A Class of Rumor Spreading Models with Population Dynamics∗

Suyalatu Dong(董苏雅拉图)and Yong-Chang Huang(黄永畅)

College of Applied Sciences,Beijing University of Technology,Beijing 100124,China

AbstractBased on the characteristics of rumor spreading in online social networks,this paper proposes a new rumor spreading model.This is an improved SIS rumor spreading model in online social networks that combines the transmission dynamics and population dynamics with consideration of the impact of both of the changing number of online social network users and different levels of user activity.We numerically simulate the rumor spreading process.The results of numerical simulation show that the improved SIS model can successfully characterize the rumor spreading behavior in online social networks.We also give the effective strategies of curbing the rumor spreading in online social networks.

Key words:online social networks,rumor spreading model,population dynamics,stability of equilibrium point

1 Introduction

In recent years,the rapid and continuing growth of global online social media pushes the number of their active users climbing to about 29%of the global human population.[1]Online social networking has become the most important platform for the exchange of information outside the traditional media.[2−3]However,the information in online social networks is also mixed with many rumors,whose spreading can make damage to individuals,or harm social stability.To explore the mechanism and establish models for rumor spreading in online social networks,to put forward the strategies of online rumor inhibition,and to assist the solution of the urgent issues of rumor hunted online information propagation,this paper propeses an improved SIS model with elaborated analytic and numerical analysis.[4−11]

Most models of rumor spreading in online social networks evolved from epidemic model. At present,the SIR[12]and the SIS epidemic model[13]are the most widely and most thoroughly studied.Many scholars have improved the traditional infectious disease models and put forward new models to suit different properties of the rumor spreading processes in online social networks.For example,based on the average field theory,Gong et al.[14]proposed the mathematical model for the network virus transmission in mobile environment.Wang et al.[15]proposed a CSR(Credulous-Spreader-Rationals,or equivalently SIR)rumor spreading model,and introduced memory effect and acceptance thresholds into the model.Zhao et al.[16]set up a rumor temporal spreading model,combining the impact of temporal lag,spatial diffusion,media coverage and other factors on the spreading of rumor in social networks.Wan et al.[17]proposed the SIERsEs rumor spreading model,analysing the steady state of the theory,and solved the threshold for the spreading of infection and the threshold for rumor clearance.He et al.[18]proposed a heterogeneous network based epidemic model that incorporates the two kinds of methods to describe rumor spreading in MSNs,with the design of two costefficient strategies to restrain the rumors.In Ref.[19],a novel susceptible-infected-removed(SIR)model is proposed,based on the mean- field theory,to investigate the dynamical behaviours of such model on homogeneous networks and inhomogeneous networks,respectively.Considering that the exposed nodes may become the removed nodes at a rate,Liu et al.[20]proposed a novel rumor propagation SEIR model on heterogeneous network,provided formula of the rumor spreading threshold for the model and analysed the globally dynamical behaviours of the rumor free equilibrium set.Tan et al.[21]proposed a novel rumor propagation model,inspired by a model of elastically collision balls,namely the elastic collisionbased rumor-propagation model(ECR Model),and investigated the dynamics of rumor propagation between network nodes,similar to the dynamics of collisions between elastic balls.Aldila et al.[22]proposed a mathematical model to explain the spreading of rumors in a closed human population,implementing in the model the government interventions to educate people about the danger of rumor along with apprehensions of fanatical people.Based on the reaction-diffusion equations,Zhu et al.[23]proposed a novel epidemic-like model with both discrete and nonlocal delays for investigating the spatial-temporal dynamics of rumor propagation.

All previous rumor spreading models were based on the compartment model of infectious diseases,which assume that the epidemic disease occurs in a closed system and lasts a relatively short period of time.It ignores population dynamic factors such as birth and death.Therefore,rumor spreading models based on the compartment model of infectious diseases can only simulate the rumor spreading processes in a closed environment for which the time of rumor spreading is relatively short.The limit of the compartment model is obvious.First of all,the cycles of some of the online information dissemination processes may be very brief,but the periods of some information transmission processes may be relatively longer,in the human society in an era of information explosion,with all information in human society being constantly updated.It is the relatively longer period dissemination processes of information that may give the public order and social stability the greater harm.As we know that,some of the rumors about politics,because of their difficulty to be effectively suppressed,would extend considerably longer time of propagation.For example,in 2016,the United States established the Information Analysis and Response Center,focusing on collecting,analysing and counterattacking foreign propaganda and fabrications that can harm US interests and undermine the relationship between the United States and its allies.Secondly,the actual online social network is an open system,not a closed system;the number of users in the online social network is dynamically changing.Therefore,we must consider the impact of changes in the total number of network users on the spreading and evolution of rumors.Finally,the real online social networks have different levels of user activity.Not all users are on line all the time.Frequently some users may go hibernation and miss the rumor propagation processes happening in the hibernating duration.Therefore,it is also necessary to consider the in fluence of user activity on rumor propagation processes.

The outline of the article is as follows.In Sec.2,an improved SIS rumor spreading model is proposed,based on the characteristics of rumor spreading in online social networks,with consideration of the population dynamically impacted rumor transmission dynamics parametrically controlled by changing number of online social network users and different levels of user activities.In Sec.3,an elaborated mathematical analysis of the equilibrium points and the stability of equilibrium solutions of the system is performed,followed by the numerical analysis of the behaviour of the system.We have simulated the rumors spreading process based on data sets in real online social network in Sec.4.An effective rumor suppression strategy is proposed and numerically analyzed in Sec.5.Section 6 is a brief summary and conclusion.

2 Rumor Propagation Mechanism and Model Building

Because the network population grows but definitely has the global human population as its upper ceiling,we have the reason to assume that the changing total user number N(t)in the online social network satisfies the logistic model

where b=(the registration rate of new users)–(deactivation rate of users)=(the net growth rate of users).By parameter b,the combined impact of the changing number of online social network users and different levels of user activity are considered.K is the environmental capacity of online social networks,referring to the maximum number of users the online social network can carry,usually a pretty large integer,say a million,10 million or 100 million,and the actual network user population should not exceed K,such that 0≤N(t)≤K.The restricting factors on the environmental capacity of online social networks should include the full population size of internet users,the popularization rate of the internet,the popularization rate of the online social network etc.We set K=1,the logistic model changes into

Thus the network population gets rescaled,

Online social network is an online realization of social structure composed of many nodes,and links between the nodes.The nodes can be individuals or organizations,and the links between nodes correspond to a variety of social relations,such as friendships.In the analysis of the problem of rumor spreading in online social network,we classify the network users,or network nodes into two categories:health node S are(the number of)those network users who are not in the in fluence of some rumor message at some time,and the transmission node I are(the number of)those network users who are in the in fluence of some rumor message at some time so that they involve in the propagation of the message.These two types of network users make up the total population of the network users in the online social network,N(t)=S(t)+I(t).We see that we also have 0≤S(t),I(t)≤1.

In our age when the total population of the social networks are rapidly growing,we assume that the new registered users are all health nodes.Model(2)can be written as

If S and I are not directly interacting with each other,except through N,the equations for S and I can be separately obtained from the above equation,

Because the transmission nodes would pass on rumor message to the healthy nodes during their interaction at some probability,and the transmission nodes also may recover from the in fluence of rumor message at some time rate,we have the following propagation rule for our SIS rumorspreading model,

These two conversion rules between two different types of nodes being taken into account,Eq.(5)of the SIS rumor spreading model with consideration of population dynamics and different levels of user activity in online social networks transforms into the following form

where β is the infection rate,at which the transmission nodes pass on the rumor message to the healthy nodes to transform them into transmission nodes;σ the cure rate,at which the transmission nodes recover from the in fluence of the rumor;and b the net growth rate of population,by which the combined impact of the changing number of online social network users and different levels of user activity are considered.These parameters are assumed to be all constants.N(t)is not an independent variable as it depends on S(t)and I(t)by N(t)=S(t)+I(t).This relation being taken into account,Eq.(6)becomes,

3 Stability of Equilibrium Solutions and Numerical Analysis

3.1 Analysis of System Equilibrium Points and Stability

To find the equilibrium points of the system,letting Eq.(7)equal to zero,we get the equilibrium equation of the system,

Solving these equations,we get two solutions for one of the unknown variables,

or

Therefore,the system has two equilibrium points.Substituting I0=0 into the system,we can get S0=1.So the first equilibrium point of the system is

This equilibrium point corresponds to the situations when the population of the network users reaches to the maximal environmental capacity and the online social networks are completely free of rumors.And it is called the rumor elimination point.Substituting I1back into Eq.(8),we get

or

Among them,S2= σ/β,if being substituted into Eq.(10)would cause I1= −σ/β and produce negative value for population variable,thus should be ignored.So we just take S1=(b+σ)/β as the meaningful solution.Substituting S1=(b+σ)/β into I1in Eq.(10),we get

Therefore,the second equilibrium point of the system is

which is a non-zero equilibrium point.

Thus,we have found the two equilibrium points of the system.Now we want to check the stability of these two equilibrium points.First of all,we remind that the condition for population variables S1,I1to be positive,and the positivity of all the parameters requires,

To check the stability of the equilibrium point,we first shift the original differential equations to that for the variables that are defined as the variations from the equilibrium solutions,

such that

for the first equilibrium point.Substituting the new variables in Eq.(18)into the original differential equations Eq.(7),

To check the stability of the rumor elimination equilibrium point,substitutinginto Eqs.(19)and(20),

For the coefficient matrix of the linear part of the above differential equations,

solving its eigenvalue equation

we can find the eigenvalues as,

We see that one of the eigenvalues is positive valued,and another one negative

Since the real parts of the eigenvalues are not all negative,we know that this rumor elimination equilibrium point is not stable.It means that it is unrealistic for us to hope for the online social networks to be completely free of any rumors.

To check the stability of the second equilibrium point,substituting

into Eqs.(19)and(20),we get

Then,we can solve the eigenvalue equation for the coefficient matrix of the linear part of these differential equations,

and the following two roots

According to Eq.(12),β≥b+σ≥0,about the positivity of population,and of all parameters,we can define a fourth positive parameter

Then,on the one hand,we have the determinant

On the other hand,because all parameters are all nonnegative,we have

Using Eqs.(33)and(34),we conclude that both of the roots of the eigenvalue equation Eq.(30),are real,and negative,

We thus show that the second equilibrium point given in Eq.(15)is stable.Also we remind that the first equilibrium point,namely the rumor elimination equilibrium point given is Eq.(11),has been shown to be unstable so that unrealistic in engineering.

3.2 Numerical Analysis

Figure 1 shows the curves of health node density and transmission node density changing with time.The parameters used in numerical simulation are:b=0.4,β=0.35,σ=0.38.Figure 1 shows that,in the initial stage,the health node density is increasing rapidly with the incoming of new registered users.At t=30,the health node density tends to be saturated.In the beginning,the transmission node density increases rapidly,reaching to the maximum value at about t=25,but gradually decreasing but gradually decreasing to 0 near t=100,with the transmission node density changing into health node.

Fig.1(Color online)Health node and transmission node curves.

When the total number of users increasing towards the environmental capacity,online social networks no longer can accommodate any more new registered users,in the meanwhile,the health node density reaches the saturation.This situation is more in line with rumor spreading characteristics in online social network.

Figure 2 shows different transmission node density curves for different infection rate β values,for example,β=0.15,0.25,and 0.35.It can be seen that,with increased β value,the peak value of the curve of transmission node density becomes larger.This shows that the greater the infection rate β,the spread of rumors in social networks is more extensive,the greater the risk of rumors,consistent with the rumor spreading characteristics in online social network.

Fig.2 (Color online)Di ff erent transmission node density curves for different infection rate β values.

Figure 3 shows different transmission node density curves for different cure rate σ values,for example,σ =0.38,0.48 and 0.58.It can be seen that,with increased cure rate σ value,the peak value of the curve of transmission node density becomes small.It shows that the higher the cure rate,the spread of rumor in the online social network is inhibited more effectively.

Fig.3 (Color online)Di ff erent transmission node density curves for different cure rate σ values.

4 Model Simulation

4.1 Data Set Description

The Facebook user data set is selected as the experimental simulation data.We select fifteen registered Facebook users and their buddy list as the initial online social network nodes,to generate a simple Facebook network containing 4039 nodes.Figure 4 shows a simple Facebook network visualization.

Fig.4 (Color online)Visualization of the simple facebook network.

As shown in Fig.5,a simple Facebook network degree distribution is obtained by using Gephi software.The average degree of the simple Facebook network nodes is 43.691.From the degree distribution map of Facebook network,we can see that the degree distribution of the network satisfies power law distribution,which shows that the simple Facebook network has scale-free feature.

Fig.5 (Color online)Degree distribution of the Simple Facebook Network.

As shown in Fig.6 that,the clustering coefficient distribution of simple Facebook network is obtained by using Gephi software.The average clustering coefficient of the simple Facebook network nodes is 0.617.From this we can see that the simple Facebook network has a high clustering coefficient,which shows that users are very likely to get to know each other through mutual friends in simple Facebook network.

Fig.6 Clustering coefficient distribution of the Simple Facebook Network.

Fig.7 Eigenvector centrality distribution of the Simple Facebook Network.

As shown in Fig.7 that,the eigenvector centrality distribution of simple Facebook network node is obtained by using Gephi software.The eigenvector centrality score of node 1 is the highest,which is 0.4279.Eigenvector centrality theory believes that the importance of a node is closely related to the importance of other nodes to which it is connected,namely for a node,if the node is connected to a number of nodes with high degree of centrality,then the node has a high degree of importance.[24]

Based on the above analysis,we can see the topological characteristics of the simple Facebook network.Here we use the Facebook data set to construct the network as a rumor evolution map,to analyze the propagation and evolution of rumors in the real online social network.

4.2 Simulation Result

In order to verify the correctness of the new model(3),based on the proposed rules of rumor spreading,we simulate the rumor spreading process in the real online social network for a long time.We refer to the methods proposed by Albert and Barabdsi in Ref.[27].According to the degree distribution,clustering coefficient and other statistical properties of the simple Facebook network with 4039 nodes,we get a suitable Facebook network generating algorithm.Based on this algorithm,we add new nodes and edges to a simple Facebook network with 4039 nodes by Networkx software,and construct an online social network with changing number of nodes.We select the network as the map of rumor dissemination and evolution.The average degree of node in the Facebook network is 43.691,so the initial state selects one node with degree 44 in the network as the transmission node;and the total number of initial nodes in the network is N0=4039.Considering the real online social network topology structure,whether or not the healthy node is connected with the transmission node is knowable,so β=1,[28]but the other parameters are chosen to be consistent with the numerical simulation of the model in Sec.3.

Figure 8 shows the time varying curves of the node densities for the rumor spreading in the Facebook network.From Fig.8 we can see that,in Facebook network,the change of each node density with time is consistent with the numerical simulation results in Fig.1.The first peak of the curve of the transmission node density appears at t=22,with its peak appears in advance in time,compared with that of Fig.1.The health node density in online social networks reaches to the environment capacity of online social network at around t=27,so the saturation for the state of the system is achieved.

In the following,we give an effective strategy to control the spread of rumor according to the topology structure of the simple Facebook network.

Fig.8 (Color online)The change of each node density with time in Facebook networks.

5 Inhibition Strategies for Rumor Spreading in Online Social Networks

The previous theoretical analysis provides some useful strategies for controlling the spreading and diffusion of rumors in online social networks.Firstly,from the perspective of restricting the infectious source,we should push the costs for the rumor spreading to be higher,and this kind of rumor control strategy can be called the passive immunization.For example,to punish the users who spread malicious rumors by account close-down,to hold them criminally responsible for resultant bad impacts,etc.,so that in the process of information dissemination the majority of users will become more cautious in forwarding unexamined information.It is equivalent to the decrease in the number of infectious nodes or the enhancement of the recovery rate.Secondly,from the perspective of strengthening the user immunity to rumors,we should establish the mechanism of refuting the rumor in the online social network,and this kind of rumor control strategy can be called active immunization.For example,Guokr and Songshuhui who enjoy very strong in fluence in Sina Weibo,have built great in fluence in the dissemination of science and the restriction of science related rumors on the internet.To refute rumor and spread popular science,online social networks(like Sina Weibo)can recommend the public broadcasting organizations with certain credit and in fluence in the network(like Guokr and Songshuhui)to current network users or set these scientific dissemination accounts as the default followings for the new users register at the network.And it is equivalent to the decrease in the infection rate.

Next,following the method of Refs.[25–26]on rumor inhibition strategy for mathematical quantitative analysis,we propose two inhibition strategies to curb the rumor spreading in online social networks:

(i) Network supervision department to punish the users who spread malicious rumors by account close-down,to hold them criminally responsible for resultant bad impacts,equivalent to the manipulation of the subtraction of f1t=νI(t)in the second equation of rumor spreading Eq.(3).

(ii)Online social networks can recommend the public broadcasting organizations with certain credit and in fluence in the network to current network users,thus implementing the immunization by the addition of f2t=ωS(t)I(t)in the first equation of Eq.(3).

In the above,the constants ν and ω represent the effect of two inhibition strategies on infected individuals and the impact on healthy individuals.The mean field equations of the control framework are listed as follows:

We call the system described by Eq.(37)the SIS model of rumor spreading with suppression.For simplicity,we do not consider the optimal strategy of restraining strategy,but only carry out a simple quantitative analysis of the inhibition strategy.

Shown in Fig.9,are the curves of I(t),the population proportionality of the infectious nodes,when no inhibition strategy is applied,or when strategy(i)or strategy(ii)is applied alone,or when both strategy(i)and(ii)are applied simultaneously,respectively.While the two restraining strategies are particularly prominent in the control of the spreading of rumor as shown in the figure,we can see that,the two restraining strategies have a clear control effect on the spreading of rumors.

Fig.9(Color online)Curve I(t)vs.time showing the effects of different inhibition strategies on suppressing the amount of infected individuals.

6 Conclusion and Discussion

This paper has studied the rumor spreading model in online social networks.According to the dynamic model of infectious diseases and population dynamics model,we set up an SIS rumor spreading model that it is in line with the rumor spreading characteristics of online social networks.By numerical simulations,the effect of different infection rate and cure rate on the behavior of the new rumor spreading model is studied.Finally,based on the results of numerical analysis and model simulation,some useful inhibition strategies on rumor spreading are proposed.

It must be pointed out that,the discussions about rumor spreading in online social networks will continue.For example,we can further consider the topological structure characteristics of the networks.In fact,the community character and self-similarity are typical characteristics of the topological structures of real online social networks.In the future work,it can be further discussed.

Acknowledgments

The authors thank Dr.Yanbin Deng for very useful discussions.