information by fusing multiple attention concentration indexes (ACIs) as well as filters the noise information caused by multi-index fusion through consensus among the indexes. Specifically, to begin with, to enrich the available long-term network evolution information, the unidirectional attention concentration indexes (UACIs) and the bidirectional attention concentration indexes (BACIs) are defined; next, the consensus attention concentration index (CACI) is selected according to the principle of minimum upper and lower bounds of link prediction bias to filter noise information; the potential SMI is determined by adaptively calculating CACI among the user to be identified, unconnected user group and their common neighbor. The validity and reliability of the proposed method are verified by the actual data of Twitter.

Keywords

Concept marketing social media influencers attention concentration index consensus smart discriminant algorithm

1 Introduction

Concept marketing is a marketing method in which a company establishes a relationship between a consumer and a brand through emotion or thought, and influences the concept of consumers [1]. It can stimulate consumers’ needs and desires, stimulate consumers to purchase conceptual products, improve customer satisfaction, and achieve corporate profitability goals. For example, Juventus Football Club, after 120 years of traditional marketing, innovatively launched a new strategy combining sports and concept marketing in 2017. The main plan was to fully link the “Jj” new team logo with every football supporter of the club, and to convey the concept of sportsmanship through the “J” brand, thereby stimulating consumers to purchase conceptual products [2]. For concept marketing, when a concept is conveyed by a reliable and prestigious individual or organization, it is more credible to consumers.

The users who occupy strategic positions on networks and have a lot of followers are seen as the social media influencers (SMIs) [3 –5]. According to traditional communication theory, the information flow is divided into two steps: first, from the mass media to the SMIs, then from the SMIs to others [6], in the latter step, SMIs are found to play an important role in promoting communication among users and improving the exchange of information [7]. Magno and Cassia [8] found that SMIs generated content strongly influenced their followers and could make the people easily accept their information or change their behavior intention. Freberg et al. [9] believed that SMIs were a new type of independent third-party spokesmen who influenced the attitude of the audience through blogs, Twitter, and other social media. Therefore, SMIs can play an important role in effectively promoting concept marketing and changing consumers’ attitudes. Marketing through the SMIs is an indispensable means of achieving concept marketing.

The networking and socialization of social media enable the user community to form an interconnected online community focusing on certain brands. In the online brand community (OBC), social media users and potential product customers help each other by sharing their experiences and ideas [10]. The OBC has a profound impact on consumers’ brand attitudes and behaviors, providing a fertile ground for the development of concept marketing [11]. Extant research mainly focuses on selecting popular and currently influential SMIs in OBC according to topological structure [12] and information dissemination [13], namely influence maximization, which is, however, not applicable to concept marketing but is suitable for viral marketing. Domingos and Richardson described the market as a social network in marketing and defined it as a Markov random field to investigate the influence maximization problem and its application on customer’s network value exploration (i.e., viral marketing) [14]. Taking advantage of the power of influential users to spread a message to millions of people in a short time span, namely viral marketing, is often used to initiate and promote a specific promotional campaign to drive sales instantly [15]. Different from viral marketing, concept marketing focuses on propagating new concept and innovative ideas which are hard to be accepted and internalized at one stroke [16]. Therefore, the formation of consumers’ values, attitudes, beliefs, interests, opinions, and viewpoints on products, services, or brand image calls for persistent propagation and persuasion [17]. Long-term employment of currently influential SMIs is expensive and risky because the influence of the SMIs may change as the network evolves [18]. On the contrary, identifying potential SMIs who have relatively lower influence at present but may have greater influence as the network evolving in the future in OBC helps to perform a long-time and effective concept marketing campaign. Therefore, to publicize a concept consistently to target customers in an effective and economical manner and guarantee long-term effectiveness, it is of great significance to discriminate potential SMIs.

In terms of identifying potential SMIs in OBC, the typical approaches are to predict the network connections over a long period and regard the nodes with the larger degree in the predicted network as the SMIs employing link analysis such as PageRank [12], HITS [19] and ExpertiseRank [20], and link prediction algorithm (LPA). However, these discriminant methods often face the problem of insufficient available information of long-term evolution of the network, therefore they suffer from low accuracy.

To make up for the lack of long-term network evolution information in identifying potential SMIs, this study proposes the consensus smart discriminant algorithm (CSDA) to identify them with the aid of attention concentration (AC) between users in the closed triadic structure, which takes the following two measures to enrich and expand the available information: the first is to put forward the attention concentration indexes (ACIs) which have important impacts on the establishment of friendship and enrich and expand the user’s AC information through multi-index fusion; the second is to filter the noise information caused by multi-index fusion by extracting the consensus among the indexes.

Specially, first of all, according to the triadic closure structure, some unidirectional attention concentration indexes (UACIs) are defined. Next, different combinations of these UACIs are used to obtain the multiple bidirectional attention concentration indexes (BACIs), and the optimal BACI is selected according to the upper and lower bounds of the prediction bias; and then, consensus attention concentration index (CACI) is obtained, which shares the highest consensus with other indexes. Finally, the possibility of a node becoming an SMI in OBC is determined by calculating the consensus attention concentration score (CACS) among the candidate nodes, other unconnected users, and their common neighbors (CNs).

The remainder of this study is organized as follows. In section2, we review related literature. Section 3 describes the potential SMIs discrimination system. Section 4 introduces the user node ACI. Section 5 provides the CSDA. Section 6 shows the experiment and results in analysis. The last section sums up the conclusion and future research direction.

2 Literature review

2.1 Attention concentration

From the perspective of human response dynamics, attention is an inherent attribute of users, which can explain the behavior of online users [21]. Users in social media usually focus on their friends or followers due to limited attention, their AC mainly depends on the number of friends [22]. Moreover, attention, as a social mechanism, can analyze users’ behavior according to their AC. For example, Hodas and Lerman [23] thought that the users’ attention was distracted and limited, and the relationship between the users’ attention and the forwarding behavior of friends and followers was mined by quantifying the users’ attention. The results showed that their AC declined with the increase of the number of their friends. In addition, Rafailidis and Weiss [24] also found that the number of meaningful relationships limited the users’ attention. For example, the fewer the number of friends, the higher the users’ AC. Lerman et al. [25] proposed a limited attention alpha centrality to measure the users’ AC and simulated the process of users’ attention effectively by identifying influential people on the network in the communication process. Zhu and Lerman [26] found that through the attention diffusion mechanism, the higher the users’ AC, the higher the attention received, and the more new followers could be attracted, so that the users were more likely to become influential people.

From the above research, we can see that AC affects the behavior of users. In fact, limited and distracted attention changes the nature of the interaction between users, so we can use AC to identify key nodes in the network and then identify potential SMIs in OBC in the study.

2.2 Potential nodes identification and link prediction algorithms

The identification of potential nodes is essential to research regarding network attacks, information dissemination, and epidemic spreading. Previous research has proposed many potential user discrimination methods in the static network, which generally adopts a series of degrees that reflect the topological structure to measure node importance in information diffusion [27, 28]. For example, link analysis, such as PageRank [12], HITS [19], and ExpertiseRank [20], are generally used. Some scholars have explored other methods. Sun et al. [29] used the weighted formal concept analysis to identify influential nodes in a complex network. Aleahmad et al. [30] proposed an algorithm called OLFinder, which categorized the core topics of demonstrations in social networks and calculated the capability and the popularity scores of each participant, and then a ranking list of SMIs were obtained. Jain and Katarya [31] proposed an improved firefly algorithm to identify key social network users. The algorithm constantly updated the attractiveness value of each combination and selected the local best as the SMI in the end. However, extant research ranks users at a certain instant or period of time employing existing data and neglects that the results can change considerably with the evolution of the network [18].

LPA is an algorithm to predict the possibility of a link between unconnected nodes in the network. Many researchers combine link prediction as a base method to perform applications in social networks, such as recommender systems, community detection, anomaly detection, and influence analysis [32]. As for methods, according to evolution rules, the complexity of calculation methods, and types of information, LPA can be divided into four categories, including similarity-based methods, algorithmic methods, probabilistic and statistical methods, and preprocessing methods [33]. Similarity-based methods are to calculate the similarity of network structure to predict links [34]. Algorithmic methods refer to the use of machine learning or optimization algorithms for link prediction [35]. Probabilistic and statistical methods assume that the network has a known structure and use statistical methods to estimate model parameters to calculate the link probability of each link to be predicted [36]. Preprocessing methods are to improve the performance of the method by reducing the “weak” or “fake” link noise in the network [37]. Among the four methods, similarity-based methods are the most widely used and effective. The existing research on LPA mainly focuses on the links that may be generated in the network in one iterative evolution cycle, moreover, little has shed light on identifying influential nodes employing LPA, and there are two differences between our research and the existing research. One is that the BACI and CACI are proposed better to describe the possibility of nodes and node group links so that to compensate for the lack of useful information. The other is that the link possibility between one node and node groups in the long-term network evolution is predicted to identify important nodes further and find the potential SMIs as the evolution of networks.

3 Potential SMIs discrimination system

Concept marketing is mainly to promote some ideas or products to the circle of friends of the SMIs in OBC so that a class of products can be more easily accepted and bought by customers. Usually, the influential users in OBC are seen as SMIs, but it is expensive to employ them to promote the products, therefore it is necessary to identify the potential SMIs from their circle of friends to stage a long-time and effective concept marketing campaign in OBC. So-called potential SMIs are the users who will be likely to become the SMIs in OBC in the future. The potential SMIs can not only plan and publicize brand-relevant ideas consistently to target customers but also can help the brand companies reduce marketing costs of spreading product concepts. Therefore, the potential SMIs discrimination is of great significance to concept marketing. In this study, the idea of AC of the members in the triadic closure structure is used to predict the network connection in the long-term and subsequently identify the potential SMIs. Generally, the higher the AC between the user and other numbers in a community, the greater possibility this community becomes the potential user’s circle of friends [38], the more likely such user can be identified as the potential SMI of the community [39].

In this study, the potential SMIs discrimination system is proposed to mine the potential SMIs in the community. The system consists of three parts: Network, Algorithm, and Application, as shown in Fig. 1. The network contains a user x to be identified (i.e., the node to be identified) and a community C with multiple circles. It is known that users in circle P have friendship with x, while users in circle Q have no friendship with x, so the users in circle Q are defined as unconnected users. The algorithm mainly calculates the AC between x and the nodes in circle Q based on the triadic closure structure. Relevant abbreviations and definitions are summarized and listed, as shown in Table 1. The process is as follows: to begin with, seven ACIs are developed, and they are unidirectional (i.e., UACIs), and then different n-ary BACIs can be obtained by different combinations of UACIs. Next, based on the prediction bias interval smallest principle, the best n-ary BACI can be screened. Then calculate the consensus of selected BACIs and find the most appropriate one as the CACI, which can be employed to predict CACS, based on which the application part determines whether the user x is an SMI. Specifically, the higher the CACS, the higher the possibility of the connection between the user to be identified and the unconnected node groups, the greater the possibility of the user becoming an SMI in the community.

Fig. 1

Potential SMIs discrimination system.

Table 1

Abbreviations and Definitions

Abbreviations	Definitions
ACI	Attention concentration index
UACI	Unidirectional attention concentration index
BACI	Bidirectional attention concentration index
CACI	Consensus attention concentration index
CACS	Consensus attention concentration score
CSDA	Consensus smart discriminant algorithm
ACS	Attention concentration score
CN	Common neighbor
UCNACI	ACI of the node to be identified and the unconnected node groups on the CNs
UNACI	ACI of the node to be identified on the CNs
NUCACI	ACI of the CNs on the node to be identified and the unconnected node groups

4 The construction of ACI

In this study, using the AC characteristics of nodes in the closed triadic structure, we overcome the lack of available information in long-term link prediction from two aspects: on the one hand, considering the topological information of multiple nodes, on the other hand, considering the bidirectional attention concentration information between nodes and their common neighbors. Therefore, three types of ACIs are constructed, which are: ACI of the node to be identified and the unconnected node groups on the CNs (UCNACI); ACI of the node to be identified on the CNs (UNACI); ACI of the CNs on the node to be identified and the unconnected node groups (NUCACI).

The triadic closure structure is the most basic local structure of the network and an important link generation mechanism, which plays a critical role in network evolution. As the smallest local structure in the network, it has the characteristics of structural balance and stability [40]. As shown in Fig. 2, node A connects with nodes B and C, so the possibility of forming the connection between nodes B and C is also very high. When node B and C are connected, the node A, B, and C jointly constitute a triadic closure structure.

Fig. 2

A triadic closure structure.

The user’s AC on the friends is defined as the reciprocal of the number of friends (i.e., degree) of the user in the social network. The ACI is proposed based on the triadic closure structure and used to describe the possibility of the connection between the node to be identified and the unconnected node groups. The basic principle of constructing ACI is that the higher the AC of the node to be identified and the unconnected nodes on their common friends, the more likely they generate the connection. In addition, the higher the AC of the common friends on the node to be identified and the unconnected nodes, the more probability these nodes are connected.

Next, the definition of ACI is introduced in detail. As shown in Fig. 1, assume that node x is an important node to be identified. Define P ={ p₁, p₂, ⋯ , p_n }, where p_i represents node i in circle P, and all nodes in P(P ⊆ C) are connected to x. Define Q ={ q₁, q₂, ⋯ , q_m }, where q_i represents node i in circle Q, and all nodes in Q(Q ⊆ C) are unconnected to x, so P ∪ Q = C. ACI refers to the overall AC between node x and node set C, which is defined in detail as follows.

4.1 ACI of the node to be identified and the unconnected node groups on the CNs (UCNACI)

Neighbor nodes refer to all nodes connected to a node in the network, and the neighbor nodes of the node groups are a union of all neighbor nodes of the groups. Γ(Q) is a set of the neighbor nodes of all nodes in Q, as shown in Equation (1). $Γ (Q) = ⋃_{i = 1}^{m} Γ (q_{i})$ (1)

If Γ(x) and Γ(y) are respectively the neighbor nodes of the node x and y, Γ(x) ∩ Γ(y) is the CNs set of x and y. Similarly, the CNs set between the node x and the node group Q is defined as Equation (2). $CN (x, Q) = Γ (x) \cap Γ (Q)$ (2)

The AC of a node pair on its neighbor nodes is closely related to the number of the neighbor nodes in addition to their own degree. That is, the more the number of the CNs nodes, the higher the node pair’s AC on the CNs. The principle is also applicable to the node groups, which derives the following three UCNACIs: $S_{xQ}^{sorensonk}, S_{xQ}^{LHNk 1} and S_{xQ}^{LHNk 2}$ , as shown in Equation (3), (4), and (5), respectively. $S_{xQ}^{sorensonk} = \frac{| CN (x, Q) |}{[\frac{k (x) + k (q_{1}) + k (q_{2}) + \dots + k (q_{m})}{m + 1}]}$ (3) $S_{xQ}^{LHNk 1} = \frac{| CN (x, Q) |}{k (x) \times \max {k (q_{1}), k (q_{2}), \dots, k (q_{m})}}$ (4) $S_{xQ}^{LHNk 2} = \frac{| CN (x, Q) |}{k (x) \times \min {k (q_{1}), k (q_{2}), \dots, k (q_{m})}}$ (5)

where, |CN(x, Q) | is the CNs number of the node to be identified x and the unconnected node group Q, k(.) represents the degree of the node. And the denominator of $S_{xQ}^{sorensonk}$ is the average degree of all nodes in x and Q, while the denominator of $S_{xQ}^{LHNk 1}$ and $S_{xQ}^{LHNk 2}$ are the product of the degree of x and the maximum or minimum degree of Q, respectively.

4.2 ACI of the node to be identified on the CNs (UNACI)

In order to further elaborate the AC among the nodes, the UNACIs: $S_{xQ}^{RA 1}$ and $S_{xQ}^{RA 2}$ are proposed. $S_{xQ}^{RA 1}$ represents the direct concentration index of the node to be identified on the CNs, and $S_{xQ}^{RA 2}$ indicates the concentration of the node to be identified on neighbor nodes which remove the neighbor node P, and thus is an indirect concentration index, as shown in Equation (6) and (7), respectively. $S_{xQ}^{RA 1} = \frac{| CN (x, Q) |}{k (x)}$ (6) $S_{xQ}^{RA 2} = \frac{| Γ (x) | - | Γ (x) \cap P |}{k (x)}$ (7)

where Γ(x) represents the neighbors set of the node x, |Q| indicates the size of set Q.

4.3 ACI of the CNs on the node to be identified and the unconnected node groups (NUCACI)

In the triadic closure structure, in addition to the AC of the node pairs on the CNs, the AC of the CNs on the node pairs also has a significant influence on whether or not the friendship among the node pairs can be established. Therefore, the NUCACIs are proposed, namely $S_{xQ}^{RA}$ and $S_{xQ}^{AA}$ . The biggest difference between these two NUCACIs is that $S_{xQ}^{AA}$ reduces the dispersion of the degree of the CNs by taking the logarithm of the degree, as shown in Equation (8) and (9). $S_{xQ}^{RA} = \sum_{z ɛ CN (x, Q)} \frac{1}{k (z)}$ (8) $S_{xQ}^{AA} = \sum_{z ɛ CN (x, Q)} \frac{1}{\lg (k (z))}$ (9)

5 The construction of CSDA

However, the above proposed indexes are not suitable for all networks, and every ACI is unidirectional and single. In order to construct appropriate ACIs according to different networks, a CSDA is proposed, which uses UACIs to construct composite ACI. The algorithm enriches AC information between nodes through multi-index fusion and filters the noise information through the consensus among the indexes, to overcome the lack of available information in long-term link prediction. The whole process is shown in Fig. 3. To begin with, the n-ary BACIs are constructed by combining UACIs (n = 1, 2, . . , N, where N represents the number of single ACIs). It should be noted that initially used ACIs are called UACIs to emphasize their unidirectional and single characteristics and distinguish them from the subsequently proposed bidirectional composite indexes. Secondly, according to the principle of minimizing the upper and lower bound of prediction bias, the optimal n-ary BACI for each n is selected. Finally, the consensus of BACIs is calculated, and the index with minimum noise information is identified, that is, the CACI with the highest consensus.

Fig. 3

The framework of CSDA.

5.1 Construction of BACI

In order to overcome the lack of available information and enrich AC information between nodes, in this section, BACI is proposed through the combination of these UACIs. The specific combination process is as follows: first of all, the best index is selected among all UACIs, and the selected UACI is combined with all UACIs in a one-one combination to obtain BACIs; secondly, BACI with the smallest upper bound of the prediction deviation is selected; finally, if there exist such optimized BACI, repeat the above process until N-ary BACI is selected or meet the termination conditions.

5.2 Screening of the high performance n-ary BACI

The constructed BACI enriches the useful information of long-term link prediction but also introduces noise information, so it is hard to ensure the high performance of each index. Therefore, the performance evaluation function of indexes and the estimation method of upper and lower bounds of prediction bias are proposed to select the high performance n-ary BACI.

(1) The performance evaluation function of an index

Generally speaking, if the attention concentration score (ACS) of the node x to be identified and the unconnected node group Qis higher, the newly generated edges which connecting them during the evolution process are more. Hence the predicted score variance (Gap) is proposed, which refers to the difference between ACI and the ratio of the number of news edges to the number of the original unconnected edges, as shown in Equation (10): ${Gap}_{i} = {(\frac{L (Q_{x})}{| Q |} - {ACI}_{i})}^{2}$ (10) where ACI_i represents the i_th ACI, and L(Q_x) represents the number of the edges between x and Q in the long-term evolved network, |Q| is the number of the nodes in Q.

(2) Estimation of the upper and lower bounds of the prediction bias

From different circles sets in the given network, a sample set for ACI_i is constructed. The dependent variable in the sample is Gap_i, and the independent variable is ACI_i. To begin with, all samples are divided into two parts which include a training sample set ${AC}_{i}^{T}$ and a test sample set ${AC}_{i}^{P}$ . In order to verify the effectiveness of the algorithm in long-term link prediction, it is necessary to delete more links in the network and use these deleted links as test data. In this study, 20% of the edges in a circle was deleted, and the deleted edges were employed as the test data in a sample to calculate $\frac{L (Qx)}{| Q |}$ in Equation (10), while the remaining 80% of the edges were reserved as training data to obtain ACI_i in Eq.(10). X_iis the independent variable vector in the training sample for ACI_i, and $x_{i}^{0}$ is the independent variable vector in the test sample. The coefficient b_i and the standard error of regression S_i are obtained by training the linear regression model on the training set ${AC}_{i}^{T}$ , and Gap of ${AC}_{i}^{P}$ is predicted via $x_{i}^{0}$ , namely $\hat{{Gap}_{i}^{0}}$ , as shown in Equation (11). $\hat{{Gap}_{i}^{0}} = x_{i}^{0^{'}} b_{i}$ (11)

The prediction variance [41] estimated based on the prediction data set is expressed in Equation (12). ${Var}_{i} = S_{i}^{2} + x_{i}^{0^{'}} [S_{i}^{2} {(X_{i}^{'} X_{i})}^{- 1}] x_{i}^{0}$ (12)

Then the upper bounds ${\hat{{Gap}_{i}^{0}}}_{U}$ and the lower bounds ${\hat{{Gap}_{i}^{0}}}_{L}$ of $\hat{{Gap}_{i}^{0}}$ for ACI_i are defined as Equation (13) and (14), respectively, where t is a threshold at a 95% confidence interval, and the prediction interval is expressed as Equation (15), which depicts the accuracy of ACI_i for mining the circle of friends. Compared with the single Gap index, the interval estimation is more comprehensive. $The upper bound : {\hat{{Gap}_{i}^{0}}}_{U} = \hat{{Gap}_{i}^{0}} + t \times \sqrt{{Var}_{i}}$ (13) $The lower bound : {\hat{{Gap}_{i}^{0}}}_{L} = \hat{{Gap}_{i}^{0}} - t \times \sqrt{{Var}_{i}}$ (14) ${\hat{[{Gap}_{i}^{0}}}_{L}, {\hat{{Gap}_{i}^{0}}}_{U}]$ (15)

In the study, we define that if prediction interval A is included in prediction interval B or the upper bounds of A is less than the upper bounds of B, the prediction interval A is better than B. Therefore, according to the prediction interval, for each n, the most suitable n-ary BACI with the best performance in most test samples can be selected. Then E_n is the optimal n-ary BACI set, the screening algorithm of the high performance n-ary BACIs is expressed as follows:

Find out ACI with the best prediction interval among the seven UACIs to form E₁, and set the element counter j = 1.

Add the ACI in E_j with the seven UACIs respectively to obtain the (j + 1)-ary BACI set U_j+1.

Find out the ACI with the best prediction interval in U_j+1, and form E_j+1.

If the prediction interval of E_j+1 is better than that of E_j, and j < 6, then j = j + 1 and repeat Step 2 to Step 4, otherwise stop.

Through the above algorithm, the optimal BACI in different combinations of UACIs can be selected, that is, E₁, E₂ … , E_n, …, E_H, H ⩽ N = 7.

5.3 Construction of the CACI and discrimination of potential SMIs

Although the discrimination performance of the combined UACI is better than that of UACI, the combination of UACIs also introduces noise information. In order to further filter noise information that affects the accuracy of the algorithm, the CACI is developed by selecting the high performance n-ary BACI from E₁, E₂ … , E_H which shares the maximum consensus with others. Specifically, based on the prediction interval of the Gap of each index, the similarity of each ACI mining results for different circles is analyzed, and then the CACI is extracted. Given a test sample set AC ^P consisting of network circles, the consensus between h-ary BACI (E_h) and other indexes are shown as Equation (16), (17) and (18). ${CSACI}_{h} = \sum_{1 ⩽ v ⩽ H} CS (E_{h}, E_{v}, {AC}^{P})$ (16) where, $CS (E_{h}, E_{v}, {AC}^{P}) = \frac{\sum_{k \in {AC}^{P}} sgn ({\hat{{Gap}_{hk}^{0}}}_{U}, {\hat{{Gap}_{vk}^{0}}}_{U})}{| {AC}^{P} |}$ (17) $sgn ({\hat{{Gap}_{hk}^{0}}}_{U}, {\hat{{Gap}_{vk}^{0}}}_{U}) = {\begin{matrix} 1, if {\hat{{Gap}_{hk}^{0}}}_{U} < {\hat{{Gap}_{vk}^{0}}}_{U} \\ 0, others, \end{matrix}$ (18) where ${\hat{{Gap}_{hk}^{0}}}_{U}$ , ${\hat{{Gap}_{vk}^{0}}}_{U}$ are respectively the upper bounds of GAP of E_h and E_v for the k_th circle in AC ^P, H(H7) is the number of n-ary BACIs obtained in section 5.2. Then the ultimate CACI can be defined as Equation (19). $CACI = \underset{E_{h}}{\arg} \max {{CSACI}_{h} | 1 ⩽ h ⩽ H}$ (19)

For an SMI, by calculating the value of its composite index which has the highest consensus (i.e., CACI), we can obtain its CACS. Defining the threshold θ, when a user’s CACS > θ, it can be recognized as the potential SMI; otherwise, it is just an ordinary user. It should be emphasized that the specific value of θ can be adjusted according to the actual situation.

6 Experiment

6.1 Experiment set up

On Twitter social media, more and more users share and exchange experiences, and enterprises manage brands, then gradually form OBC [42]. A circle (i.e. community) is mainly composed of fans of a brand. 1,000 ego-net data were obtained from Twitter, including 5,400 circles and 81,362 users, where the size of ego-net ranged from 10 to 4,964 nodes [43]. The purpose of the experiment is to use CACI to determine how many edges are generated between the node to be identified and other unconnected nodes, and then the SMIs are identified through the degree of the node to be identified.

In order to train the optimal CACI, 80% of the 5400 circles were used as the training set and the remaining 20% of the circles were used to calculate the prediction interval of Gap and select CACI.

6.2 Accuracy measurement of algorithms

6.2.1 Comparison of the accuracy of different circle structures

The accuracy of the proposed algorithm was compared with the traditional single link prediction indexes (SIs) on various circles data. The most representative 7 SIs (i.e. S1 ∼ S7) are selected, which are: Salton, Sorenson, Hub promoted index (HPI), LHN, Adamic-Adar (AA), Resource allocation (RA), and average commute time (ACT). For each SI, the scores of each node pairs in a circle were calculated and the average values of these scores were standardized to replace the ACI in Equation (10). Then the Gap of each SI was calculated according to Equation (10), which is denoted as Gap(Si) (1 ⩽ i ⩽ 7), and in the same way, the CACI and its Gap were calculated, namely Gap(CACI), as shown in Fig. 4. As can be seen from the definition of Gap, the lower the value of Gap, the higher the accuracy of the algorithm, and vice versa. It can be observed from the experimental results in Fig. 4 that the average value of Gap(CACI) is lower than the average value of Gap(Si), so the accuracy of the algorithm proposed in this study is higher.

Fig. 4

Comparison of the average value of Gap(CACI) and Gap(Si).

To further illustrate the applicability of the algorithm, we classified all circles into ten categories according to the number of nodes in them, as shown in Fig. 5. In ten categories of circles, the average value of Gap(CACI) and the average value of Gap(Si) were calculated respectively, and the results are shown in Fig. 6 and Table 2. Compared with Gaps of other traditional link prediction indexes, Gap(CACI) is the smallest in all circles. To further test the prediction accuracy of the proposed algorithm, five classical fusion algorithms, such as k-nearest neighbors (KNN) [44], support vector machines (SVM) [45], random forest (RF) [46], generalized linear model (GLM) [41], classification discriminant (CD) [46], were respectively used to fuse the above SIs and calculate the corresponding value of the composite index for every node pair in each circle. Averaged and standardized these composite indexes values to replace the ACI in Equation (10) to calculate Gap(KNN), Gap(SVM), Gap(RF), Gap(GLM), and Gap(CD) in the circle, respectively. Similarly, Gap(CACI). in each circle was calculated. The average of Gaps generated by each fusion algorithm under different categories of circles is shown in Fig. 6 and Table 2. Compared with Gaps of other fusion algorithms, Gap(CACI). is the smallest in all circles. In conclusion, the proposed algorithm has great advantages for overcoming the lack of information in long-term link prediction. Besides, our algorithm can achieve excellent performance in both large circles with dense nodes and small circles with sparse nodes. This shows that the proposed algorithm has strong robustness and is not affected by the sparse or dense features of the network.

Fig. 5

Number distribution of circle nodes.

Fig. 6

Comparison of precision under different structural circles.

Table 2

Comparison of precision under different structural circles

Number of Degree	(0,5]	(5,10]	(10,15]	(15,20]	(20,25]	(25,30]	(30,35]	(35,40]	(40,45]	(45,+∞)
CACI	0.164	0.147	0.138	0.105	0.110	0.090	0.102	0.091	0.056	0.132
S1	0.248	0.201	0.243	0.222	0.218	0.189	0.236	0.323	0.120	0.300
S2	0.327	0.262	0.319	0.270	0.274	0.245	0.350	0.491	0.151	0.448
S3	0.274	0.252	0.283	0.226	0.250	0.247	0.199	0.266	0.134	0.335
S4	0.199	0.179	0.158	0.159	0.219	0.144	0.121	0.166	0.098	0.160
S5	0.218	0.224	0.240	0.213	0.180	0.225	0.216	0.273	0.114	0.218
S6	0.205	0.205	0.179	0.209	0.178	0.164	0.151	0.172	0.101	0.190
S7	0.323	0.355	0.421	0.305	0.302	0.314	0.326	0.365	0.128	0.249
KNN	0.296	0.249	0.323	0.264	0.281	0.227	0.109	0.388	0.134	0.188
SVM	0.334	0.580	0.747	0.836	0.877	0.561	0.913	0.925	0.937	0.943
RF	0.241	0.255	0.300	0.215	0.178	0.227	0.109	0.156	0.135	0.188
GLM	0.212	0.621	0.768	0.836	0.877	0.896	0.913	0.925	0.937	0.943
CD	0.304	0.620	0.581	0.836	0.877	0.896	0.913	0.925	0.937	0.943
Number of circles	2704	971	321	106	75	45	34	12	40	45

6.2.2 Analysis of the effect of adopting consensus discrimination

In the process of establishing the CACI, there are two main steps: the first step is to screen the high performance n-ary BACIs, and the second step is to obtain the CACI based on attention concentration consensus. To illustrate the validity and necessity of computing the concentration consensus, the accuracy of CACI and BACI obtained by only screening the high performance UACI, as shown in section 5.2, were compared, that is, Gap(CACI). and Gap(BACI). were compared, as shown in Fig. 7. It can be seen from Fig. 7 that CACI has a lower Gap. value in the most structural circles, which indicates that CACI is more accurate. Moreover, in sparse circles with fewer nodes, the advantages of CACI are more obvious, while in the large circles with dense nodes, for example, the number of circle nodes is between 20 and 45, CACI still has more obvious advantages compared with BACI. These findings strongly demonstrate that the introduction of CACI helps improve the performance of the algorithm.

Fig. 7

Accuracy comparison between Gap(CACI) and Gap(BACI).

6.3 Potential SMIs discrimination for concept marketing

In this study, three typical structural circles with a small to a large number of nodes were selected, which respectively contained 5, 13, and 39 nodes. Then, some nodes in the circle were selected as candidate SMIs, which were arranged in descending order according to their degrees. Usually, the node with a higher degree is chosen as the SMI, but there are many such nodes. In order to determine which node has more potential as an SMI, the proposed CSDA was used to calculate CACS of the users to be identified and the unconnected users, which are shown in Table 3. The node with the largest degree is often chosen as the SMI, for example, in a small circle, node 1 with the highest degree is regarded as an SMI, but its CACS is ranked as 3, therefore node 2 whose rank is 1 is chosen as an SMI. And there are similar situations in the middle circle. Only in the large circle, the SMI judged by degree is consistent with the SMI determined by CACS ranking (i.e. node 1). Therefore, the proposed method has greater application value for those developing middle circles and small circles.

Table 3
Ranking of influence potential of different circle nodes

Range of circles Node ID CACS Importance ranking

Small circle 1 0.239951 ③

2 0.999953 ①

3 0.219645 ④

4 0.988947 ②

Middle circle 1 0.658799278 ⑤

2 1.809234578 ②

3 1.674397154 ③

4 1.332321414 ④

5 2.152969861 ①

Large circle 1 1.00003785 ①

2 0.547559149 ⑥

3 0.549509252 ⑤

4 0.259900771 ⑧

5 0.684113272 ③

6 0.597234231 ④

7 0.691563257 ②

8 0.519769996 ⑤

Range of circles	Node ID	CACS	Importance ranking
Small circle	1	0.239951	③
	2	0.999953	①
	3	0.219645	④
	4	0.988947	②
Middle circle	1	0.658799278	⑤
	2	1.809234578	②
	3	1.674397154	③
	4	1.332321414	④
	5	2.152969861	①
Large circle	1	1.00003785	①
	2	0.547559149	⑥
	3	0.549509252	⑤
	4	0.259900771	⑧
	5	0.684113272	③
	6	0.597234231	④
	7	0.691563257	②
	8	0.519769996	⑤

7 Discussion and conclusions

In OBC, the way to influence the customer’s consuming behavior by conveying the product concept will make a far-reaching impact on the customers. Especially, it is highly credible to spread product or service ideas through influential SMIs. To ensure the sustainability of concept marketing, tapping the potential SMIs is more advantageous than choosing the popular SMIs. This study aims at the dilemma of the lack of available information in the discrimination of the potential SMIs and proposes CACI to make the long-term predictions of the links in the circle of friends, to identify potential SMIs. To be specific, the ACIs are proposed to measure the AC among the node to be identified, an unconnected node group, and their CNs. Then, the CSDA is proposed to identify the potential SMIs according to these ACIs. The proposed algorithm mainly includes the following steps: Firstly, unidirectional ACIs are combined into different n-ary BACIs to overcome the lack of available information and enrich the AC information between nodes. Subsequently, the upper and lower bounds of their prediction bias are calculated and the n-ary BACI with the highest performance is selected according to the minimum of the upper and lower bounds. Then, the consensus of each n-ary BACI is calculated to find the most suitable CACI, which filters the noise information and improves the accuracy of the discriminant algorithm. Finally, the potential SMIs are identified according to CACI, and the experimental results verify the accuracy of the CACI in different circle structures.

There are still some shortcomings that need to be improved in future research. In this study, we only employ the social relationship between network nodes to analyze the trend of network evolution, without considering the impact of external factors on network evolution. In future research, we will introduce the external factors to predict the network evolution. Besides, the emotional attitude of SMIs toward concept marketing is not taken into consideration, for example, positive, negative, or neutral, which heavily affect the benefits of concept marketing. The emotional attitude of SMIs for concept marketing should be fully considered in future research to improve marketing efficiency.

Footnotes

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 71871135).

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