^{1}

^{*}

^{1}

^{1}

^{1}

The rumor spreading has been widely studied by scholars. However, there exist some people who will persuade infected individuals to resist and counterattack the rumor propagation in our social life. In this paper, a new
SICS
(susceptible-infected-counter-susceptible) rumor spreading model with counter mechanism on complex social networks is presented. Using the mean-field theory the spreading dynamics of the rumor is studied in detail. We obtain the basic reproductive number
r
and equilibriums. The basic reproductive number is correlated to the network topology and the influence of the counter mechanism. When
*ρ***＜1**
, the rumor-free equilibrium is globally asymptotically stable, and when
*ρ*＞1
, the positive equilibrium is permanent. Some interesting patterns of rumor spreading involved with counter force have been revealed. Finally, numerical simulations have been given to demonstrate the effectiveness of the theoretical analysis.

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 [

Rumor can be viewed as an “infection of the mind”, and its spreading shows an interesting similarity to the epidemic spreading [

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.

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

The SICS model has the flow diagram given in

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

Thus, the dynamic mean-field reaction rate equations can be written as

The probability

the probability

where

In this section, we present an analytic solution to the deterministic equations describing the dynamic of the (SICS) rumor spreading process.

Theorem 1. Let.

Proof. To get the equilibrium solution

where

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,

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

Therefore, the positive equilibrium

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

Theorem 2. If

Proof. We rewrite the system (1) as

The Jacobian matrix of system (13) at

where

By mathematical induction method, the characteristic equation can be calculated as follows

where

The stability of

Note that

So, we have obtained

When

Now we will prove that

Now we consider the comparison equation with the condition

integrating from 0 to t yields

Since

According to the comparison theorem of functional differential equation, we can get

Thus,

Theorem 3. If

Proof. We will use the result of Thieme in Theorem 4.6 [

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

Denote

where

It is easy to verify that system (13) has a unique equilibrium _{ }is a covering of

where

Since

For

For all

The derivative of V along the solution is given by

Hence

In this section, several numerical simulations are presented to illustrate our analysis. We consider the system (1) on a complex social network with

In

In

In

In

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

This work was supported in part by the National Natural Science Foundation of China under Grant 60973012.

Chen Wan,Tao Li,Yuanmei Wang,Xiongding Liu, (2016) Rumor Spreading of a SICS Model on Complex Social Networks with Counter Mechanism. Open Access Library Journal,03,1-11. doi: 10.4236/oalib.1102885