Research on twin-SIR rumor spreading model in online social network

Abstract

In the study filed of rumor spreading, kill rumor or dispel rumor is very important in order to control rumor spreading and reduce the bad influence of the rumor. In the previous studies, rumor clarification is mostly finished by relying on external media or news reports instead of intervening and controlling from inside the network, which causes that the speed of rumor clarification is far lower than the speed of rumor spreading, and it is not ideal for the effect of rumor clarification. In this paper, a new Twin-SIR spreading model is proposed, in which, a rumor clarification node named as “rumor dispeller” with the spreading ability is introduced. The rumor dispeller is involved in the spreading process of the model together with the rumor spreader to control the spreading of rumor and thus to achieve the purpose of clarifying rumor. At the same time, during the process of building the model, we also apply the traditional media as a spreading parameter to the spreading process of the model. We built the mean-field equation of the model and then implemented further analysis of the model on homogeneous networks and heterogeneous networks. Through experimental simulations, the “rumor dispeller” was found to have the ability to reduce the spread of rumor spreading, and that the selection of the initial “rumor dispeller” node can affect the effect of rumor spreading, and at the same time, the external media have an important influence on rumor clarification. These conclusions have a new function for guiding us to study the mechanism of rumor spreading.

Keywords

Online social network spreading model rumor dispelling twin-SIR model

1 Introduction

With the rapid development of online social media, social service networks, QQ, Weibo, Facebook, etc. accelerated the information sharing and large-scale information spreading, but they also facilitated the spreading of rumor. All of these rumors affect the normal life order of people and bring a serious negative impact on many areas of society. For example, on Apr.23, 2013 hacker invaded the Twitter account of the Associated Press and published the false news that “Two explosions occurred in the White House and Barack Obama was injured.” It resulted in a shock in the US capital market immediately, which caused the dropping of 14 points for 500 types of the stock index for Standard & Poor within 5 seconds, and thus the market value evaporated US$136.5 billion. In Dec. 2017, the annual top ten types of rumors in food safety, health care, and false information summarized by People’s Daily had also caused different degrees of negative impact on people. Therefore, it is more important to study rumors and eliminate the negative effects brought by them.

For the purpose of dealing with the negative impact of rumors effectively, domestic and international scholars have to implement research on issues such as rumor recognition [1], rumor detection [2], and rumor spreading models [3, 4], with the expectation to reveal the law of rumor spreading so as to facilitate the control of spreading process to achieve the final purpose of killing the rumor or dispelling the rumor [5].

At present, the researches for rumor spreading models based on two classic models. One type of model takes the D-K model [6] and its improved model [6 –9] as the basis, which lay emphasis on theoretical analysis, has high abstractness and rigorous logicality. However, their description of the process of spreading is not intuitive enough, and the model cannot be solved [10]. According to the similarity between rumors spreading and disease transmission, another type of model uses the infectious disease model, which is the macro mathematical model based on mathematical statistics, to model the spreading of rumor [11]. In this type of model, population individuals are abstracted into three types (states) of Susceptible (S), Infected (I), and Recovered (R). It can build three basic models of SI [12], SIS [13] and SIR [14] according to the transformation relation between states. Many scholars have built and improved many spreading models based on the SIR model and its methods of research [15 –29]. These achievements build the spreading model from two perspectives of closed network and dynamic network. Most of these models implement control of rumor through changing the rules for rumor spreading, including individual immunization strategies, strategies such as government’s use of media, etc. for implementing interventions to the external environment. It does not apply these immunization strategies by these models as a model factor to the building of the model. In other words, in the rumor spreading model that has been built, the completion of rumor clarification strategies are more relied on the external media or news reports, instead of intervening and controlling it from the inside of the network. Therefore, the speed of rumor clarification is far lower than the speed for rumor spreading, and the effect of rumor clarification is unsatisfactory. Recently, Li et al. [30] has proposed to add a type of “rumor-killer” node to the model to control the spreading of rumor, which has had a good effect, but the function of “rumor-killer” is just “kill” rumor spreaders instead of being involved in the process of spreading, so its function is limited.

To solve the above challenges, we propose a Twin-SIR to express the rumor propagation process in social networks more realistically. In this model, we first add a “rumor dispeller” node into the model to clarify rumors effectively, “rumor dispeller” is involved in the rumor spreading process, and we consider two factors into the model, the external immunity probability and the external rumor dispelling infection probability, which is caused by the intervention of the traditional media. Experimental results show that the Twin-SIR model is more consistent with the characteristics of rumor propagation in social networks.

The main contributions of this paper are as follows.

First, this article innovatively puts forward the concept of the “rumor dispeller” for the first time, when rumor spreaders begin to spread rumors in the network and when the rumors perpetuate, a “rumor dispeller” will begin to appear and publish rumor clarification information. This outcome will involve the “rumor dispeller” in the process of rumor spreading as a type of spreading node that reduces the speed of rumor spreading to achieve the effect of rumor clarification.

Second, a twin-SIR model is proposed, which includes four types of nodes (rumor ignorant, rumor spreader, rumor dispeller and rumor immune), and the rules for the conversion of the node state are defined, which fully considers the impacts of media reports on rumor spreading.

Third, there are some qualitative analyses for the twin-SIR model inhomogeneous networks and heterogeneous networks.

The remaining structure of this paper is as follows: Section 2 introduces the related work. Section 3 analyzes the spreading mode of rumor in social networks, which is the basis for building the model of this paper. Section 4 gives a detailed description of the model proposed in this paper. Firstly, it provides a description of the model. Then, it provides the mean-field equation for the model on the basis of model assumptions, after that, it implements further analysis to the model respectively on homogeneous networks and heterogeneous networks, including effective spreading ratios and spreading thresholds. Section 5 provides the experimental simulations, which verify the validity of the model. The last section is the conclusion of the work in this paper.

2 Related work

The study of the rumor spreading model has been started in the 1960 s. Delay and Kendall firstly built the D-K rumors spreading model in a closed homogenous mixed population and studied the dynamical equation of this model by the approach of the random process [6]. It does not fully comply with the actual process of rumor spreading by this model, but it is reasonable under certain approximate conditions. Maki and Thompson implemented modification to the spreading mechanism of the D-K model and thought that the rumors are spread through the bidirectional contact between the spreader and other people in the crowd, and thus obtained a new famous M-T rumor spreading model [7]. The rumored model based on mathematics is mainly concentrated on the theoretical analysis, and it is not intuitive in the process description, the rumor spreading that is approaching to display can be expressed in a mathematical model, but it is impossible to be solved. On the other hand, Sudbury [11] used communicable disease models to implement the modeling to the spreading of rumor for the first time by disease transmission model according to the similarity between rumor spreading and disease transmission, according to the conversion relations between statuses of populations, Moreno [14] built SIR model. Afterward, scholars have conducted studies on the rumor spreading models and their immune mechanisms in closed networks and dynamic networks.

The study of the rumor spreading model on the dynamic network can be implemented provided that the links can be added to the network topology at a certain rate and also can be removed from the spreading network at a certain rate. Dong and Deng built an online social SEIR network rumor spreading model, which had different sizes of the population. The model indicated that the entry of new users has a great impact on the rumors in the system, and can cause rumor spreading in a wider range. At the end of the paper, it pointed out that the spreading features of rumor spread in online social networks could be able to more realistically reflected [23]. Based on scale-free networks, Wan and Li proposed a new SEIR rumor spreading model, which got the basic regenerative factors and balance points of the model through the mean-field theory analysis and therefore provided a reliable strategic basis for preventing the rumor spreading [24]. Jiang et al. [25] built a new type of online social network (OSNs) rumor spreading model, which adopted the degree of a link to describe the dynamic changes of the number of rumor spreaders, and can deem as an extension of the traditional SIR model. This paper provided a discussion of the stability of the model and proposed a model with an immune structure to explore the control method of rumor spreading. The implementation of immunization to susceptible individuals is an effective method to control rumor. It provides new views to control the spreading of rumors in OSNs.

The study of the spreading model on a closed network is implemented under the assumption that it does not consider the moving in and out of users in the process of spreading of rumor information, namely, the total numbers N in the network is unchanged. Wang et al. [26] built a new SIR model and found that the existence of link recognition lowers the spreading extent of rumor slightly under the impact of delay time, and the longer the delay time, the worse the immunity effect of immune strategies. Liu et al. [27] introduced exposed nodes of hesitant in the model by considering the hesitation mechanism in the feelings of individuals. Then the author believed that the exposed nodes became the immune nodes by a certain probability, and thus proposed the SEIR rumor spreading model, to get rumor spreading threshold value, they also took into account the influence of adding immune mechanisms on the model. Huo et al. [28] indicated that each node in the network turns between a high active state and low active state at a certain probability, and introduced a dynamic spreading model, to get an equilibrium point which is locally close to a stable state. The random rumor spreading model was rebuilt by Jia F et al. [29] to analyzed the sufficient condition for extinct and continuing rumor, and they got the boundary condition between extinct and continuing rumor. Li and Wang et al. [30] proposed a new CSER rumor spreading model and introduced the node of “rumor killer” in the model, which set up the differential equation for the dynamics of spreading by adopting the complicated network theoretic mean-field theory. It was found through the study that the “rumor killer” reduced the impaction of rumors spreading in the network, which provides a new method for us to control the rumor that spread within the network.

With the fast development of network technology, although the spreading of network information has challenged the monopoly status of traditional media to a certain extent, there is high credibility for news reports of traditional media to the vast Internet users. Therefore, media reports have played a key effect in effectively controlling the spread of rumor information in the network. According to Huo and Huang et al. [31], during the process of spreading and diffusion of rumor, it was taken into account that the factors such as media reports, the transparency of government information, etc. extend the D-K rumor spreading model. Besides, the optimal control was also used to discuss the optimal control strategy for spreading of rumor. Aiming at the impact of government intervention on the spreading of rumor, Zhu, Zhao and Wang [32] built the rumor spreading model of space-time network controlled and feedback by the government, and discussed the stability and oscillation of the network. Zhao and Zhu [33] proposed a rumor spreading model of space-time network with the effects of media reports, and analyzed the impact of in-depth reports of media on the density of rumor spreaders in the network and on the stable area of the system. Dhar and Jain et al. [34] studied the dynamic behavior of news dissemination in social networks and proposed a model of SEI news dissemination, which took the media awareness as a control strategy to reduce the spreading of rumor. Li [35] proposed a delay rumor model with a saturation control function and discovered that the stability of the system can be affected by the government’s regulation ability. Through studying the impact of two media on the rumor spreading, and calculating the equilibrium value of the model, Wang and Song [36] found that the scale of the spreader was directly affected by the conversion rate of the ignorant between the two media, and the dynamic behavior of spreading was significantly affected by different media.

3 Analysis of rumor spreading model in online social network

In social networks, one node represents one user of the social network, and the following relation among users was regarded as the edges of the network, then the information is spread along the edges between nodes. The precondition for studying the rumor clarification model in social networks is that the rumors have already existed in the network and have begun to spread. Therefore, there are both rumor spreading mode and rumor clarification spreading mode in the rumor clarification model. It displays a rumor spreading mode with rumor dispellers in a social network with 10 nodes (users) and 14 edges in Fig. 1.

Fig. 1

Rumor spread mode in the social network.

Under the initial condition, A got the rumor information through external media and believed in the information, or fabricated a rumor by himself, and became the first “spreader” of the rumor for the first time to spread the rumor information. B, D, G can get the rumor information, but among them, B, D were more interested in the content of the rumor, and they spread the rumor with a certain probability, and therefore they also became the spreader of the rumor. G is not interested in the content of the rumor, therefore he did not forward the rumor, and thus they became the direct immune node about the rumor. After spreading the rumor information for a period of time, D began to lose interest in the rumor with a certain probability and thus became the immune node.

At the same time when the rumor spreads throughout the network, the national news media or associated agencies denied the rumor through network media or by binding the “Big V” users forcibly after they obtained the truth by investigating the content of the rumors, then the “dispeller” A + appeared and spread the information about clarifying rumors. F, C, H, and I can obtain the information on clarifying rumors. Among them, C and I spread the rumor dispelling information, and then became the “rumor dispeller”, F did not have interest in such kind of information and thus became the immune node of the rumor, B obtained the truth from C after spreading the rumor information, and then B changed into “rumor dispeller” from the “spreader” of the rumor. This moment, both rumor information and rumor clarification information are spread throughout the social network.

From the above analysis, it is perceptible that the rumor clarification model in social networks is that rumor spreader A and dispeller A + keep continuous competing for network resources, namely, it is the process of people and edges (relations) in the network. If the speed of rumor spreading is faster than that of rumor clarification in social networks, then more and more people will become a spreader of rumor, resulting in instability and major economic losses or imbalances of society. On the contrary, if the speed of rumor clarifying is faster than that of rumor spreading, it will have the rumors controlled effectively in the early period. The main work of this paper is to set up a rumor clarification model with rumor spreading, with the intention of spreading the rumor clarification information at a faster speed while slowing the speed of rumor spreading in order to minimize the effect of rumors in the social network.

4 Twin-SIR model

In social networks, users can release information freely which can be accessed by the friends of the user easily and conveniently. Therefore, a piece of information in social networks can be obtained and spread by thousands of users quickly. Based on the SIR model, this paper set up the rumor clarification model of the social network. S (susceptible) refers to the ignorant who is susceptible to rumor and represents the user node which has not obtained any information about the rumors, including all user nodes for obtaining rumor information or rumor clarification information. I(Infected) refers to the infected person in the model and consists of two parts. One part is the rumor spreader I₁, representing a type of user node who obtains some rumor information and spreads the rumor information. The other part is the rumor dispeller I₂, representing a type of user node who clarifies a certain rumor, rumor spreader and rumor dispeller appear one after another, that the source of the “Twin” concept of the model in this paper. The rumor immune R(removed) represents the user node who has no interest in the rumors, and such types of users are neither involved in spreading rumor nor the clarifying rumor. About certain rumor information, the status of each node in the social network will change between S, I₁, I₂, and R at a certain probability. According to the analysis of Section 2, we purpose a Twin-SIR model of the paper shown as Fig. 2.

Fig. 2

Twin-SIR model.

The model based on the following assumptions:

It does not consider the moving in and out of users in social networks in the spreading process of the rumor information, namely, the total number; N in the network is unchanged.

We define the meaning of the formula as follows:

N(t) is the total number of nodes in the social network at the time of t at the time t.

S(t) is the proportion of rumor ignorant(rumor susceptible), who do not know the rumor at time t.

I₁(t) and I₂(t) represents the proportion of rumor infected. I₁(t) represents the proportion of rumor spreader who obtained and spread the rumor information at the time of t.

I₂(t) represents the proportion of rumor dispeller who know the truth and clarify the rumor at time t.

R(t) refers to the proportion of rumor immune who do not have interest in the rumor at time t.

Therefore, they do not spread the rumor or clarify the rumor. The proportion is the density, then S (t) + I1 (t) + I2 (t) + R (t) =1.

The rules for the conversion of node state are defined as follows:

Through contacting the internal rumor spreaders or clarifies of social networks, a rumor susceptible S respectively changes into a rumor spreader I₁ with one internal infection probability P_SI1, or S changes into a rumor dispeller with another internal infection probability P_SI2.

Through seeing the rumor clarification information by external media such as television news reports, newspapers, etc., a rumor susceptible S changes into rumor a rumor dispeller I₂ with the external infection probability β.

A rumor susceptible S, who has obtained the rumor information or rumor clarification information through contacting the rumor spreader or dispeller, but is not interested in this topic, changes into immune R of the rumor with the probability δ.

A rumor spreader I₁ who has lost interest in rumor after spreading the rumor for a period, then changes into a rumor immune R with the immune probability P_I1 _R.

A rumor spreader I₁ changes into a rumor dispeller node I₂ with the immune probability α through contacting rumor dispeller or forcible intervention of policy after spreading the rumor for a period.

A rumor dispeller I₂ who has lost interest in rumor after spreading the rumor for a period of time, changes into the rumor immune R with the immune probability of P_I2 _R.

According to the description to Fig. 2, the mean-field equation for the process of rumor diffusion in the social network is obtained as Equation (1).

${\begin{matrix} \frac{dS (t)}{dt} = - P_{{SI}_{1}} S (t) I_{1} (t) - P_{{SI}_{2}} S (t) I_{2} (t) - β S (t) - δ S (t) (I_{1} (t) + I_{2} (t)) \\ \frac{{dI}_{1} (t)}{dt} = P_{{SI}_{1}} S (t) I_{1} (t) - α I_{1} (t) I_{2} (t) - P_{I_{1} R} I_{1} (t) \\ \frac{{dI}_{2} (t)}{dt} = P_{{SI}_{2}} S (t) I_{2} (t) + β S (t) + α I_{1} (t) I_{2} (t) - P_{I_{2} R} I_{2} (t) \\ \frac{dR (t)}{dt} = δ S (t) (I_{1} (t) + I_{2} (t)) + P_{I_{1} R} I_{1} (t) + P_{I_{2} R} I_{2} (t) \end{matrix}$ (1)

4.1 Basic regeneration system of model and analysis of its stability

For the design of a new analysis of traditional cultural symbols in the field of visual communication design of the calculation, we have a high application of the code form of the calculation model. In general, the design of the computational model of our common genetic algorithm is encoded using the form of binary encoding [12]. However, the calculation of this paper requires much data to study, but it is difficult to deal with the computational research of large-scale computational items. So in this paper, the form of floating-point encoding is adopted in the research after the analysis of the way it codes. Floating-point coding can fully demonstrate the computational requirements of this article, but it will cause us to take too long in the design of the model [13]. Chart 1 below in which we analyze the advantages and disadvantages of floating-point encoding and binary coding [14 –16].

The gradual increase in the number of rumor spreaders and dispellers in the whole process from the appearance of rumor to the success of rumor clarification is the most intuitive change of rumor spreading in social networks. After it reaches a peak, and when it begins to fall to a certain threshold (The proportion of rumor spreader to dispeller is generally believed as 2:8 in the industry), the rumor clarification will be considered successful. At this moment, it reaches a relatively stable state in the system. Assume that the number of immunes is R when the rumor clarification is successful, then $R = final {R (t)} = lim_{t \to \infty} R (t) = R (\infty)$ (2)

R is used to measure the effect of rumor spreading, namely, the density of effective spreading. For large-scale networks, it is usually assumed that there are only one or a few individuals are infected and there is no immune population at the initial time, thus S (0) ≈ -1, I₁ (0) ≈1, I₂ (0) ≈1, R (0) ≈ 0. Then,

$\begin{matrix} \frac{dR (t)}{dS (t)} = \\ \frac{δ S (t) (I_{1} (t) + I_{2} (t)) + P_{I_{1} R} I_{1} (t) + P_{I_{2} R} I_{2} (t)}{- P_{{SI}_{1}} S (t) I_{1} (t) - P_{{SI}_{2}} S (t) I_{2} (t) - β S (t) - δ S (t) (I_{1} (t) + I_{2} (t))} \end{matrix}$ (3)

According to the condition that S (0) =1, R (0) =0, Equation (3) can be changed into Equation (4).

$\frac{dR (t)}{dS (t)} = \frac{P_{I_{1} R} I_{1} (t) + P_{I_{2} R} I_{2} (t)}{- P_{{SI}_{1}} I_{1} (t) - P_{{SI}_{2}} I_{2} (t) - β - δ (I_{1} (t) + I_{2} (t))} \frac{1}{S (t)}$ (4)

According to the condition that S (∞) =1 - R (∞) = 1 - R, the transcendental equation can be obtained as Equation (5) $R = 1 - e^{- τ R}$ (5)

Where, $τ = \frac{P_{SI 1} - P_{SI 2} - α - β - δ}{P_{I 1 R} + P_{SI 2}}$

Equation (5) is the transcendental equation of the Twin-SIR model.

Conclusion: τ is the spreading threshold of the model. If τ<1, then R = 0, which means that the rumor cannot be spread. If τ>1, then R > 0, and, as the value τ increases, the value of R will also increase, which means that the rumor is spread widely throughout the network.

4.2 Analysis of a model of a homogeneous network

In the homogeneous network, we assume that the degree of each node is approximately equal to the average degree < k> of the network, then the mean-field equations of the Twin-SIR model inhomogeneous networks can be obtained according to the mean-field theory as Equation (6).

${\begin{matrix} \frac{dS (t)}{dt} = - P_{{SI}_{1}} 〈 k 〉 S (t) I_{1} (t) - P_{{SI}_{2}} 〈 k 〉 S (t) I_{2} (t) - β S (t) - δ 〈 k 〉 S (t) (I_{1} (t) + I_{2} (t)) \\ \frac{{dI}_{1} (t)}{dt} = P_{{SI}_{1}} 〈 k 〉 S (t) I_{1} (t) - α 〈 k 〉 I_{1} (t) I_{2} (t) - P_{I_{1} R} I_{1} (t) \\ \frac{{dI}_{2} (t)}{dt} = P_{{SI}_{2}} 〈 k 〉 S (t) I_{2} (t) + β S (t) + α 〈 k 〉 I_{1} (t) I_{2} (t) - P_{I_{2} R} I_{2} (t) \\ \frac{dR (t)}{dt} = δ S (t) 〈 k 〉 (I_{1} (t) + I_{2} (t)) + P_{I_{1} R} I_{1} (t) + P_{I_{2} R} I_{2} (t) \end{matrix}$ (6)

Now, we change to use ρ(t) the express the density of infected individuals at the time of t. When the time t runs to infinity, the steady-state density of infected individuals is expressed as ρ. Based on the theory of the mean-field theory, when the scale of the network runs to infinity, by ignoring the correlation of degrees between different nodes, the reaction equation can be obtained as follows Equation (7). $\frac{\partial ρ (t)}{\partial t} = - ρ (t) + τ 〈 k 〉 ρ (t) [1 - ρ (t)]$ (7)

The above formula has physical meaning as follows; the first item on the right side of the equal sign is the transformation of infected individuals to immune individuals at unit speed. The second term on the right of the equal sign is the average density of newly infected individuals generated by a single infected individual. As our concern is the spreading status of rumor under the condition ρ(t) = 1, the other high-order correction terms are ignored in the above formula.

Let the right of Equation (7) be equal to zero, and the steady-state density ρ of the infected individual is solved as follows Equation (8). $ρ = {\begin{matrix} 0, τ < τ_{c} \\ \frac{τ - τ_{c}}{τ}, τ ⩾ τ_{c} \end{matrix}$ (8)

Where, the spreading threshold is $τ_{c} = \frac{1}{〈 k 〉}$ .

It indicates that if the spreading rate τ < τ_c, then the number of spreaders will decrease exponentially, and the rumor is not able to be spread, if τ ⩾ τ_c, the rumor spreaders will spread the rumor and have the network in balanced state finally.

4.3 Network analysis of a model of heterogeneous

In accordance with [37], the social network is a scale-free network. Therefore, it can divide the user nodes of the social network into several communities, and each node in each community has a degree k (k = 1, 2,...). So, the individual users in each community are classified into four categories (S, I₁, I₂, and R) according to their state.

Set N(k,t) as the total user number in the network community with degree k at time t, and S(k,t), I₁(k,t), I₂(k,t), and R(k,t) respectively represent the proportion (namely the density) of rumor ignorant (susceptible nodes), rumor spreader, rumor dispeller, and rumor immune. then S(k,t)+I₁(k,t)+I₂(k,t)+R(k,t) = N(k,t), and then Equation (1) can be further modified to Equation (8).

${\begin{matrix} \frac{dS (k, t)}{dt} = - P_{{SI}_{1}} S (t) θ_{1} (t) - P_{{SI}_{2}} S (t) θ_{2} (t) \\ - β S (t) - δ S (t) (θ_{1} (t) + θ_{2} (t)) \\ \frac{{dI}_{1} (k, t)}{dt} = P_{{SI}_{1}} S (t) θ_{1} (t) - α θ_{1} (t) θ_{2} (t) - P_{I_{1} R} θ_{1} (t) \\ \frac{{dI}_{2} (k, t)}{dt} = P_{{SI}_{2}} S (t) θ_{2} (t) + β S (t) + α θ_{1} (t) θ_{2} (t) \\ - P_{I_{2} R} θ_{2} (t) \\ \frac{dR (k, t)}{dt} = δ S (t) (θ_{1} (t) + θ_{2} (t)) + P_{I_{1} R} θ_{1} (t) \\ + P_{I_{2} R} θ_{2} (t) \end{matrix}$ (9)

In Equation (9), θ₁(t) is the probability that any edge of the network is connected to the spreading node I₁ at time of t, and θ₂(t) is the probability that any edge of the network is connected rumor dispeller node I₂ at time of t. $θ_{1} (t) = \frac{\sum_{k} kp (k) I_{1} (k, t)}{< k >}, θ_{2} (t) = \frac{\sum_{k} kp (k) I_{2} (k, t)}{< k >}$ (10)

Where, P(k) is the degree distribution function of the social network, and < k> is the average degree of the nodes of the network.

When a node with k degree is provided, Equation (11) describes the transformation probability between users with degree k. ${\begin{matrix} P_{S \to I_{1}} (k, t) = P_{{SI}_{1}} S (t) θ_{1} (t) + β S (t) \\ P_{S \to I_{2}} (k, t) = P_{{SI}_{2}} S (t) θ_{2} (t) + β S (t) \\ P_{S \to R} (k, t) = δ S (t) (θ_{1} (t) + θ_{2} (t)) \\ P_{I_{1} \to I_{2}} (k, t) = α I_{1} (t) θ_{2} (t) \\ P_{I_{1} \to R} (k, t) = P_{I_{1} R} I_{1} (t) \\ P_{I_{2} \to R} (k, t) = P_{I_{2} R} I_{2} (t) \end{matrix}$ (11)

Therefore, the transformation probability of a state in a social network can be obtained according to the following Equation (12). ${\begin{matrix} P_{S \to I_{1}} (t) = \sum_{k} P_{S \to I_{1}} (k, t) \\ P_{S \to I_{2}} (t) = \sum_{k} P_{S \to I_{2}} (k, t) \\ P_{S \to R} (t) = \sum_{k} P_{S \to R} (k, t) \\ P_{I_{1} \to I_{2}} (t) = \sum_{k} P_{I_{1} \to I_{2}} (k, t) \\ P_{I_{1} \to R} (t) = \sum_{k} P_{I_{1} \to R} (k, t) \\ P_{I_{2} \to R} (t) = \sum_{k} P_{I_{2} \to R} (k, t) \end{matrix}$ (12)

5 Experimental simulation

Based on the built Twin-SIR model, this section implemented experimental simulation respectively on homogeneous social networks and heterogeneous social networks, implemented a comparison experiment with the SIR spreading model, and thus verifies the effectiveness of the model.

5.1 Homogeneous networks

This setup of the experiment intends to conduct the efficiency experiment and comparison experiment of the model on homogeneous networks. Firstly, we generate a microcosm network with 1000 nodes and an average degree of 6 for a node by Matlab simulation software. Afterward, implement the comparison experiment between the Twin-SIR model and the SIR model in the generated microcosm network.

Fig. 3 shows the comparison between the SIR model and the Twin-SIR model.

Fig. 3

Comparison with the baseline.

Fig. 3(a) shows the density curves of the three types of nodes in the SIR model that change with time during the rumor spreading process. It can be seen from Fig. 3 that the density of the susceptible nodes S(t) keeps attenuating until it reaches 0. The density of the spreading nodes I(t) increases rapidly in the initial stage, reaches its peak value of 0.34 when t = 6, and then falls rapidly until it reaches 0. However, the density of the immune nodes R(t) increases rapidly in the initial period of the topic spreading, but it gradually trends toward a steady-state after it reaches its peak value and finally approaches 1.

Fig. 3(b) shows the density curves of the four types of nodes in the Twin-SIR model as they change with time during the rumor spreading process. The changing trend in Fig. 3(b) is similar to that in Fig. 3. However, with the introduction of the internal rumor dispelling mechanism, the dispellers and spreaders are present in the network at the same time, and the density I(t) of the spreading nodes reaches its peak value of 0.28 at approximately t = 12. This finding indicates that the number of spreaders decreases due to the participation of dispellers and thus reduces the negative impact of rumors in the network.

Fig. 4 shows the influence R of rumor spreading about SIR ant Twin-SIR. It’s perceptible from Figure that, when the rumor appears in the network in the beginning, as the appearance of the dispellers has the issue of the time lag, the influence of rumor spread almost has no difference in the early period. Subsequently, with the appearance of vast dispellers, the influence of the spreaders in the model in this paper is significantly lower than that of the SIR model. It also indicates the effectiveness of this model’s inhomogeneous networks.

Fig. 4

Influence of R.

5.2 Heterogeneous networks

This setup of the experiment intends to conduct the efficiency experiment and comparison experiment of the model on heterogeneous networks.

(1) Data set

It adopts the data sets of Facebook users as the experiment data, which includes 4,039 nodes, 88,234 edges. We get the topological features of the network as follows by Usenet, average degree < k> = 78.92, the maximum degree of a node is 638, the average clustering coefficient is 0.2056 (indicating that the relation of nodes in the network is more close), the number of communities is 3, the degree of the module is 0.2127 (indicating that the network has a clear community structure), the average path is 4.71 (indicating that the network has obvious characteristics of the microcosm).

Assume that all types of nodes in the network at the initial time are S(0) = N-1, I₁(0) = 1, I₂(0) = 0 and R(0) = 0 respectively. The proportions of the number of susceptible nodes, rumor spreading nodes, rumor dispeller nodes and in the network are expressed by S(t), I₁(t), I₂(t) and R(t) respectively, that is, $\begin{matrix} S (t) = \sum_{k} kS (k, t) \\ I_{1} (t) = \sum_{k} {kI}_{1} (k, t), \end{matrix}$ $\begin{matrix} I_{2} (t) = \sum_{k} {kI}_{2} (k, t) \\ R (t) = \sum_{k} kR (k, t) . \end{matrix}$

(2) The influence of initial rumor spreader’s degree in the spreading process

In this experiment, we select and fix the node with a degree of 200 as the initial rumor clarification node, and select the degrees of initial rumor spreading nodes as k₀ = 400, 162, 12 separately to implement the experiment. Figure 6 shows the influence of initial spreading nodes with different degrees on the spreading process.

Figure 5 shows that the size of the degree of the initial spreading node has little influence on the curve of the spreading process, but has a significant influence on the curve when it reaches the peak. The main manifestation is that the greater the degree of the initial spreading node, the faster the spreading process reaches the peak value, and the faster the spreading effect displayed. Conversely, the slower the spreading effect displayed.

Fig. 5

Influence of the degree of initial spreading nodes on the spreading process.

Fig. 6

Curve of influence of the degree of initial rumor dispeller node on the spreading process.

(3) The influence of the degree of initial rumor dispeller node on the spreading process

In this experiment, we select and fix the node with a degree of 154 as the initial rumor spreading node, and select the degrees of initial rumor dispeller nodes as k₀ = 364, 102, 19 separately to implement the experiment. Figure 7 shows the influence of the degree of initial rumor dispeller node on the spreading process.

Fig. 7

Density of rumor spreaders and rumor dispellers (The degree of initial rumor dispeller node is selected as 154).

It is perceptible from Fig. 6 that the size of the degree of initial rumor dispeller node has an influence on the spreading process. When it is larger for the degree of the initial rumor dispeller node, it increases the proportion of a number of people who know the rumor information in the network, among which, the proportion of dispeller increases largest, conversely, it becomes slower in the increase in the proportion of rumor dispellers. The result indicates that if we want to dispel the rumor quickly, we should choose rumor dispellers with a larger degree (that is, a large number of neighbors), which is consistent with the current guideline for rumor clarification.

(4) Validity experiment of model

In this experiment, we conducted an experiment by comparing the proportion of the number of rumor infected persons (including spreaders and dispellers) in the SIR model with that in the Twin-SIR model along with the variation of time (including narrators and arbitrators). The experimental results are shown in Fig. 7.

It is perceptible from Fig. 7 that, as for the SIR model, rumor dispellers almost do not exist in the network, because there is no rumor clarification mechanism, and it intervenes in the influence of rumor only relying on external media such as traditional media, news reports, etc. However, in the model set by this paper, although the density of the rumor spreaders has a peak value with the variation of time, it is significantly smaller than the peak value in the SIR model, at the same time, as the rumor dispellers have been involved in the spreading process, and the number of rumor dispellers has increased dramatically. The density of dispellers is finally greater than that of spreaders. The rumor clarification will be successful when the density ratio of rumor dispellers to rumor spreaders is equal to 2:8. Fig. 7 indicates that this model is effective in heterogeneous networks.

6 Conclusion

In this paper, based on the SIR model, we added a new type of node named the “rumor dispeller” to build a new Twin-SIR model. The core purpose of this model is to intervene in and control rumors from the inside of the network by adding the “rumor dispeller” node to clarify rumors effectively. Further, considering that the traditional media, such as TV news reports and newspapers, are still a major part of the rumor dispelling mechanism in the present social situation, we added two factors into the built model, namely, the external immunity probability and the external rumor dispelling infection probability, which is caused by the intervention of the traditional media. This approach allows better compliance of our model with that of the actual network. We also implemented some theoretical analyses of the model and defined a success index for rumor dispelling to measure the effectiveness of the model. Finally, the simulation experiments implemented inhomogeneous networks and heterogeneous networks show that compared with the SIR model, the model proposed by us can better control the number of rumor spreaders in the network (i.e., the peak value of spreading) and the spreading of rumors in the shortest time to achieve the purpose of clarifying the rumor.

Footnotes

Acknowledgments

This work is partially supported through grants from the Key R & D project of Shandong Province (2019JZZY010129) , Shandong Provincial Social Science Planning Project (18CHLJ02,18CHLJ09, 19BJCJ51,18CXWJ01,18BJYJ04), and Shandong Province Social Science Popularization and Application Research Project (2020-SKZZ-51).

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