Subject Areas: Network Modeling and Simulation
Nowadays, more and more SNS (Social Networking Services) networks are emerging in our social life, such as Facebook, WeChat, LinkedIn and so on, which are seemingly like cobwebs to connect people from different places. With the rapid increase of the number of SNS users, rumor will be quickly into people’s horizons. Each coin has its two sides, as the rumors spread on the impact of our social lives. Sometimes, the rumor spreading may play a positive role, for instance, we can let more people to concern about something and take pertinent precaution measures by utilizing the rapid and efficient characteristic of rumor spreading   . However, most rumors induce public panic, social disarray and severe economic loss, etc.   . Therefore, it is very important to investigate the mechanism of rumor spreading and how to effectively control the rumor.
Rumor can be viewed as an “infection of the mind”, and its spreading shows an interesting similarity to the epidemic spreading  -  . Daley and Kendal  first proposed the classic DK model of rumor spreading. Since then, most of the studies are based on DK model  -  . In order to overcome the weaknesses of DK model, more and more researchers consider the topological characteristics of underlying networks that they have started to study the problems of rumor spreading on complex networks  -  . Nekovee and Moreno et al.  derived a conclusion that scale-free social networks were prone to the spreading of rumors. In Ref.  , the authors found that the degree distribution influenced directly the final rumor size. Recently, researchers  -  started to take full into account of the role of human behaviors and different mechanisms in the rumor spreading. Zhao et al.  presented a novel model by introducing the forget mechanism. Wang et al.  presented a novel SIR model by introducing the trust mechanism between the ignorant nodes and the spreader nodes. Han et al.  presented a novel model based on the heat energy theory to analyze the mechanisms of rumor propagation on social networks.
However, most of the previous models didn’t consider that people may not agree with the rumor and counterattack it strongly. Based on some realistic perspectives, different people may have different views to the rumor on social networks. Some people may be in conflict with their beliefs when they hear rumor. They will persuade infected individuals to resist and counterattack the rumor propagation. In order to study this phenomenon, we present a SICS (susceptible-infected-counter-susceptible) rumor spreading model with counter mechanism on complex social networks to explain it. Obviously, the counter mechanism can change the contacts among people, i.e. network topology structure. Within the counter mechanism of the SICS model, when an infected individual contacts a counter individual, it may become a counter individual with a certain probability.
The rest of this paper is organized as follows. In Section 2, we present a SICS rumor spreading model and derive the corresponding mean-field equations to describe the dynamics of the model. In Section 3, the basic reproductive number obtained at first. Then we analyze the globally asymptotic stability of rumor-free equilibrium and the permanence of the rumor in detail. Simulation results of the proposed model are shown in Section 4. Finally, we conclude the paper in Section 5.
2. Model Formulation
As mentioned earlier, we present a SICS rumor spreading model. The population is divided into three classes: susceptible individuals who have ambiguous attitude about the rumor; infected individuals who believe and spread it actively; counter individuals who reject the rumor, refute the rumor and persuade neighbors don’t believe in it. Taking into account the heterogeneity induced by the presence of vertices with different connectivities, let be the densities of susceptible, infected and counter individuals of connectivity k at time t, respectively.
The SICS model has the flow diagram given in Figure 1. In the course of rumor spreading, a susceptible individual is infected with probability if it is connected to an infected individual. When a counter individual
Figure 1. The flow diagram of the SICS model.
contacts an infected individual, the counter individual can persuade infected individual to resist and counterattack the rumor, so the infected individual becomes a counter node with probability. A susceptible individual transform into a counter individual with probability. Due to some own reason, an infected individual turns into a counter individual with probability. However, some counter individuals, due to loss of counterattack ability, join the susceptible individuals again, i.e., moving back to susceptible state, with probability. We assume that the immigration rate and emigration rate are both constant l in the spreading process of rumor. All recruitment is into the susceptible class.
Thus, the dynamic mean-field reaction rate equations can be written as
The probability describes a link pointing to an infected individual,
the probability describes a link pointing to a counter individual which satisfies the relation
where is the average degree within the network. And is the density of infected individuals in the whole network, is the density of counter individuals in the whole network, is the connectivity distribution.
3. Stability Analysis
In this section, we present an analytic solution to the deterministic equations describing the dynamic of the (SICS) rumor spreading process.
Theorem 1. Let.. There always exists a rumor-free equilibrium and when, then system (1) has a positive equilibrium solution .
Proof. To get the equilibrium solution, we need to make the right side of system (1) equal to zero. Then the equilibrium should satisfy
where, , one has
According to the following normalization condition for all k:
We can obtain:
Inserting Equation (6) into Equation (2), we obtain the following equation
Inserting Equation (7) into Equation (3), we obtain the following equation
Equation (9) divided by Equation (8), we obtain the following equation
Inserting Equation (10) into Equation (8), we can obtain
Obviously, is a solution of Equation (11), i.e.,. To ensure Equation (11) have a nontrivial solution, i.e., the following conditions must be satisfied
We can obtain the basic reproductive number
So, a nontrivial solution exists if and only if.
Substitute the nontrivial solution of (11) into (6), we can get. By (5) and (6), we can easily obtain
Therefore, the positive equilibrium is well-defined. Hence, when, one and only one positive equilibrium of system (1) exists. This completes the proof.
Remark. The basic reproductive number is obtained by Equation (12), which depends on the fluctuations of the degree distribution and the influence of counter mechanism. The can affect the basic reproductive number.
Theorem 2. If, the rumor-free equilibrium of the system (1) is globally asymptotically stable.
Proof. We rewrite the system (1) as
The Jacobian matrix of system (13) at is a as follows
By mathematical induction method, the characteristic equation can be calculated as follows
The stability of is only dependent on
So, we have obtained
When, all real-valued eigenvalues are negative. Hence, is locally asymptotically stable if.
Now we will prove that is globally attractive. From the second equation of system (1) we can get
Now we consider the comparison equation with the condition as follows
integrating from 0 to t yields
Since, we obtain as.
According to the comparison theorem of functional differential equation, we can get, for all.
Thus, as, which implies as, for. It follows that the rumor-free equilibrium is globally attractive. This completes the proof.
Theorem 3. If, the rumor is permanent on complex social networks, i.e., there exists a, such that
Proof. We will use the result of Thieme in Theorem 4.6  to prove it. Define
In the following, we will show that (1) is uniformly persistent with respect to.
Obviously, X is positively invariant with respect to system (1). If, and for, then, and for all. Since, , and, we have, . Thus, is also positively invariant. Furthermore, there exists a compact set B in which all solutions of (1) initiated in X will enter and remain forever after. The compactness condition (C4.2) in Thieme  is easily verified for this set B. Denote
where is the omega limit set of the solutions of system (1) starting in. Restricting system (1) on gives
It is easy to verify that system (13) has a unique equilibrium in X. Thus is the unique equilibrium of system (1) in. It is easy to check that is locally asymptotically stable. This implies that is globally asymptotically stable for (13) is a linear system. Therefore. And is a covering of, which is isolated and is acyclic (since there exists no solution in which links to itself). Finally, the proof will be done if we show is a weak repeller for, i.e.
where is an arbitrarily solution with initial value in. By Leenheer and Smith (2003, Proof of Lemma 3.5,  ), we need only to prove where is the stable manifold of. Suppose it is not true, then there exists a solution in, such that
Since, we can choose such that
For, by (15) there exists a such that
For all and. Let
The derivative of V along the solution is given by
Hence as, which contradicts to the boundedness of. This completes the proof.
4. Numerical Simulations
In this section, several numerical simulations are presented to illustrate our analysis. We consider the system (1) on a complex social network with, where the parameter satisfies.
In Figure 2, the parameters are chosen as then the basic reproductive number. We can see that when, grows to zero, i.e., the infectious individuals will ultimately disappear.
In Figure 3, we choose thus the basic reproductive number. We can see that when, the rumor is persist and the infected individuals’ number will converge to a positive constant respectively.
Figure 2. The time series of system (1) with and initial values,
Figure 3. The time series of system (1) with and initial values,
In Figure 4, numerical simulations show the spread of SICS model on complex social networks with and. The condition of, that different leading to different states. In addition, it is also found that the larger the is, the rumor dies out faster.
In Figure 5, numerical simulations show the spread of SICS model on complex social networks with and. The condition of, that different leading to different states. In addition, it is also found that the larger the is, the positive equilibrium will be lower. The simulations indicate that the numerical results are well consistent with the theoretical analysis.
Figure 4. The prevalence versus t corresponding to different with identical initial value.
Figure 5. The prevalence versus t corresponding to different with identical initial value.
In summary, we present a new SICS rumor spreading model with counter mechanism on complex social networks. By using the mean-field theory, we obtain the basic reproductive number and equilibriums. Theoretical results indicate that the basic reproductive number is significantly dependent on the topology of the underlying networks and the counter mechanism. The basic reproductive number is in direct proportion to. So, network heterogeneity makes rumor easy to spread. Moreover, we found that the greater can decrease the basic reproductive number, i.e., lower average rumor density and shorter rumor prevalent decay time. The global stability of rumor-free equilibrium and the permanence of rumor are proved in detail. Our theoretical and numerical simulation results give a novel explanation for rumor spreading. This study has valuable guiding significance in effectively preventing rumor spreading.
This work was supported in part by the National Natural Science Foundation of China under Grant 60973012.
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