Spreading dynamics of a 2SIH2R, rumor spreading model in the homogeneous network

In the era of the rapid development of the Internet, the threshold for information spreading has become lower. Most of the time, rumors, as a special kind of information, are harmful to society. And once the rumor appears, the truth will follow. Considering that the rumor and truth compete with each other like light and darkness in reality, in this paper, we study a rumor spreading model in the homogeneous network called 2SIH2R, in which there are both spreader1(people who spread the rumor) and spreader2(people who spread the truth). In this model, we introduced discernible mechanism and confrontation mechanism to quantify the level of people's cognitive abilities and the competition between the rumor and truth. By mean-field equations, steady-state analysis and numerical simulations in a generated network which is closed and homogeneous, some significant results can be given: the higher discernible rate of the rumor, the smaller influence of the rumor; the stronger confrontation degree of the rumor, the smaller influence of the rumor; the large average degree of the network, the greater influence of the rumor but the shorter duration. The model and simulation results provide a quantitative reference for revealing and controlling the spread of the rumor.


Introduction
With the continuous emergence of social media platforms, the traditional media era has gradually turned into the self-media era, and information dissemination has become faster, wider in scope, and deeper than ever [1]. Rumors, as a special kind of information, have greatly increased the possibility of artificial release of rumors due to their own confusion, timeliness and psychological satisfaction to the people who spread the rumor. Coupled with the self-media era, the threshold for spreading rumors is further lowered [2]. In today's society, there are some people who use people's public psychology to create rumors to obtain benefits from it [3][4]. This behavior will cause public panic and harm society. Therefore, in order to reveal the law of rumors dissemination and reduce the negative impact of rumors on society, it is necessary to establish a suitable mathematical model to analyse the characteristics and mechanisms of rumors dissemination process.
In the 1960s, Daley and Kendall [5] proposed the classic rumor spreading model, the DK model. The model divides the population into three categories: people who have never heard of rumors (Ignorant), people who spread rumors (Spreader), and people who have heard the rumors but do not spread (Stifler). The form to reflect the probability of an individual to In the above studies, many have made great contributions to the theoretical research on the process of rumor spreading on complex networks. However, there are two shortcomings in the theory to need to be improved. The first one is that, in reality, the discernible degree of the rumor is an important variable, but most previous studies did not quantify this. Allport and Postman [25] believe that there are three conditions for the generation and spread of rumors: the first one is the lack of information; the second one is people's anxiety; the third one is that the society is in crisis. Based on this, they proposed a classical formula: rumors i a = (where i represents the importance of information, and a represents the degree of unknowability of the event). The other improvement to be made is that no research has been done on the spread of truth, the opposite of rumors. With the rumor, there is also the truth. In reality, there are always some wise men who can reveal the rumor and spread the truth, in which time there will be a confrontation relationship between the rumors and the truth [24]. Based on this, we divide the population into six categories: people who have never 3 heard of rumors or truth (ignorant), people who spread rumors (spreader1), people who spread truth (spreader2), people who have heard the rumors but do not spread temporarily (hesitant1), people who have heard the rumors but do not spread(stifler1), people who have heard the truth but do not spread(stifler2), and propose the 2SIH2R model with the discernible mechanism and the confrontation mechanism.
The organization of the paper is the following. In Section 2, the 2SIH2R model is defined, and the mean-field equations of the model are established in the homogeneous network [16,26]. In Section 3, we study the rumor spreading threshold of model propagation by changing initial conditions and parameters, and extend the spreading threshold under special circumstances [27] to general conditions. In Section 4, through simulation, we study the influence of discernible mechanism, confrontation mechanism, and average degree on the rumor. In Section 5, conclusions of the paper and future work are given.

2SIH2R Rumor Spreading Model
We consider a closed and mixed population composing of N individuals as a complex network, where individuals and their contacts can be represented by vertexes and edges. This network can be described by an undirected graph ( ) where V denotes the vertexes and E represents the edges. At each time t , the people in the network can be divided into: 1 S , 2 S , I , H , 1 R , 2 R , separately, representing for people who spread the rumor, people who spread truth, people who have never heard of the rumor or truth, people who have heard the rumor but do not spread temporarily, people who have heard the rumor but do not spread, people who have heard the truth but do not spread. The rumor spreading process of the 2SIH2R can be seen in Figure 1. In Figure 1, the solid/dotted line from the "Ignorant" represents that the ignorant contact with spreader1/spreader2, and the rumor spreading rules of the 2SIH2R model can be summarized as follows.
(1) We use m to describe the discernible rate of the rumor, and () fm to describe people's ability to reveal rumors. The function of f is to map the characteristics of the rumor to the characteristics of the people. The greater the m , the greater probability that the rumor will be revealed. The greater the () fm, the greater probability that the people will not believe the rumor immediately. We assume that there is a positive correlation between m and () fm.
(2) When an ignorant encounters a spreader1, there are three possible outcomes: (i) the ignorant may believe the rumor and spread it with probability 1 (1 ( )) fm − . The 1  namely is rumor spreading rate; (ii) the ignorant may not believe the rumor immediately and hesitate to spread it with probability ( ) f m  . The  namely is potential spreading rate; (iii) the ignorant may have no response to the rumor with probability , because of having no interest of the truth. Since the truth is generally issued by an authoritative organization, or there is evidence to support it, the truth is relatively more objective, accurate, and clear. It is easier for the ignorant to judge, and not easy to become a hesitant. (4) We consider that the hesitant1 have desire to spread the information, because of the suspicion of the rumor and the environmental impact when they receive the rumor, they didn't spread the rumor immediately. In the time of hesitation, hesitant1 may believe the rumor to spread it, or they may discover the truth and spread it. So, we assume that at each step, the hesitant1 will spontaneously become people who spread the rumor(spreader1) with probability 1  , and people who spread truth(spreader2) with probability 2  . (5) When a spreader1(spreader2) encounters another spreader1(spreader2), he/she could think the rumor(truth) is widely known. So, the spreader1(spreader2) may lose spreading enthusiasm and become a stifler1(stifler2) with rumor(truth) losing-interest rate 1  ( 2  ).
(7) At each step, the stifler1 will spontaneously become stifler2 with probability  , because of the improvement of their own cognitive level. (8) When a spreader1 encounters spreader2, the spreader1 will believe the truth rather than the rumor with probability  , because of the confrontation mechanism between the truth and rumor. The  namely is confrontation rate.
Moreover, the 2SIH2R model is applied to a generated network which is a closed and homogeneous population consisting of N individuals [17,28]. We use 1 () Rt, separately, to represent the densities of spreader1, spreader2, ignorant, hesitant1, stifler1, stifler2, and at any step, we have the normalization condition: According to the rumor spreading rules, the mean-field equation of 2SIH2R model can be expressed as follows: Where k  represents the average degree of the generated network.

Steady-state Analysis
In this section, we will consider the three situations of the model.

S t S t k S t S t I t
6

Steady-state Analysis of Rumor
At the beginning of model spreading, in this situation, we assumed that there is only one spreader1 who spreads the rumor, and there is no truth. So, the initial condition can be given: After a while the number of spreader1 will increase to the top, then it reduces to zero at which time the system reaches stability.   fm , the following condition will be satisfied: Due to the confrontation mechanism, when 11 0 c   , the rumor must not spread widely in the generated network.

Steady-state Analysis of Truth
It is assumed that the government or authoritative media have already begun to spread the truth before a rumor event occurs. Then when a rumor event occurs, there will be no rumor spreader in the population. So, in this situation, the initial condition can be given: 1

Steady-state Analysis of 2SIH2R model
In this part, we consider a relatively general situation. At the beginning of model spreading, in this situation, we assumed that there is one spreader1 who spreads the rumor, and one spreader2 who spreads the truth. So, the initial conditions can be given: Moreover, it is worth noting that in this situation, due to the complicated spreading process, we can not follow the proof process in the previous part to get the condition of 2SIH2R model spreading threshold. Therefore, this part re-starts from the initial conditions and gets the condition of 2SIH2R model spreading threshold.
We use () it , 1 () rt, 2 () rt, separately, to represent from Eq.(1) to Eq.(6), and from the initial conditions we can know: So, From Eq.(19) we have: When N →, the following result can be obtained: 12 12 (1 So, if the rumor and the truth can spread widely in the generated network which is closed and homogeneous, the  , ( )=0 fm , we can get the Eq.(10). So, we can conclude that the third general situation contains the first two special situations.

Numerical simulation
In this section, through numerical simulation, we study the influence of discernible mechanism, confrontation mechanism, and average degree on the rumor. According to the 2SIH2R Rumor Spreading Model and existing research results [29][30][31], we perform numerical simulation in a generated homogeneous network, where 8 k  = , 5 10 N = . It is assumed that there are one spreader1 and one spreader2 at the time 0 t = . So, Unless otherwise specified, the above parameters are used in this section. It can be seen from Figure 2 that the density of the ignorant decreases rapidly and the other 5 categories increases to their peak, separately in a short time. As the model spreads further, the densities of spreader1 and spreader2 will continue decreasing until it reaches zero, which means the 2SIH2R model gets into the steady state. , as m increases, the final size of stifler1 also decreases. But the time to peak of spreader1 and stifler1 has not changed significantly. In Figure 5, it also can be seen that the final size of stifler1 decreases with increasing () fm, but the stifler2 increases. In summary, as m increases, the instantaneous maximum influence and the final influence range of the rumor will decrease but the truth increase.  Figure 6 displays the change of density of spreader1, under the change of parameter  . It can be seen that the greater  , the stronger confrontation mechanism, the smaller impact of the rumor, because of the decreasing peak. At the same time, from Figure 7, as  increases, the final size of stifler2 increases, because some spreader1 change into stifler2 by the confrontation mechanism. In Figure 8, it can also be seen that the final size of stifler1 decreases with increased  , but the stifler2 increases. In summary, as  increases, the instantaneous maximum influence and the final influence range of the rumor will decrease, while the final influence range of the truth will increase.  Figure 9 displays the change of density of spreader1, under the change of parameter k  . It can be seen that the greater k  , the more people can be contacted by spreader1, the greater impact of the rumor, because of the increased peak and the shortened time of reaching the peak. Moreover, we can find that with increased k  , the shape of the solid line becomes wider, which means that the duration of the rumor event is decreasing. In summary, as k  increases, the velocity and the range of the rumor spreading will increase, which means the influence of the rumor will increase significantly. But the duration of the rumor event will decrease.  Figure 10 displays the change of density of stifler1 and stifler2, under the change of parameter  . It can be seen that the change of  causes a huge impact on the rumor. As long as  changes from 0 to 0.1, almost only stifler2 exist in the network when it reaches a steady-state, which means the rumors will not have a significant impact on us. So, this paper mainly studies the situation where =0  .
13 Figure 10 Density of stifler1 and stifler2 for different values of  at steady-state Figure 11 and Figure 12 display the final size R ( which is the sum densities of stifler1 and stifler2 at steady-state) with 1  and 2  . The redder the color is, the greater the value of R . In Figure 12, under the parameter ( ) 0.5 fm= , =0.1  , 12 0.8  == , the spreading threshold condition can be distinguished roughly by the shade of color (as the black solid line denoted in the figure) which is basically consistent with the steady-state analysis from the previous section (as the black dash line denoted in the figure). In the future, a further study of 2SIH2R rumor spreading model will be conducted in the heterogeneous and some real networks. In this paper we assume that the social network is homogeneous, but in reality, lots of social networks have a more complex structure. And also, the real data may be analysed, because there are many subjective assumptions in the model and parameters setting process. The significance of the model can be better demonstrated through real data.

Data Availability
The generated data used to support the findings of this study have not been made available because the data is randomly generated according to the rules.

Conflicts of Interest
The author(s) declare(s) that there is no conflict of interest regarding the publication of this paper.

Funding Statement
This paper is financially supported by Beijing Municipal Natural Science Foundation