Opinion dynamics with intermittent-influence leaders on the signed social network

In this paper, the leader-follower architecture is constructed by combining intermittent-influence leaders with a signed social network. Unlike a typical network with leaders where leaders are supposed to continuously influence followers, in this article, the leaders intermittently influence followers. Furthermore, the number of influences is limited. We focus on how the intermittent-influence leaders impact the evolution of followers’ opinions. The relationship between followers’ opinions and the number of leader broadcasts is analyzed in detail. Then, the number of broadcasts is regarded as the cost, and the changing trend of the revenue per broadcast is obtained. The results show that as the number of broadcasts increases, the revenue per broadcast decreases gradually. Finally, the concept of assimilation is introduced to weigh the costs and benefits, and the minimum number of broadcasts required for the leader to assimilate the followers is derived. Two examples are given to demonstrate the validity of the main conclusions.


INTRODUCTION
In recent years, opinion dynamics, as an international hot research topic in the field of systems and control science, has been widely discussed. 1-4 Opinion dynamics is concerned with a social network closely related to human life, in which agents in a network can be countries, social groups, or living individuals. 5,6 The individuals' evolution of behaviors and opinion in social networks is specifically and deeply studied in opinion dynamics. 7 Opinion dynamics in society is scientifically modeled and analyzed, which can not only reveal the laws of development of human society and animal groups in nature (such as zero-sum games in futures, the migration of animal populations), but also facilitate the development of man-made complex networks such as the Internet and transportation networks. [8][9][10] To characterize the opinion evolution in social networks, a classical linear time-invariant model, also known as the DeGroot (DG) model, was proposed in Reference 11. In this model, each individual's opinion at this time is determined by the weighted average of his own and his neighbors' opinion at the previous moment. In the past decades, the research on the consensus of the DG model has never stopped. For any individual in the network, all the remaining individuals are not necessarily its neighbors. Despite this, the network can achieve the opinion consensus as long as the network connectivity is strong enough. When each node has a self-loop, the network can achieve the opinion consensus if and only if the network has a spanning tree. 12 Later, in order to describe the interaction of opinions in more detail, a series of variants of the DG model such as the Friedkin-Johnsen (FJ) model 13 and the Hegselmann-Krause (HK) model 14 were proposed, and many outstanding results were obtained. [15][16][17] The DG model is widely used in reality, such as company board of directors and jury panels. 18 It is worth noting that in the classical DG model, only cooperative relationship between individuals is considered. However, whether in nature or human society, the confrontation (or antagonism) exists widely. 19,20 In nature, animals compete for food and territory. Meanwhile, plants compete for sunlight and water. For human society, people compete for a series of valuable things such as social resources and natural resources. Therefore, opinion dynamics with antagonistic relationship has attracted a lot of attention. [21][22][23] Typically, the antagonistic relationship between individuals can be modeled by negative ties, which makes the network topology corresponding to a social network with antagonism into a signed graph. 22 For a signed graph, different control protocols may lead to polarization, fluctuation, and neutrality. 24 In Reference 20, the antagonism was introduced into the continuous-time DG model, and the necessary and sufficient conditions for opinion polarization were obtained when the signed graph was structurally balanced. However, since the structurally unbalanced network had a very complex structure, the structurally unbalanced networks were less discussed. 25 Although antagonism may be a source of inconsistency, the authors of Reference 25 pointed out that opinions could achieve a consensus in a signed network. In Reference 26, the sign-consensus of opinion was studied when opinion can not reach an agreement. In fact, the opinion evolution on the signed graph was more complicated than the unsigned graph due to the existence of antagonism. 20 Hence, more efforts should be made to explore the antagonistic impact on opinion evolution.
It should be pointed out that in all the opinion dynamics models mentioned above, each individual has exactly the same characteristics, that is, while they influence others' opinions, their opinions are also influenced by others' opinions. But in real life, there are special individuals called opinion leaders who are not influenced by other individuals. 27 Considering the existence of opinion leaders, the leader-follower framework was used to model opinion evolution. 28,29 The leader-follower architecture divides individuals in a social network into leaders and followers. Leaders, as special individuals, influence the opinions of followers while keeping their own opinions unchanged. In other words, the opinions of all followers are influenced by the leaders and will gradually tend to the opinions of the leaders.
Recently, the lead-follower architecture has been widely discussed in social networks. 30,31 The article 32 pointed out that opinion leaders can make the group's opinion converge faster. The opinion leaders were introduced to the fractional opinion formation model in Reference 33 and sufficient conditions for all followers to converge to the leader's opinion were obtained. In Reference 34, the authors got a sufficient condition to ensure that the followers' opinions move at the same speed as the dynamic leader's opinion (or the opposite of the dynamic leader's opinion). In the above-mentioned articles, leaders are continuously involved in the evolution of group opinions in a social network. This persistent influence truly characterizes the evolution of opinions in small groups. However, with the rapid development of communication technology and Internet technology, the way humans obtain and interact with information has undergone major changes. On the Internet, the influence of the leaders on the followers is often intermittent rather than continuous. 35 The intermittent-influence leaders are quite common on the Internet. For example, popular information exchange platforms such as Twitter or Weibo can be regarded as social networks. On this network, famous people such as celebrities can be regarded as leaders, and correspondingly their subscribers are regarded as followers. Famous people, through their tweets or blogs, influence their subscribers intermittently. In particular, the number of influences is limited. In this case, the follower's opinion often cannot converge to the leader's opinion, and there will be more complex phenomena in the signed social networks. In this article, these phenomena and the underlying factors that determine them are analyzed and discussed. This is undoubtedly an interesting thing.
The purpose of this study is to analyze the influence of intermittent-influence leaders on the evolution of followers' opinions in a signed social network. This intermittent influence of the leader on the followers is called broadcast. Without loss of generality, in this paper, we assume that followers have sufficient time to interact with opinions to achieve a modulus consensus of opinions after the leader's last broadcast. The main contributions of this paper are as follows.
First, we design intermittent influence leaders for a signed social network. Whereas, Zhao et al. 35 considered leaders with intermittent influence for an unsigned network. In References 34,36, the influence of leaders is continuous. However, in our paper, the broadcasts are intermittent and limited. This is more realistic.
Second, since broadcasts are intermittent and limited, we generally think that the opinions modulus consensus of followers is related to the number of broadcasts and the time of broadcast. For single-leader and multi-leader situations, a sufficient and necessary condition and a sufficient condition are given respectively to ensure that the opinions of the modulus consensus of followers which are only related to the number of broadcasts and not to the time of broadcast. Also, an expression of the relationship between the opinions of followers' modulus consensus and the number of broadcasts is obtained.
Finally, as the number of broadcasts increases, the follower's opinion will gradually approach the leader's opinion (or the opposite of its opinion value), and the number of broadcasts can be regarded as the cost. Through analysis, we can conclude that as the number of broadcasts increases, the revenue of each broadcast decreases gradually. Then, after introducing the concept of assimilation, we discuss how to weigh the costs and benefits.
This paper is organized as follows. In Section 2, we first introduce the notions and the relevant knowledge of graph theory. Then we introduce the DG model with the antagonistic relationship. Finally, we conduct a brief introduction of the model studied in this paper. The main results and proofs are in Section 3. Section 4 provides two examples to verify our conclusions. Finally, we give our conclusion in Section 5.

Notions
In this article, the following rules regarding symbols are given. R n×m represents a real matrix with n rows and m columns, and R n represents a real space of n dimensions. For any square matrix W = [w ij ] ∈ R n×n , (W) denotes its spectral radius and every element in |W| is the absolute value of each element in the matrix W. W ≥ 0 means that every element in matrix W is not less than 0. A nonnegative n-dimension square matrix W is called a stochastic matrix if its row sums are all equal to 1. For any column vector represents a diagonal matrix whose main diagonal elements are the elements in the vector b. 1 n (0 n ) is the n-dimension column vector with all elements being 1 (0). I n is an n-dimension identity matrix. ∅ is used to represent the empty set. ⟨℘⟩ is used to denote the number of elements in the set ℘. ⌈m⌉ represents the first integer greater than m. Δy(k) represents the forward difference of the function y(k), that is, Δy(k) = y(k + 1) − y(k).

Graph theory
Let G(W) = G(, , W) represent a weighted directed graph, where the node set  = (v 1 , … , v N ), the edge set  ⊆  × , and W = [w ij ] ∈ R N×N represents the weighted adjacency matrix corresponding to this graph. An edge from v j to v i can be denoted as e ji = (v j , v i ) ∈ , it indicates that node i can receive information from node j. And e jj = (v j , v j ) indicates that node j has a self-loop. w ij ≠ 0 if and only if e ji ∈ . A directed path from v i to v j is a sequence of nodes starting at v i and ending at v j , where any node is distinct. In a directed graph, if there is a node v o that has directed paths to all other nodes, then the node v o is called the root node. A directed graph has a spanning tree if there is at least one root node in the graph. When all nodes in a directed graph are root nodes, the graph is said to be strongly connected.

DeGroot model with antagonism
Let us consider a social network consisting of N individuals. The interactions of these N individuals can be represented by a weighted directed graph G(W). In this directed graph, the signs of edges can be negative, so G(W) is a signed graph.
In the signed matrix W, w ij > 0 (w ij < 0) means that individual i trusts (distrusts) individual j, so there is a cooperative (antagonistic) relationship between individual i and individual j. Naturally, w ij = 0 and w ji = 0 indicate that there is no relationship between individual i and individual j. Let z i (k) represent the opinion of individual i at time k. z i (k) can be positive, negative, or zero. When z i (k) is positive (negative), it means that individual i holds a positive (negative) attitude on the topic at time k. When z i (k) = 0, it means that the individual i is neutral on the topic at time k. The opinion of individual i at time k is affected by its own opinion and the opinion of its neighbors at time k − 1, that is, (1) , then Equation (1) can be written in the following compact form: Definition 1. The signed network (2) can achieve modulus consensus if there exists ∈ R such that for any i, j = 1, 2, … , N. In particular, if ≠ 0, the signed social network (2) can achieve bipartite consensus.
In the signed social networks, the bipartite consensus is the polarization of individuals' opinions into two opposite groups. In order to deal with this kind of signed graph more conveniently, we also need the following definition and assumptions.
The sign of the edges between the vertices in  ′ ( ′ ′ ) is positive, while the sign of the edges formed by the nodes in  ′ and  ′ ′ is negative.

G(W) is structurally balanced if and only if there exists a diagonal matrix
structurally balanced and has a spanning tree. When there is a leader, the leader maintains a cooperative relationship with the followers in set  ′ ( ′ ′ ) and an antagonistic relationship with the followers in set  ′ ′ ( ′ ). Similarly, when there are multiple leaders, multiple leaders are cooperative with the followers in set  ′ ( ′ ′ ) and antagonistic with the followers in set  ′ ′ ( ′ ).

Assumption 2. For all
Remark 1. When the network G(W) composed of followers is ( ′ ,  ′ ′ ) structurally balanced, the leader must maintain a cooperative relation with individuals in one set ( ′ or  ′ ′ ) and an antagonistic relation with individuals in the other set, which is necessary to ensure that the social network containing leaders is structurally balanced. If not, the relationship between individuals in the network becomes extremely complicated and difficult to describe, and the evolution of opinion is full of uncertainty due to a sea of different scenarios one needs to analyze. On the other hand, when leaders no longer influence followers, it is necessary for the network to contain a spanning tree about modulus consensus. So, in Assumption 1, the social network is required to have a spanning tree.
Next, we discuss the evolution of individuals' opinions in the social network from the perspective of structurally balanced and structurally unbalanced networks, respectively.
When the G(W) is ( ′ ,  ′ ′ ) structurally balanced, its node set  can be divided into two vertex sets  ′ and  ′ ′ . There must be a diagonal matrix is a row random matrix, and the left and right eigenvectors corresponding to the eigenvalue of 1 are l and 1 N , respectively. In addition, we have l T 1 N = 1. For this social network to achieve modulus consensus, we assume that G(W) has a spanning tree. Let g(k) = Γ 2 z(k). Then we have Obviously, G(Γ 2 WΓ 2 ) has a spanning tree and the elements of its main diagonal are all greater than 0. In this case, 1 is the only maximum modulus eigenvalue of Γ 2 WΓ 2 and its algebraic multiplicity is 1. 12 Naturally, we have lim k→∞ (Γ 2 WΓ 2 ) k = 1 N l T . 37 Then, lim k→∞ g(k) = 1 N l T g(0) and lim k→∞ z(k) = Γ 2 1 N l T g(0). Let = l T g(0), we have lim k→∞ g(k) = 1 N and lim k→∞ z(k) = Γ 2 1 N . It can be seen from the above equations that when there are no leaders participating on a signed social network, if ≠ 0, the opinions of individuals in a social network are polarized into two distinct groups and − .
When the G(W) is structurally unbalanced and strongly connected, (W) < 1. 25 Naturally, lim k→∞ W k = 0, where 0 is a matrix with all elements being 0. At this time, for any initial value z(0), we have lim k→∞ z(k) = 0 N . Eventually, all individuals remain neutral on the topic.

Model formulation
In this section, we introduce the intermittent-influence leaders into the DeGroot model with antagonism, and propose a leader-follower architecture on a signed social network. This type of leaders is an abstraction of some of the stars on the Internet platform. Stars can influence subscribers through text and video, while their own opinions are not influenced by subscribers. In short, subscribers are not neighbors of stars and stars are neighbors of subscribers, that is, the vertices of leaders are not influenced by anyone else.
For the convenience of description, the behavior of the leader influencing the followers is called broadcast, and these intermittent moments when the leader influences the followers are called broadcast moments. The opinions of followers evolve independently of the leaders at most moments, which are known as silence moments. In this paper, we suppose that the influences of leaders on followers is synchronous. the leaders can broadcast their opinions to all followers at a certain moment. This assumption is mainly based on the following two reasons. Firstly, this situation does exist in real social networks, for example, the board of directors held by the company from time to time. In this case, the chairman can be regarded as the leader. Other members can be regarded as followers. At the board meeting, the chairman can broadcast his/her information to all followers. Therefore, the broadcast is synchronous. Secondly, this assumption is amicable for us to analyze the model theoretically. If the broadcast is asynchronous, we need to make more efforts in theoretical analysis. This will be a direction of our future work.
The influence factor b i ∈ [−1, 1] represents the influence of the leader on the follower i at the broadcast moment. When individual i receives the influence of the leader, the influence of the rest of i's neighbors on i is weakened, so the influence of i's neighbors on the individual i becomes (1 − |b i |) times the original influence. Without loss of generality, the broadcast factor is assumed to be finite, that is, When there is only one leader, we use and x to represent the leader and the opinion of leader, respectively.
At this time, the state vector in Equation (2) is augmented to z(k) = [z , z T (k)] T , and the adjacency matrix is augmented to T (F) at the broadcast moments (silence moments). The specific forms of T and F are as follows: When there are multiple leaders, we use superscripts to number the leaders for convenience. Suppose there are p leaders, denoted by 1 , 2 , … , p , respectively. Correspondingly their opinions are denoted as z = T represents the influence of the leader d on N followers. B d = diag(|b d |). At this time, the opinion vector in Equation (2) is augmented to z(k) = [z T , z T (k)] T ∈ R N+p , and the adjacency matrices at the broadcast moments and silence moments are respectively augmented to the following forms: Since the broadcast moments are intermittent and limited, without loss of generality, the set of broadcast moments is represented by ℘ = {t 1 , … , t k }, and the number of broadcasts is denoted as ⟨℘⟩. Let z * denote the final opinion vector when leaders participate in the opinions' evolution of social network. It is supposed that after the last broadcast, the followers can still have sufficient discussions. Naturally, the evolution of a social network consisting of leaders and followers can be expressed as: where Remark 2. F * represents a phenomenon in which the followers' opinions evolve independently of the leaders after the last broadcast. This phenomenon fits our real life. After browsing the information on Weibo, Twitter and other network platforms, individuals often share, communicate and discuss information online or offline with friends, relatives, colleagues, and so forth.

MAIN RESULTS
Inspired by the fact that internet famous use text or video to influence their subscribers intermittently on Twitter or other network platforms, we introduce intermittent-influence leaders into the DG model with antagonistic relationship. With the participation of these particular leaders, the evolution of the follower's opinions has different special properties than those described in the existing literature. This article will focus on these special properties and the determinants behind them.
Since the number of broadcasts is limited, the followers' opinions cannot completely converge to the leader's opinion (or its opposite value), but only tend to the leader's opinion (or its opposite value) to a certain extent. It is worth noting that the degree of this tendency is related to the broadcast moments and the number of broadcasts. Therefore, we use mathematical expressions to give the relationship between the two in structurally balanced and structurally unbalanced networks, respectively. As the number of broadcasts increases, so does the corresponding control cost. Trends in the change in marginal revenue per broadcast are analyzed. And after introducing the concept of assimilation, we give our unique insights on how to weigh the cost and benefit.
Theorem 1. Assume that Assumptions 1 and 2 are satisfied. If there is only one leader and ⟨℘⟩ is given, then opinion evolution of followers is independent of the set of broadcast moments ℘ if and only if |b i | =b for ∀i = 1, 2, … , N. Furthermore, if ⟨℘⟩ = k, the final opinion vector of follower has the following form: where = l T g(0), = 1 or −1.
In the Theorem 1, a control strategy is proposed. Under this control strategy, then opinion evolution of followers is independent of the set of broadcast moments.and related to the number of broadcast. It should be pointed out that it is almost impossible for leaders to make opinions of all followers converge with the leader' opinions when antagonistic relationships is considered. In this paper, we assume that the number of broadcasts is limited. So when the network of all followers is structurally balanced and contains a spanning tree, all followers will eventually be divided into two clusters. 20 The absolute values of opinions for individuals in different clusters are the same, but the signs are opposite. This means that no matter what control strategy leaders use, leaders can only let a few followers converge with their opinions. In this case, a control strategy for assimilating some individuals is given in Theorem 1. The relationship between followers' opinions and the number of leader broadcasts has been analyzed investigated in Theorem 1. Unfortunately, in this case, some individuals will go to the opposite of leaders. Theorem 3 later tells us when the network of all followers is structurally unbalanced and strongly connected, all followers remain neutral on the topic. This means that leaders can not influence any individual only through limited broadcasts. For the proof of theorem 1, see Appendix A.
Remark 3. According to Definition 2 and Assumption 1, all followers are divided into two camps  ′ and  ′ ′ , and the single leader is also in one of the two camps. In signed graphs G(T) and G(F), each corresponding individual is in the same camp, and the interaction between individuals in different camps is different. The element on the main diagonal of the diagonal matrix Γ is 1 or −1 depending solely on the camp of each individual, so the matrix Γ is the same for both T and F. Remark 4. In Equation (A14), or − is the final opinion value of followers when no leader is involved. The constant z is the leader's opinion value. When = 1, the modulus of each follower's opinion is: When = −1, similarly, the modulus of each follower's opinion is: It can be seen from the above results that the modulus consensus opinion of followers is only related to the number of broadcasts and has nothing to do with the broadcast moments. [1 − (1 −b) k ]z represents the influence of the leader on the followers. The more broadcast times, the greater the influence of the leader on the followers. When the number of broadcasts is close to 0, lim k→0 (1 −b) k = 1. Naturally, z * = Γ 2 1 N , and the evolution of the followers' opinions is independent of the leader, that is, each follower's opinion is either or − . When the number of broadcasts tends to infinity, lim k→∞ (1 −b) k = 0. After that, z * = Γ 2 1 N z . In this case, each follower's opinion is either z or −z . Next, we will specifically study which individuals in the followers tend to z and which tend to −z as the number of broadcasts k tends to infinity. Corollary 1. It is assumed that the conditions and assumptions of Theorem 1are satisfied. When the number of broadcasts tends to infinity, if = 1 ( = −1), the opinions of followers in  ′ ( ′ ′ ) converge to z , and the opinions of followers in the set  ′ ′ ( ′ ) converge to −z .
Proof. By Equation (A14), lim k→∞ z * = Γ 2 1 N z . When = 1, lim k→∞ z * = Γ 2 1 N z . Naturally, the opinion of followers in set  ′ converges to z , while the opinion of followers in set  ′ ′ converges to −z . When = −1, similar to the above discussion, it can be concluded that the opinion of followers in set  ′ ′ converges to z , and the opinion of followers in set  ′ converges to −z . So far, the proof of this Corollary is completed. ▪ Remark 5. Corollary 1 points out that in a structurally balanced network, when the number of broadcasts tends to infinity, the followers in the cluster that cooperated with the leader would tend to the leader's opinion, while the followers in the other cluster would tend to the opposite value of the leader's opinion, that is, the opinions of followers achieve bipartite consensus. In Reference 34, a continuous-time model of such opinion separation problems with signed networks was investigated. The authors established a necessary and sufficient condition, which ensure all opinions finally converge to bipartite consensus as long as the network topology is structurally balanced. In Reference 36, the authors examined an eigenvector-based peer selection strategy and show how the strategy can be beneficial in the spreading efficiency of stubborn individuals' opinions. For the signed social networks with stubborn individuals, this strategy can ensure that the network achieves bipartite consensus. Although all followers' opinions reach bipartite consensus in References 34 and 36, the influence of leaders on followers is continuous and infinite. Corollary 1 shows that our results are consistent with the results of References 34 and 36 if the broadcasts are assumed to be continuous.
Combining Theorem 1 and Corollary 1, when = 1, the more the number of broadcasts, the closer the followers' opinion in  ′ is to the leader's opinion z , and the closer the followers' opinion in  ′ ′ is to −z . The less the number of broadcasts, the closer the followers' opinion in  ′ is to the opinion when no leader is involved, and the closer the followers' opinion in  ′ ′ is to − . Similarly, when = −1, the more the number of broadcasts, the closer the opinion of followers in  ′ and  ′ ′ is to −z and z , respectively. The less the number of broadcasts, the closer the followers' opinion in  ′ and  ′ ′ is to and − , respectively.

Theorem 2.
Suppose there are multiple leaders and Assumptions 1 and 2 are satisfied. When ⟨℘⟩ is given, for ∀i, j ∈ , the modulus consensus opinion of followers is independent of the set of broadcast moments ℘. Further, if ⟨℘⟩ = k, the final opinion vector of followers has the following form: Proof. Similar to the proof of Theorem 1, we havẽ where Γ 1 = I p or −I p , Γ 1 = I p if multiple leaders are cooperative with the followers in set  ′ and antagonistic with the followers in set  ′ ′ , Γ 1 = −I p if multiple leaders are cooperative with the followers in set  ′ ′ and antagonistic with the followers in set  ′ . When ∀d = 1, 2, … , p, Since Γ 2 WΓ 2 is row-random, , that is,TF =FT. Naturally, TF = ΓTΓΓFΓ = ΓTFΓ = ΓFTΓ = FT. In this case, the modulus consensus opinion of followers is independent of the broadcast moments ℘. If ⟨℘⟩ = k, Equation (A4) has the following form: In the end, the opinion vector of followers is as follows: j z j , the modulus of each follower's opinion can be expressed as j=1b j z j , the modulus of each follower's opinion can be expressed as Comparing (21) and (22) with (11) and (12) respectively, it is not difficult to find that the multi-leader situation can be equivalently regarded as the single-leader situation with x = ∑ p j=1b j x j and the influence factor isb ′ . At this time, the modulus consensus opinion of followers is jointly influenced by all leaders.
In the above discussion, we have studied the case that the signed graph is structurally balanced. For the structurally unbalanced network, we have the following result.

Theorem 3. Assume that Assumption 2 is satisfied. When ⟨℘⟩ is given, if G(W) is structurally unbalanced and strongly connected, eventually all followers remain neutral on the topic.
Proof. When G(W) is structurally unbalanced and strongly connected, (W) < 1. Naturally, lim k→∞ W k = 0. After that, Substituting F * into Equation (8) At this point, we have investigated the evolution of the opinion of followers for the structurally balanced and unbalanced networks, respectively. Next, we focus on the marginal effect of each broadcast by the leader. As the number of broadcasts in structurally unbalanced network has no effect on the evolution of followers' opinions, so the marginal effect is specifically analyzed only in the structurally balanced network. For this, we have the following results. Proof. When = 1, the opinion of followers in  ′ lies between the opinion z of the leader and the opinion of the followers without the participation of the leader. The opinion of followers in  ′ ′ is between −z and − . At this time, combined with Equation (11), we can use |[1 − (1 −b) k ]z + (1 −b) k − | to represent the deviation of the opinion. The larger is the deviation of this opinion, the closer the opinion of followers in  ′ is to z , and the closer the opinion of followers in  ′ ′ is to −z . Therefore, this deviation can be used to measure the benefits of broadcasts. Without loss of generality, the number of broadcasts k is used to denote the cost. After that, we can define a cost-performance function y(k) whose specific form is as follows: Taking forward difference of y(k), we can get Δy(k) = y(k + 1) − y(k) Let x(k) = (kb + 1)(1 −b) k − 1, then Since k is a positive integer and 0 <b < 1, Δx(k) < 0. So, the maximum value of the monotonically decreasing function x(k) is x(1) = (b + 1)(1 −b) − 1 = −b 2 < 0. Naturally, Δy(k) < 0, the cost-performance function y(k) is monotonically decreasing. When = −1, combined with Equation (12), deviation of the opinion can be denoted as | − [1 − (1 −b) k ]z + (1 −b) k − |. At this point, the cost-performance func- Similarly, we can conclude that the cost-performance function is still decreasing.
Theorem 4 states that when the network includes a leader, the marginal revenue per broadcast decreases gradually as the number of broadcasts increases. In fact, this phenomenon also applies to multiple leaders. When extending the single-leader situation to the multiple-leader situation, we just need to replaceb and z withb ′ and ∑ p j=1 (b j ∕b ′ )z j , respectively. Later, the same conclusion can be drawn. ▪ Remark 7. For a cooperative social network, it was obtained that the marginal revenue of broadcasting decreases gradually with the increase of the number of broadcasts. 35 Theorem 4 shows that it is still true for a structurally balanced social network with cooperation and antagonism.
From the above discussion, it can be concluded that in a structurally balanced leader-follower network, when the leader maintains a cooperative relationship with the followers in set  ′ ( ′ ′ ) and an antagonistic relationship with the followers in set  ′ ′ ( ′ ), with the increase of the number of broadcasts, the opinion of the followers in  ′ ( ′ ′ ) will gradually approach that of the leader. but the marginal revenue per broadcast is decreasing. The number of broadcasts represents the cost, and without loss of generality, it is assumed that the cost of each broadcast is the same. Naturally, how to weigh the cost and the benefit of broadcasts is an intriguing topic. To this end, we introduce the following concept of assimilation.
In a leader-follower network, the leader's opinion is assumed to be z . Taking the opinion z as the center, the constant is the neighborhood radius. Followers are said to be assimilated by the leader if their opinions are located within the neighborhood. In particular, the neighborhood radius is called the assimilation limit. The minimum number of broadcasts required to assimilate followers is inextricably linked to the assimilation limit. Next, under the premise that is the assimilation limit, the minimum number of broadcasts required for assimilation is specifically analyzed.

Corollary 2.
It is assumed that the conditions and assumptions of Theorem 4 are satisfied. Then, the following conclusions hold.
1. If = 1, the minimum number of broadcasts required by the leader to assimilate the followers in  ′ is ⌈ ⌉.
2. If = −1, the minimum number of broadcasts required by the leader to assimilate the followers in Proof. Item 1: When = 1, assume the minimum number of broadcasts required by the leader to assimilate all followers is m, then we have Afterwards, Taking the logarithm of both sides of the above two inequalities, we have Finally, Item 2: When = −1, assume the minimum number of broadcasts required by the leader to assimilate all followers is m, then we have Similar to the processing of Item 1, we have

NUMERICAL EXAMPLES
In this section, we will give several examples to observe the evolution of the systems state in order to verify our conclusions.
Example 1. Consider a social network ℑ 0 with four followers, and the associated adjacency matrix W is given by: Obviously, G(W) has a spanning tree and is structurally balanced. Individuals 1 and 2 belong to set  ′ , and individuals 3 and 4 belong to set  ′ ′ . At this time, through simple calculation, we can obtain that the corresponding left eigenvector is l = [1, 0, 0, 0] T when the eigenvalue of matrix ΓWΓ is 1. Assume that the initial value of the four followers is z(0) = [0.2, 0.6, −0.7, 0.32] T . Naturally, when the leader is absent, each follower's opinion is = 0.2 or − = −0.2.
After that, we construct a leader-follower network ℑ 1 by introducing the leader into network ℑ 0 . Suppose the leader maintains a cooperative relationship with the followers in the set  ′ and its opinion is z = 1.  Figure 1. In Figure 1, the black lines represent the evolution of followers' opinions when no leader is involved ( and − ), and the green lines represent the evolution of followers' opinions when the leader continues to influence followers (z and −z ). The red lines, the cyan lines, the pink lines, and the blue lines represent the evolution of the followers' opinions when the leader broadcasts 1, 2, 3, and 4 times, respectively. It is not difficult to see that with the increase of the number of broadcasts, the opinions of followers in  ′ gradually tend to 1, while those of followers in  ′ ′ gradually tend to −1. Next, we consider the marginal revenue per broadcast in network ℑ 1 . When the number of broadcasts is 1, 2, 3, and 4, the corresponding marginal revenue is clearly represented in Table 1. Obviously, the marginal revenue per broadcast decreases gradually as the number of broadcasts increases.
Example 2. Consider a network ℑ 2 with four followers, and the adjacency matrix associated with it is as follows: Clearly, G(W) is strongly connected and aperiodic and structurally unbalanced. After that, we introduce three leaders into the network ℑ 2 to construct a leader-follower network  Figure 2. The red lines, the cyan lines, the pink lines, and the blue lines represent the evolution of the followers' opinions when the leader broadcasts 1, 2, 3, and 4 times, respectively. In Figure 2, regardless of the number of broadcasts (the number of broadcasts is finite), the four individuals end up being neutral on the topic.

F I G U R E 2
The opinion evolution for four followers in Example 2.

CONCLUSIONS
This paper investigates the influence of intermittent-influence leaders on followers' opinions in a signed social network. First, for a structurally balanced network, we analyze the relationship between the followers' final opinions and the number of broadcasts, and extend the single-leader case to the multiple-leader case. Second, in a structurally unbalanced network, it is concluded that all followers remain neutral on the topic regardless of the number of broadcasts. Finally, by analyzing the cost-performance function for a structurally balanced network, the fact that the marginal revenue per broadcast decreases gradually as the number of broadcasts increases is obtained. Therefore, the concept of assimilation is introduced to solve the problem of how to weigh the costs and benefits, and the minimum number of broadcasts required by leaders to assimilate followers has been calculated. Finally, two examples are used to verify our conclusions.
In the future, we will try to consider consider the impact of asynchronous broadcasts on the evolution of follower's opinions.

ACKNOWLEDGMENTS
This work was supported in part by the National Natural Science Foundation of China under grants 62273004 and 12271003, in part by the Scientific Research Project of Colleges and Universities of Anhui under grants KJ2021A0513 and KJ2020A0368.

CONFLICT OF INTEREST STATEMENT
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. The authors declare no conflicts of interest.

DATA AVAILABILITY STATEMENT
Data sharing not applicable to this article as no data sets were generated or analyzed during the current study. Let g(k) = Γz(k), accordingly, Equation (8) has the following version: This shows thatFT =TF. At this point, the evolution of followers' opinions is independent of the sequence of broadcast moments ℘. Necessity:FT =TF implies that Γ 2 WΓ 2 Γ 2 b = Γ 2 b . If b = 0 N , then b i = 0 for i = 1, 2, … , N. If b ≠ 0 N , it means that Γ 2 b is the corresponding right eigenvector when the eigenvalue of Γ 2 WΓ 2 is 1. Since Γ 2 WΓ 2 is row random, the corresponding right eigenvector when its eigenvalue is 1 is 1 N , and Γ 2 b is linearly related to 1 N . So, Γ 2 b =b1 N , |b i | =b for any i = 1, 2, … , N.
At this point,T andF are commutative. When the number of broadcasts ⟨℘⟩ = k, it can be concluded that Regardless of how the broadcast moments are chosen, the final opinion vector is as follows: . (A12)