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In this paper, we present a SARS (susceptible-adopted-removed-susceptible) social information spreading model with overlapping community structures on complex networks. Using the mean field theory, the spreading dynamic of the model has been studied. At first, we derived the spreading critical threshold value and equilibriums. Theoretical results indicate that the existence of equilibriums is determined by threshold value. The threshold value is obviously dependent on the topology of underlying networks. Furthermore, the globally asymptotically stable equilibriums are proved in detail. The overlap parameter of community structures can't change the threshold value, but it can influence the extent of the social information spreading. Numerical simulations confirmed the analytical results.

Complex networks can be described by many real-world systems [

With the study of network structural properties, the spread of an epidemic over complex networks has investigated very mature [

In sum, the rumor or disease transmission model contributes to understanding the intrinsic mechanisms of those spreading processes and designing efficient control strategies. However, information spreading has difference from disease infections because of its specific features, such as time decaying influence [

In Section 2, we present a SARS social information spreading model with overlapping community structures and introduces related work on complex networks. In Section 3, we analyze the globally asymptotically stable equilibriums in detail. In Section 4, numerical experiments and simulation results are given to illustrate the theoretical results. Finally, conclusions and future works are drawn in Section 5.

In this article, we discussed the social information spreading on complex networks with the overlapping community structure. Overlapping community structure is mainly to describe the network topology relatively strongly linked to the internal part of the node and the external characteristic of contact relatively sparse. We use a SARS model to illustrate the proposed social information spreading process. In this model, we assume that social information spreading is disseminated by direct contacts of adopted nodes with others, and the population is divided into three groups: susceptible (S), adopted (A), removed (R), where S, A, R represent the people who never heard the information (Susceptible), those who are spreading information (Adopted), and the ones who heard the information but have lost interest in diffusing it (Removed). From now on, we refer to the SARS model as the information spreading model. On the size of the N in the social information network, we suppose there are two communities with the same size A and B. We defined v is an overlap parameter. The probability of each adopted nodes connect to any node in the community A by v, with the probability of

For the SARS model on scale-free network, taking into account the heterogeneity included by the presence of

vertices with different connectivity, let

adopted and removed nodes of degree k at time t respectively. With these signs and symbols, the dynamics mean-field reaction rate equations can be written as

The dynamics of SARS subsystems are coupled through the function

where

So

where

Definition. The equilibrium is an information-free equilibrium if

Theorem 1. Let

Proof. To get the equilibrium solution

This leads to

Substituting (2.4)

Clearly,

We can obtain the threshold value:

where

Hence the system (2.1) has an permanent equilibrium

In this section, the globally asymptotically stable

Theorem 2. The information-free equilibrium

Proof. First, we prove that

We rewrite system (2.1) as

After the linearization, we write the system (3.1) as

Then the Jacobian matrix of (3.2) at

where

Using induction on n, the characteristic equation can be expressed as

The characteristic equation have n eigenvalues for

Since

All the eigenvalues of J are negative if

Next, we will prove that the equilibrium

Now we consider the comparison equation with the condition

Integrating from 0 to t yields,

According to the comparison theorem of functional differential equation, we have

Therefore,

We now prove the globally asymptotically stable of equilibrium

Theorem 3. When

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

Obviously, X is positively invariant with respect to system (2.1). If

where

(2.1) starting in

It is easy to verify that system (3.4) has a unique equilibrium

which is isolated and is acyclic (since there exists no solution in

where

By Leenheer and Smith [

fold of

Since

For

for all

V represent the proportion of all adopted individuals to all individuals. The derivative of V along the solution of system (2.1) is given by

There

In this section, we will give some numerical simulations to illustrate the theoretical analysis. We consider the system (2.1) on a scale-free network with the degree distribution

In

In

In

In

In this paper, a SARS social information spreading model with the overlapping community structures on complex networks has been presented. By mean-filed theory, we have proved that there exists a threshold value

This work was supported in part by the National Natural Science Foundation of China under Grants 60973012, 6147234.

Xiongding Liu,Tao Li,Yuanmei Wang,Chen Wan, (2016) Spreading Dynamics of a Social Information Model with Overlapping Community Structures on Complex Networks. Open Access Library Journal,03,1-11. doi: 10.4236/oalib.1102701