Spreading dynamics of an online social information model on scale-free networks

Physica A 514 (2019) 497–510

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Physica A journal homepage: www.elsevier.com/locate/physa

Spreading dynamics of an online social information model on scale-free networks ∗

Xiongding Liu, Tao Li , Hao Xu, Wenjin Liu School of Electronics and Information, Yangtze University, Jingzhou 434023, PR China

highlights • • • •

Present a new ICST online social information spreading model on scale-free networks. Study the stability of equilibriums and the permanence of information spreading. Comment mechanism and effective comment rate can affect information spreading. Adaptive weight can affect the information spreading and the sustained level.

article

info

Article history: Received 1 June 2018 Received in revised form 4 August 2018 Available online 26 September 2018 Keywords: ICST model Comment mechanism Heterogeneity Stability Permanence

a b s t r a c t In order to study the influence of the comment mechanism and the heterogeneity of underlying networks on the spreading of online social information, we present a new ICST (ignoramus-commentator-sharer-stifler) online social information spreading model based on scale-free networks. By using the mean-field theory, the spreading dynamics of the model is analyzed in detail. Then, the basic reproductive number R0 and equilibriums are derived. Theoretical results show that the basic reproduction number is significantly dependent on the topology of the underlying networks. The relationships among the basic reproduction number R0 , sharing rate, effective comment rate are studied. Furthermore, the global stability of the information-elimination equilibrium, the permanence of online social information spreading and the global attractivity of information-prevailing equilibrium are proved in detail. In addition, we study the influence of weight in networks and analyze the corresponding of dynamics behaviors. The adaptive weight cannot change the basic reproductive number, but it can weaken the information spreading. Numerical simulations confirmed the analytical results. © 2018 Elsevier B.V. All rights reserved.

1. Introduction With the rapid development of the information technology, the Internet has penetrated into every area in our daily lives. While network technology has brought us convenience, we also need to control and supervise the Internet [1–5]. In the internet networks, people use social platforms, such as Facebook, WeChat and Microblog to communicate and spread information [6,7]. However, compared with the forms of traditional social contact, users in online social platforms are no longer a media audience of passive accept information, but it focuses on building virtual communities to share files and information resources, becomes information producers, distributors and spreaders. So, the study of online social information spreading is of great significance to restrain the spread of malicious information and strengthen the supervision of public opinion on the internet. ∗ Corresponding author. E-mail address: [email protected] (T. Li). https://doi.org/10.1016/j.physa.2018.09.085 0378-4371/© 2018 Elsevier B.V. All rights reserved.

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X. Liu et al. / Physica A 514 (2019) 497–510

Fig. 1. The flow diagram of the ICST model.

In the spreading dynamics of complex networks, researchers have made some achievements in rumor spreading. Daley and Kendall mathematically formalized the classical DK rumor spreading model in 1965s [8]. Maki and Thomson [9] modified and developed another model by changing spreading mechanism based on DK model. Whereafter, based on the effect of topologies properties on complex social interactions, Zanette [10,11] and Buzna et al. [12] studied the rumor spreading model on small-word networks and put forward the critical threshold. Considering the effect of latent state, Zhang Y [13] presented a variant SIR (spreader–ignorant-remove) rumor spreading model on scale-free networks. We all know that rumor is a kind of social information, the theories and methods of studying rumors also apply to the spread of social information. Some researchers began to study the rule of information spreading from control and the influence of people’s behaviors on information spreading. The trust mechanism, information pushing, response time and multi-messages spreading have been taken into account [14–18]. With the further development of networks technology, the form of information spreading is more diversified, especially in online social information. Li A researched on the CSER rumor spreading model in online social networks and analyzed the heterogeneous of networks [19]. Meanwhile, some researchers focused on the complex topology of social interaction. It is well known that a fundamental characteristic of social networks is scale-free property [20,21]. Obviously, the online social networks also has scale-free property. In online social networks, nodes represent individuals and edges represent the relationships of people. Liangjian Zhang described the information spreading on a small world networks [22]. Wan et al. studied an e-commercial preferential information spreading on scale-free networks and analyzed the permanence of preferential information spreading [23]. In addition, the effects of rewiring strategies on information spreading in complex dynamics networks has been studied [24,25]. From most of the research work mentioned above, people’s comments on online social information spreading are not considered. In fact, with the spreading of information, people pay more attention to people’s comments to verify the credibility of information. Ref. [26] proposed a comment mechanism in information spreading model on online networks. However, the global attractivity of information-elimination equilibrium and analyze the influence of effective comment rate are not researched in details. Meanwhile, the degree of intimacy between nodes, that is weight, also plays an important role in online society information spreading. Motivated by the above, we establish a novel ICST online social information spreading model on scale-free networks and comprehensively prove the stability of the model in detail. The remaining part of the paper has been arranged as follows: Section 2 presents the ICST online social information spreading model on scale-free networks. In Section 3 the basic reproduction number and equilibriums are obtained. Section 4 analyzes the globally asymptotic stability of information-elimination, the permanence of the online social information spreading and the global attractivity of information-prevailing equilibrium in detail. In Section 5, the weight of ICST online social information spreading model is introduced and the corresponding of dynamical behaviors is studied. In Section 6, numerical simulations are presented to illustrate our main results. Finally, we give some conclusions and discussions in Section 7. 2. Model formulation In the process of online social information spreading, we divide the whole population into four distinct classes. People who never see the information are called ignoramus (I), people who know about the information and comment the information are called commentator (C ), people who share the information are called sharer (S), people who have no response to the information are called stifler (T ). The information spreading model has the flow diagram given in Fig. 1 with the following assumptions. When an ignoramus encounters a sharer, there are three possible outcomes: (i) the ignoramus may comment on the information with probability α, namely commenting rate; (ii) the ignoramus may be interested in the information so that he/she shares the information with probability β, namely sharing rate; (iii) the ignoramus may have no response to information with probability γ , namely stifling rate. A commentator may have interest in the information and become a sharer with probability η. Due to the recession of the information interest for people and the decrease of heat for information, sharers may become stiflers with probability δ . The new registered users of the network’s growth rate is p. The number of logout rate of each individual state is µ. In this model, we assumed µ equal to p. Let Ik (t), Ck (t), Sk (t) and Tk (t) be the relative

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densities of ignoramus, commentators, sharers and stiflers nodes of degree kat time trespectively. With these assume, the dynamics mean-field equations of the ICST model can be written as follows:

⎧ ⎪ dIk (t) ⎪ ⎪ = p − µIk (t) − α kΘ (t)Ik (t) − β kΘ (t)Ik (t) − γ kΘ (t)Ik (t), ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ dC (t) ⎪ ⎪ ⎨ k = α kΘ (t)Ik (t) − ηCk (t) − µCk (t), dt

(2.1)

⎪ ⎪ dSk (t) ⎪ ⎪ = β kΘ (t)Ik (t) + ηCk (t) − δ Sk (t) − µSk (t), ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dTk (t) ⎩ = δ Sk (t) + γ kΘ (t)Ik (t) − µTk (t). dt

Θ (t)denotes the probability of an ignorant node with k links points to a sharer node at time t, which satisfies the relation Θ (t) =

n 1 ∑

⟨k⟩

kP(k)Sk (t).

(2.2)

i=1

∑n

∑n

P(k) is ∑ the probability that a node has degree k and thus k=1 P(k) = 1; ⟨k⟩ = k=1 kP(k) denotes the average degree. n S(t) = P(k)S (t) is the total density of sharers on scale-free networks. Clearly, these variables obey the normalization k k=1 condition: Ik (t) + Ck (t) + Sk (t) + Tk (t) = 1.

(2.3)

The initial conditions for system can be given as follows Tk (0) = 1 − Ik (0) − Sk (0) − Ck (0) ≥ 0, Tk (0) ≥ 0, Sk (0) ≥ 0,Ck (0) ≥ 0. 3. The basic reproduction number and existence of equilibriums In this section, we present an analytic solution to the deterministic equations describing the dynamics of the ICST online social information spreading process. ⟨k2 ⟩(β (η+µ)+ηα )

Theorem 1. Consider the system (2.1), define R0 = ⟨k⟩(δ+µ)(η+µ) . There always exists an information-elimination equilibrium E0 (1, 0, 0, 0) when R0 < 1. When R0 > 1, the system (2.1) has an information-prevailing equilibrium E ∗ (Ik∗ , Ck∗ , Sk∗ , Tk∗ ). Proof. One can easily find that E0 (1, 0, 0, 0) is always an equilibrium of system (2.1), which is called the informationelimination equilibrium. To get the information-prevailing equilibrium solution E ∗ (Ik∗ , Ck∗ , Sk∗ , Tk∗ ), we need to make the right side of system equal to zero, it should satisfy

⎧ p − µIk∗ − (α + β + γ )kΘ ∗ Ik∗ = 0, ⎪ ⎪ ⎨α kΘ ∗ I ∗ − (η + µ)C ∗ = 0, k

k

⎪ β kΘ ∗ I ∗ + ηCk∗ − (δ + µ)Sk∗ = 0, ⎪ ⎩ ∗ k δ Sk + γ kΘ ∗ Ik∗ − µTk∗ = 0, ∑n ∗ where Θ ∗ = ⟨1k⟩ i=1 kp(k)Sk (t). One has ⎧ ⎪ (µ + δ )(η + µ) ⎪ ⎪Ik∗ = S∗, ⎪ ⎪ ∗ k ( η + µ ) + ηα) k Θ ⎪ (β ⎪ ⎪ ⎨ α (µ + δ ) Ck∗ = Sk∗ , ⎪ ( η + µ ) + ηα) (β ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∗ δ (β (η + µ) + ηα) + γ (µ + δ )(η + µ) ∗ ⎪ ⎩Tk = Sk . µ (β (η + µ) + ηα) Considering the following normalization condition Ik (t) + Ck (t) + Sk (t) + Tk (t) = 1 for all k, we obtain kΘ ∗ µ (β (η + µ) + ηα) ( ). Sk∗ = (µ + δ ) (β (η + µ) + ηα) kΘ ∗ + (αµ + γ (η + µ)) kΘ ∗ + µ(η + µ)

(3.1)

(3.2)

(3.3)

Inserting into (2.2), we obtain that

Θ∗ =

n 1 ∑

⟨k⟩

i=1

kp(k)

kΘ ∗ µ (β (η + µ) + ηα) (µ + δ ) ((β (η + µ) + ηα) kΘ ∗ + (αµ + γ (η + µ)) kΘ ∗ + µ(η + µ))

.

(3.4)

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Let Θ ∗ ≜ f (Θ ∗ ) , clearly, Θ ∗ = 0 is a solution of Eq. (3.3). To ensure the equation has a nontrivial solution, the following condition must be satisfied

⏐ ⏐ ⏐ ⏐

df (Θ ∗ ) ⏐ dΘ ∗

>1

and

f (1) ≤ 1

(3.5)

Θ ∗ =0

we can obtain the basic reproductive number

⟨k2 ⟩ (β (η + µ) + ηα) > 1, ⟨k⟩(δ + µ)(η + µ) ∑n 2 where ⟨k2 ⟩ = i=1 i P(i). So, a nontrivial solution exists if and only if R0 > 1.

(3.6)

R0 =

Inserting the nontrivial solution of (3.4) into Eq. (3.3), we can obtain Sk∗ . By (3.2) and (3.3) we can easily get 0 < Ik∗ < 1, 0 < Ck∗ < 1, 0 < Sk∗ < 1, 0 < Tk∗ < 1. Thus, the equilibrium E ∗ (Ik∗ , Ck∗ , Sk∗ , Tk∗ ) is well-defined. Hence, when R0 > 1,only one positive equilibrium E ∗ (Ik∗ , Ck∗ , Sk∗ , Tk∗ ) of system (2.1) exists. The proof is completed. Define. c = αη, c represents the effective comment rate to measure the influence of the parameter η on information spreading.

Remark. The basic reproductive number R0 is obtained by Eq. (3.6), which depends on the fluctuations of the degree distribution and some model parameters. Obviously, as the effective comment rate c increases, the basic reproductive number R0 increases. In Section 6, their effects will be explored by detailed numerical calculation. If α = 0 and η = 0, then the system (2.1) become the standards model with R0 = β⟨k2 ⟩/(β⟨k2 ⟩) , which consists with Ref. [27]. 4. Stability analysis of the equilibrium Theorem 2. The information-elimination equilibrium E0 of the system (2.1) is locally asymptotically stable when R0 < 1, and it is unstable when R0 > 1. Proof. For Ik (t) + Ck (t) + Sk (t) + Tk (t) = 1, i.e., if the values of Ik (t), Ck (t) and Sk (t) are fixed there is only one corresponding Tk (t), we will discuss the first three equations of (2.1).

⎧ ⎪ dIk (t) ⎪ ⎪ = p − µIk (t) − α kΘ (t)Ik (t) − β kΘ (t)Ik (t) − γ kΘ (t)Ik (t), ⎪ ⎪ dt ⎪ ⎪ ⎨ dCk (t) = α kΘ (t)Ik (t) − ηCk (t) − µCk (t), ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dSk (t) = β kΘ (t)I (t) + ηC (t) − δ S (t) − µS (t), k

dt

k

k

(4.1)

k

where the Jacobian matrix of the information-elimination equilibrium E0 of the system (4.1) is A11 ⎢A21

A12 A22

.

.. .

··· ··· .. .

An1

An2

···

⎡

J =⎢ ⎣ ..

Ann

⎡ −µ ⎢ 0 =⎣ 0

0

A1n A2n ⎥ Ann

⎤

.. ⎥ ⎦ .

−(µ + η) η

0 ⎢0

⎡ ,

where Aij = ⎣ 0

3n×3n

0 0 (β (η + µ) + ηα) i · j · P(j)

0 0 0

0 0

⎤

⎥ (β (η + µ) + ηα) i · j · P(j) ⎦ (i ̸= j), (η + µ)⟨k⟩ ⎤

− (µ + δ )

⎥ ⎦,

(i, j = 1, 2, . . . , n).

(η + µ)⟨k⟩ By mathematical induction method, the characteristic equation can be calculated as

{

(λ + µ)n (λ + η + µ)n (λ + δ + µ)n−1 λ + (δ + µ) −

} ] (β (η + µ) + ηα) [ 2 1 P(1) + 22 P(2) · · · n2 P(n) . (η + µ)⟨k⟩

(4.2)

This equation has a negative root −µ with multiplicity n, a negative root −µ − η with multiplicity n and a negative root −µ − δ with multiplicity n − 1. Note that ⟨k2 ⟩ = 12 P(1) + 22 P(2) + · · · + n2 P(n).

λ + (δ + µ ) −

(β (µ + η) + ηα) ⟨k2 ⟩ = 0, (η + µ)⟨k⟩

i.e. λ = (δ + µ)(R0 − 1).

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If λ = R0 < 1 then λ < 0 and if λ = R0 > 1, then λ > 0. Thus, E0 is locally asymptotically stable when R0 < 1, and it is unstable when R0 > 1. In the following, we consider globally asymptotically stable of E0 and the global attractivity of E ∗ , which is one of the most important topic in the study of information spread. Lemma 4.1 ([28]). If a > 0, b > 0 and dt ≥ b − ax, when t ≥ 0 and x(0) ≥ 0, we have limt →∞ inf x(t) ≥ dx(t) and dt ≤ b − ax, when t ≥ 0 and x(0) ≥ 0, we have limt →∞ sup x(t) ≤ ba . dx(t)

b , a

if a > 0, b > 0

Theorem 3. The information-elimination equilibrium E0 (1, 0, 0, 0) of system (2.1) is globally asymptotically stable if R0 < 1. Proof. Let us consider a non-negative solution (Ik (t), Ck (t), Sk (t)) of system (4.1). We first claim that limt →∞ Sk (t) = 0. From the first equation for the system, it follows that dIk (t)

= p − µIk (t) − α kΘ (t)Ik (t) − β kΘ (t)Ik (t) − γ kΘ (t)Ik (t) ≤ p − µIk (t). dt By Lemma 4.1, we derive that p =: Ik0 . lim sup Ik (t) ≤

(4.3)

(4.4)

µ

t →∞

Thus, for arbitrarily enough small ε1 > 0, there exists t1 > 0 such that Ik (t) ≤ Ik0 + ε1 for t > t1 . If t > t1 , it follows that dSk (t)

≤ β kΘ (t) Ik0 (t) + ε1 + ηCk (t) − (δ + µ)Sk (t), dt Now, we consider the comparison system with the condition Uk (0) = Sk (0) ≥ 0, as follows:

(

)

(4.5)

dUk (t)

( ) ≤ β kΘ ′ (t) Ik0 (t) + ε1 + ηCk (t) − (δ + µ)Uk (t), (4.6) ∑ n where Θ ′ (t) = ⟨1k⟩ k=1 kP(k)Uk (t). We will then show that positive solutions of (4.6) tend to zero as t goes to infinity. ∑n k(η+µ) Let us consider the Lyapunov function V (t) = > 0. Then we have k=1 hk Uk (t), where hk = ⟨k⟩ dt

n ⏐ ∑ ) ] [ ( hk β kΘ ′ (t) Ik0 (t) + ε1 + ηCk (t) − (δ + µ)Uk (t) ⏐=

dV ⏐ dt

=

k=1

n [ ∑ kP(k) (β (η + µ) + αη) k

(Ik0 (t) + ε1 )Θ ′ (t) −

kP(k)(η + µ)(δ + µ)

⟨k⟩ ) 2 ⟨ k ⟩ (β (η + µ) + αη) =Θ ′ (t) R0 + ε1 − 1 . ⟨k⟩

⟨k⟩

k=1

] Uk (t)

(

Since R0 < 1, we can choose an small enough ε1 > 0, such that R0 + ⟨k2 ⟩ (β (η + µ) + αη) ε1 ⟨k⟩−1 < 1. This ensures that R0 < 1, dV ≤ 0 for all Uk (0) ≥ 0, and that dV = 0 only if Uk (0) = 0. Therefore, we have the solutions of (4.6) tend to zero dt dt as t → +∞, that is, limt →+∞ Uk (t) = 0. By the comparison theorem, we have 0 ≤ Sk (t) ≤ Uk (t), for all t > 0. Therefore, Sk (t) = 0 as t → +∞, for k = 1, 2, . . . , n. Combining with the second equation of system (4.1), it obviously follows that Ck (t) = 0 as t → +∞, for k = 1, 2, . . . , n. Next, we will show Ik (t) = Ik0 . Since limt →+∞ Sk (t) = 0 and limt →+∞ Ck (t) = 0, for arbitrarily enough small ε2 > 0, there exists t2 > 0 such that 0 ≤ Ck (t) ≤ ε2 , 0 ≤ Sk (t) ≤ ε2 for t > t2 . From the first Eq. (4.1), we have dIk (t) dt where M =

≥ p − µIk (t) − (α + β + γ )Ik (t)M ε2 . ∑n 1 ⟨k⟩

i=1

lim Ik (t) ≥

t →+∞

kP(k), by Lemma 4.1, we have limt →+∞ Ik (t) ≥

p

µ

= Ik0 .

From (4.4) and (4.7), it is clear that limt →+∞ Ik (t) = Ik0 = globally asymptotically stable R0 < 1. The proof is completed.

p

µ+(α+β+γ )M ε2

. Setting ε2 → 0, it follows that (4.7)

p

µ

= 1. This prove that the equilibrium E0 of system (4.1) is

Next, the global attractivity of the information-prevailing equilibrium is discussed. The main result is given in the following theorem. Theorem 4. Suppose that (Ik (t), Ck (t), Sk (t)) is a (solution of system)(4.1) satisfying initial conditions Ck (t) > 0 or Sk (t) > 0. ( ) If R0 > 1, then limt →∞ (Ik (t), Ck (t), Sk (t)) = Ik∗ (t), Ck∗ (t), Sk∗ (t) , where Ik∗ (t), Ck∗ (t), Sk∗ (t) is the information-prevailing equilibrium of (4.1) satisfying (3.2) for k = 1, 2, . . . , n.

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Proof. In the following, k is fixed to be any integer in (1, 2, . . . , n). By Theorem 4, there exists a sufficiently small constant ξ (0 < ξ < 1) and a larger enough constant T > 0 such that Sk (t) ≥ ξ for t > T , therefore Θ (t) > ξ Θ for t > T . Submit this into the equation of (4.1) gives Ik′ (t) ≤ p − µIk (t) − (α + β + γ )kΘ ξ Ik (t),

t > T.

(4.8)

By the standard comparison theorem in the theory of differential equations, for any given constant 0 < ξ < there exists a t1 > T , such that Ik (t) ≤ (1) Ak

=

p

µ + (α + β + γ )kΘ ξ

(1) Ak

− ξ1 for t > t1 , where

(α+β+γ )kΘ ξ , 2(µ+(α+β+γ )kΘ ξ )

+ 2ξ1 < 1.

From the second equation of (4.1), it follow that Ck′ (t) ≤ α kΘ (1 − Ck (t)) − (µ + η)Ck (t),

t > t1 .

(4.9)

Hence, for any given constant 0 < ξ2 < min{1/2, ξ1 , (η + µ)[2(µ + η + α kΘ )] (1) Ck (t) ≤ Bk − ξ2 for t > t2 , where (1)

Bk =

−1

}, there exists a t2 > t1 , such that

α kΘ + 2ξ2 < 1. α kΘ + η + µ

Then, it follows from the third equation of (4.1), Sk′ (t) ≤ β kΘ (1 − Sk (t)) + η(1 − Sk (t)) − (µ + δ )Sk (t),

t > t2 .

(4.10)

Similarly, for any given constant 0 < ξ3 < min{1/3, ξ2 , (δ + µ)[2(µ + η + δ + β kΘ )] (1) Sk (t) ≤ Dk − ξ3 for t > t3 , where

−1

}, there exists a t3 > t2 , such that

β kΘ + η + 2ξ3 < 1. β kΘ + η + µ + δ ∑n Since Θ (t) ≤ ⟨1k⟩ i=1 iP(i) =: H, we substitute this into the first equation of (4.1) (1)

Dk =

Ik′ (t) ≥ p − µIk (t) − (α + β + γ )kHIk (t),

t > T.

(4.11)

So for any given enough small constant 0 < ξ4 < min{1/4, ξ3 , p[2(µ + (α + β + γ )kH)] (1) Ik (t) ≥ ak + ξ4 for t > t4 , where p

(1)

ak =

µ + (α + β + γ )kH

−1

}, there exists a t4 > t3 , such that

− 2ξ4 > 0.

It follows that (1)

Ck′ (t) ≥ α kΘ ξ ak − (µ + η)Ck (t),

t > t4 .

(4.12) (1)

So for any given enough small constant 0 < ξ5 < min{1/5, ξ4 , α kΘ ξ ak [2(µ + η)]−1 }, there exists a t5 > t4 , such that (1) Ck (t) ≥ bk + ξ5 for t > t5 , where (1)

(1)

bk = α kΘ ξ ak (µ + η)−1 − 2ξ5 > 0. From the third equation of (2.1) implies that (1)

(1)

Sk′ (t) ≥ β kΘ ξ ak + ηbk − (µ + δ )Sk (t),

t > t5 .

(4.13)

So for any given enough small constant 0 < ξ6 < min{1/6, ξ5 , [β kΘ ξ (1) that Sk (t) ≥ dk + ξ6 for t > t6 , where (1)

(1)

(1) ak

+η

(1) bk

](2(µ + δ )) }, there exists a t6 > t5 , such −1

(1)

dk = [β kΘ ξ ak + ηbk ] (2(µ + δ ))−1 − 2ξ6 > 0. (1)

(1)

(1)

(1)

(1)

(1)

Due to ξ is a small positive constant, we can derive that 0 < ak < Ak < 1, 0 < bk < Bk < 1 and 0 < dk < Dk < 1. Let q(j) =

n 1 ∑

⟨k⟩

j=1

(j)

iP(i)di , Q (j)

n 1 ∑

⟨k⟩

(j)

iP(i)Di ,

j = 1, 2, . . . , n..

(4.14)

j=1

We can easily get 0 < q(j) ≤ Θ (t) ≤ Q (j) < H, t > t4 . Again, from the first equation of (4.1), it has Ik′ (t) ≤ p − µIk (t) − (α + β + γ )kq(1) Ik (t),

t > t4 .

(4.15)

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Hence, for any given constant 0 < ξ7 < min{1/7, ξ6 }, there exists a t7 > t6 , such that (2)

(1)

Ik′ (t) ≤ Ak ≜ min{Ak − ξ1 , p[µ + (α + β + γ )kq(1) ]−1 + ξ7 },

t > t7 .

Then, from the second equation of (4.1), we have (1)

Ck′ (t) ≥ α kQ (1) Ak − (µ + η)Ck (t),

t > t7 .

(4.16)

So, for any given constant 0 < ξ8 < min{1/8, ξ7 }, there exists a t8 > t7 , such that (2)

(1)

(2)

Ck′ (t) ≤ Bk ≜ min{Bk − ξ2 , α kQ (1) Ak (µ + η)−1 + ξ8 },

t > t8 .

Consequently, from the third equation of (4.1), we have (2)

(1)

Sk′ (t) ≤ β kQ (1) Ak + ηBk − (µ + δ )Sk (t),

t > t8 .

(4.17)

Hence, for any given constant 0 < ξ9 < min{1/9, ξ8 }, there exists a t9 > t8 , such that (2)

(1)

(2)

(2)

Sk′ (t) ≤ Dk ≜ min{Dk − ξ3 , (β kQ (1) Ak + ηBk )(δ + µ)−1 + ξ9 },

t9 > t8 .

Turning back, one has Ik′ (t) ≥ p − µIk (t) − (α + β + γ )kQ (2) Ik (t),

t > t9 .

(4.18)

So, for any given enough small constant 0 < ξ10 < min{1/10, ξ9 , p[2(µ + (α + β + γ )kQ (2) )]−1 }, there exists a t10 > t9 , such (2) that Sk (t) ≥ ak + ξ10 for t > t10 , where (2)

(1)

ak = max{ak + ξ4 , p µ + (α + β + γ )kQ (2)

(

)−1

− 2ξ10 }.

It follows that (2)

Ck′ (t) ≥ α kq(1) ak − (µ + η)Ck (t),

t > t10 .

(4.19)

So for any given enough small constant 0 < ξ11 < min{1/11, ξ10 , α (2) Ck (t) ≥ bk + ξ11 for t > t10 , where (2)

(2) kq(1) ak

[2(µ + η)]−1 }, there exists a t11 > t10 , such that

(1)

(1)

bk = max{bk + ξ5 , α kq(1) ak (µ + η)−1 − 2ξ11 }. From the third equation of (4.1) implies that (2)

(2)

Sk′ (t) ≥ β kq(1) ak + ηbk − (µ + δ )Sk (t),

t > t11 .

(4.20)

So, for any given enough small constant 0 < ξ12 < min{1/12, ξ11 , [β (2) such that Sk (t) ≥ dk + ξ12 for t > t12 , where (2)

(2) bk

+η

(i)

(i)

](2(µ + δ ))−1 }, there exists a t12 > t11 ,

(2)

(2)

(1)

(2) kq(1) ak

dk = max{[dk + ξ6 , β kq(1) ak + ηbk ](µ + δ )−1 − 2ξ12 }. (i)

(i)

(i)

(i)

Repeating the above analyses and calculation, we get six sequences Ak , Bk , Dk , ak , bk , dk , i = 1, 2, . . . , n. Due to the first three are monotone decreasing sequences and the last three are monotone increasing, there exists a sufficiently large positive integer L ≥ 2, such that l ≥ L: p

(l)

Ak =

(l)

Dk = (l)

bk =

µ + (α + β + γ )kq(l−1)

(l) β kQ (l−1) A(l) k + η Bk + ξ6l−3 , δ+µ

α kq(l−1) a(lk −1) − 2ξ6l−1 , µ+η

(l)

+ ξ6l−5 ,

Bk =

α kQ (l−1) A(l) k + ξ6l−4 , µ+η p

(l)

ak = (l)

dk =

µ + (α + β + γ )kQ (l)

− 2ξ6l−2 ,

(4.21)

(2) β kq(1) a(2) k + η bk − 2ξ6l . µ+δ

We can easy get that (l)

(l)

(l)

(l)

(l)

(l)

ak ≤ Ik (t) ≤ Ak , bk ≤ Ck (t) ≤ Bk , dk ≤ Sk (t) ≤ Dk ,

t > t6l .

(l) liml→∞ ∆k

(4.22) (l) ∆k

(l) Ak

(l) Bk

(l) Dk

(l) ak

(l) bk

(l) dk

(l) Qk

(l) qk

Since the sequential limits of (4.21) exist, let = ∆k , where ∈ { , , , , , , , } and ∆k ∈ {Ak , Bk , Dk , ak , bk , dk , Qk , qk }. Noting that 0 < ξ1 < 1/l, one has ξ1 → 0 as l → ∞. In the six sequences of (4.21), by taking l → ∞, it follows from (4.21) that (l)

Ak =

p

µ + (α + β + γ )kq

,

(l)

Bk =

α kQAk , µ+η

(l)

Dk =

β kQAk + ηBk δ+µ

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p

(l)

ak =

µ + (α + β + γ )kQ

(l)

,

bk =

α kqak , µ+η

(l)

dk =

β kqak + ηbk , δ+µ

(4.23)

where q=

n 1 ∑

⟨k⟩

iP(i)di ,

Q =

i=1

n 1 ∑

⟨k⟩

iP(i)Di ,

i=1

further,

((µ + η)β + αη) pkQ , (δ + µ)(µ + η) (µ + (β + γ + α )kq)

(l)

Dk =

((µ + η)β + αη) pkq

(l)

dk =

(δ + µ)(µ + η) (µ + (β + γ + α )kQ )

.

(4.24)

Substituting (4.24) into q and Q , respectively, one has n (µ + η)β + αη p ∑

( 1=

)

⟨k⟩(δ + µ)(µ + η)

i=1

n (µ + η)β + αη p ∑

( 1=

)

⟨k⟩(δ + µ)(µ + η)

µ + (β + γ + α )iQ )( ), µ + (β + γ + α )iq µ + (β + γ + α )iQ

(4.25)

µ + (β + γ + α )iq )( ). µ + (β + γ + α )iQ µ + (β + γ + α )iq

(4.26)

i2 P(i) (

i2 P(i) (

i=1

By subtracting (4.25) and (4.26), it arrives at

) n ∑ ( )( ) i3 P(i). (4.27) ⟨k⟩(δ + µ)(µ + η) µ + (β + γ + α )iQ µ + (β + γ + α )iq i=1 ∑n 3 It is obviously that q = Q , so ⟨1k⟩ i=1 i P(i)(Di − di ) = 0, which sees that Di = di , for i = 1, 2, . . . , n. From (4.21) and (4.22), (µ + η)β + αη p(β + γ + α )(Q − q)

(

0=

it follows that lim Ik (t) = Ak = ak ,

lim Ck (t) = Bk = bk ,

t →∞

t →∞

lim Sk (t) = Dk = dk .

t →∞

Finally, substituting q = Q into (4.22), in view of (3.2) and (4.23), it obtains Ik = Ik∗ , Ck = Ck∗ , and Sk = Sk∗ . The proof is completed. 5. The analysis of ICST model with weight Consider that in real contact in online social information spreading, there are some nodes with a high density of edges or lower density of link between nodes in networks. The varieties of linking within a contact networks can be described by link weights, which can represent the intimacy between individuals. People are always first to tell his family or friends when something unexpected or good news happens. The large weight between two nodes means they have close connection, it is easier for susceptible individuals to be infected through the edge [29]. However, with the spreading of various public opinion and false information, some individuals tend to be more rational about the information in the internet and make some reflects to reduce the intimacy such as ignoring the information, blocking or adding it to the blacklist. The weight of link can be changed by such behaviors, which can be described as an adapt weight. Based on the above observations and model (2.1). We present the modified model with weight. First, the contact transmission rate between ignoramuses and sharers can be changed as follows:

Θ (t) =

∑ i

λik

ϕ (i) i

P(i⏐k)Si (t),

⏐

(5.1)

λik denotes the transmission rate from nodes with degree i to nodes with degree k, ϕ (i) is the sharer with degree i. 1/i represents the probability that one of the infected neighbors of a node, with degree i, will contact this node at present time step. P(i|k) represents the probability that a node of degree k is connected to a node of degree i. In this paper, we focuses on scale-free networks. Hence, the condition probability satisfies P(i|k) = iP(i)/⟨k⟩. Weight in networks are usually described by a function of their degree [30–33], i.e., ωij = ω0 (ij)m , where the basic parameter ω0 and m depend on the particular networks. Here, we use ωij = ω0 l(i)l(j) to present the edge weight function between two nodes and assume that l(k) is an increasing function of k due to the nodes with more connections will be more powerful and gain more weight. For convenience, it is assumed that ωij ∈ [0, 1], ∀i, j ∈ (1, 2, . . . , n). When ωij =0, it means that the edge between them is broken. The initial weights in the whole networks are assumed to be either 0 or 1 [34]. From the weight between two connected edge ωij , a node

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with degree k also∑ can be measured by weight, which can be obtained by summing the weights of the links that connected to it, thus, Nk = k i P(i|k)ω(i, k). On scale-free networks, we can obtain Nk : Nk = ω0 kiP(i)li lk /⟨k⟩ = ω0 ⟨ili ⟩klk /⟨k⟩. Here, we assume that the node with degree k has a fixed total a transmission rate which is given by λk, and the transmission rate on the edge from the i-degree node to the k-degree node will be redistributed by the proportion of the k-degree node’s strength that the edge’s weight accounts for [34]. So, λik can be defined as follows:

λik = λk

ωik Nk

=

λlk ⟨k⟩ . ⟨klk ⟩

(5.2)

Furthermore, if we consider individual behavior in information spreading, the value of a weight function will be less as the information spreading progresses. Thus, the weight evolution can be expressed as l′ (k) = l(k) · exp (−f (k)S(t)), where f (k) is a increasing functions of k. The corresponding λik becomes

( ) λ⟨k⟩l(k) · exp −f (k) · S(t) ( ) . λik = λk = Nk ⟨k · l(k) · exp −f (k) · S(t) ⟩ ωik

(5.3)

Substituting (5.2) and (5.3) into (5.1),

Θ 1 (t) =

λlk ∑ ϕ (i)P(i)Si (t), ⟨klk ⟩

(5.4)

i

′

Θ 1 (t) =

λl(k) · exp (−f (k) · S(t)) ∑ ϕ (i)P(i)Si (t), ⟨kl(k) · exp (−f (k) · S(t))⟩

(5.5)

i

when f (k) = 0, (5.5) reduce to (5.4). Substituting (5.4) into (2.1), we obtain the fixed weight system

⎧ λklk Ik (t) dIk (t) ⎪ ⎪ = p − µIk (t) − (α + β + γ )θ (t), ⎪ ⎪ ⎪ dt ⟨klk ⟩ ⎪ ⎪ ⎪ ⎪ ⎪ αλklk Ik (t) dC (t) ⎪ ⎪ k = θ (t) − (η + µ)Ck (t), ⎨ dt ⟨klk ⟩ ⎪ dSk (t) βλklk Ik (t) ⎪ ⎪ ⎪ = θ (t) + ηCk (t) − (δ + µ)Sk (t), ⎪ ⎪ dt ⟨klk ⟩ ⎪ ⎪ ⎪ ⎪ ⎪ γ λklk Ik (t) ⎪ dTk (t) ⎩ = δ Sk (t) + θ (t) − µTk (t). dt ⟨klk ⟩

(5.6)

Substituting (5.5) into (2.1), we obtain the adaptive weight system

⎧ λl(k)k · exp (−f (k) · S(t)) dIk (t) ⎪ ⎪ = p − µIk (t) − (α + β + γ )θ (t), ⎪ ⎪ ⎪ dt ⟨kl(k) · exp (−f (k) · S(t))⟩ ⎪ ⎪ ⎪ ⎪ ⎪ dCk (t) λl(k)k · exp (−f (k) · S(t)) ⎪ ⎪ = θ (t) − (η + µ)Ck (t), ⎨ dt ⟨kl(k) · exp (−f (k) · S(t))⟩ ⎪ dSk (t) λl(k)k · exp (−f (k) · S(t)) ⎪ ⎪ ⎪ = θ (t) + ηCk (t) − (δ + µ)Sk (t), ⎪ ⎪ dt ⟨ kl(k) · exp (−f (k) · S(t))⟩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dTk (t) = δ Sk (t) + λl(k)k · exp (−f (k) · S(t)) θ (t) − µTk (t), dt ⟨kl(k) · exp (−f (k) · S(t))⟩ ∑ where θ (t) = ϕ (k)P(k)Sk (t). Theorem 5. Consider the system (5.6) and (5.7). Define the base reproduction number R1 = statements hold: (1) There always exists an information-elimination equilibrium E0′ (1, 0, 0, 0). ′ ′ ′ ′ ′ (2) There is a unique information-prevailing equilibrium E ∗ (Ik∗ , Ck∗ , Sk∗ , Tk∗ ) if R1 > 1.

(5.7)

λ(β (η+µ)+ηα )⟨kϕ (k)lk ⟩ , then the following ⟨klk ⟩(δ+µ)(η+µ)

Proof. We can easily find that E0′ is always an equilibrium of system (5.6) , which is named the information-elimination ′ ′ ′ ′ ′ equilibrium. In order to obtain the equilibrium solution E ∗ (Ik∗ , Ck∗ , Sk∗ , Tk∗ ), we let the right side of the system (5.6) to be

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equal to zero. Thus,

⎧ ′ λklk Ik∗ (t) ′ ⎪ ′ ⎪ ⎪ (α + β + γ )θ ∗ (t) = 0, p − µIk∗ (t) − ⎪ ⎪ ⟨ kl ⟩ k ⎪ ⎪ ⎪ ′ ⎪ ⎪ αλklk Ik∗ (t) ∗′ ′ ⎪ ⎪ θ (t) − (η + µ)Ck∗ (t) = 0, ⎨ ⟨klk ⟩ ′ ⎪ βλklk Ik∗ (t) ∗′ ′ ′ ⎪ ⎪ θ (t) + ηCk∗ (t) − (δ + µ)Sk∗ (t) = 0, ⎪ ⎪ ⎪ ⟨klk ⟩ ⎪ ⎪ ′ ⎪ ⎪ γ λklk Ik∗ (t) ∗′ ⎪ ′ ′ ⎪ ⎩δ Sk∗ (t) + θ (t) − µTk∗ (t) = 0. ⟨klk ⟩

(5.8)

A direct calculation yields

⎧ ⎪ (µ + δ )(η + µ)⟨klk ⟩ ′ ⎪ ∗′ ⎪ Ik∗ = ⎪ ′ Sk , ⎪ ⎪ [β (η + µ) + ηα ] λklk θ ∗ ⎪ ⎪ ⎨ α (µ + δ ) ′ ′ Ck∗ = Sk∗ , ⎪ [ ] β ( η + µ ) + ηα ⎪ ⎪ ⎪ ⎪ ⎪ δ (β (η + µ) + ηα ) + γ (µ + δ )(η + µ) ∗′ ⎪ ′ ⎪ ⎩Tk∗ = Sk , µ [β (η + µ) + ηα ]

(5.9)

where ′

λklk θ ∗ µ(β (η + µ) + ηα ) . ′ ′ (µ + δ )((β (η + µ) + ηα )λklk θ ∗ + (αµ + γ (η + µ))λklk θ ∗ + ⟨klk ⟩µ(η + µ))

′

Sk∗ =

Inserting into (5.5), we obtain that n

′

λlk ∑ λklk θ ∗ µ(β (η + µ) + ηα ) ). ( ϕ (k)p(k) ′ ′ ⟨klk ⟩ (µ + δ ) (β (η + µ) + ηα) λklk θ ∗ + (αµ + γ (η + µ)) λklk θ ∗ + µ(η + µ)⟨klk ⟩ i=1

′

Θ∗ = ′

′

′

Let Θ ∗ ≜ f (Θ ∗ ), Clearly, Θ ∗ = 0 is a solution of equation. To ensure the equation has a nontrivial solution, the following condition must should satisfied ′

df (Θ ∗ ) ⏐ ′ dΘ ∗

⏐ ⏐

′

Θ ∗ =0

> 1 and f (1) ≤ 1.

We can obtain the reproductive number R1 =

λ(β (η + µ) + ηα )⟨kϕ (k)lk ⟩ . ⟨klk ⟩(δ + µ)(η + µ)

(5.10)

By computing model (5.7), the reproductive number R1 is also given by (5.10), which implies that the adaptive weight cannot change the reproductive number. 6. Simulation results and analyses First, we perform some sensitivity analysis of the basic reproduction number R0 in terms of the model parameters on scale-free networks. Obviously,

∂ R0 ⟨k2 ⟩ = , ∂β (δ + µ)⟨k⟩

∂ R0 η⟨k2 ⟩ = , ∂α (δ + µ)(η + µ)⟨k⟩

∂ R0 (β (η + µ) + ηα )⟨k2 ⟩ =− , ∂δ (δ + µ)2 (η + µ)⟨k⟩

∂ R0 αµ⟨k2 ⟩ = , ∂η (δ + µ)(η + µ)2 ⟨k⟩

∂ R0 (β (η + µ)2 + (α + η + 2µ)ηα )⟨k2 ⟩ =− . ∂µ [(δ + µ)(η + µ)]2 ⟨k⟩

It can be found some interesting results, which manifest from Fig. 2. In Fig. 2 (a), the parameters are chosen as η = 0.16, µ = 0.2, δ = 0.5. We can see that larger α or β can leads to larger R0 , β works more effectively, that is to say, a higher share rate or comment rate makes the information easier spread. In Fig. 2(b), the parameters are chosen as δ = 0.5, µ = 0.2, α = 0.05, it shows larger η or β can leads to larger R0 , in other word, people’s comments on information can influence whether people believe in and spread information. In Fig. 2(c), the parameters are chosen as η = 0.1, µ = 0.2, δ = 0.2.R0 increase as δ decrease or β increase, and β has a great influence on R0 . So, lower stifler rate makes information easier to spread. Fig. 2(d) shows the effect of changing the number of users on R0 . Variance of degree distribution ⟨k2 ⟩ manifests the diversity in contact patterns. Particularly, the ratio ⟨k2 ⟩/⟨k2 ⟩ is the parameter defining the level of heterogeneity of the networks [35]. So, it is clear that the heterogeneity favors information spreading via enlarging the basic reproductive number.

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Fig. 2. The relationship between the basic reproduction number R0 and the parameters in scale-free networks.

Fig. 3. Each compartment population changes over time when R0 < 1 (a) and R0 > 1 (b).

Next, numerical simulation are presented to verify our results of the model (2.1), we take the scale-free networks with degree distribution is P(k) = ξ k−τ (2 < τ ≤ 3) and the constant ξ satisfies

τ = 3 and n = 1000.

∑n

k=1

P(k) = 1. Consider system (2.1), we choose

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Fig. 4. Dynamics behavior of sharer with different degree when R0 < 1 (a) and R0 > 1 (b)

Fig. 5. The prevalence S100 versus t corresponding to different sharing rate in (a) and effective comment rate in (b).

Fig. 6. Dynamics behavior of sharer with different degree in fixed weight when R1 < 1 (a) and R1 > 1 (b).

In Fig. 3(a), we choose β = 0.1, η = 0.2, δ = 0.5, α = 0.1, µ = 0.2, γ = 0.8, thus the basic reproduction number R0 = 0.98. The figure shows that when R0 < 1, the sharers will ultimately disappear, which means that the online social information propagation will fade out. In Fig. 3(b). The parameters are chosen as η = 0.3, δ = 0.45, α = 0.4, µ = 0.2,

X. Liu et al. / Physica A 514 (2019) 497–510

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Fig. 7. The prevalence S100 versus t corresponding to different weight when R1 < 1 (a) and R1 > 1 (b).

γ = 0.25, β = 0.35, thus the basic reproduction number R0 = 4.1. We can see that when R0 > 1, the online social information spreading is persist and the number of sharers will maintain at a positive constant. Fig. 4 shows the dynamics behavior of sharers with different degree when the reproduction number R0 < 1 and R0 > 1. We find that the larger degree leads to larger value of the online social information spreading level. In Fig. 5(a), the prevalence S100 versus t corresponding to different sharing rate β , which are chosen 0.8, 0.5,0.3, 0.1 from bottom to top. This figure indicates that a larger β can increase the online social information spreading. In Fig. 5(b), the prevalence S100 versus t corresponding to different effective comment rate c, which are 0.54, 0.35, 0.1,0.01 from bottom to top. This figure shows a larger c can increase the online social information spreading. It means that the positive share and comment contribute to online social information spreading. Fig. 6 shows the dynamics behavior of online social information spreading with different degree in weight networks. Let ϕ (k) = kx , l(k) = km , where x and m are positive constants. In Fig. 6(a), we choose x = 0.8, α = 0.35, µ = 0.2, γ = 0.35, η = 0.4, β = 0.3, δ = 0.3, m = 1, thus the basic reproduction numbers R1 = 0.44. In Fig. 6(b), we choose η = 0.4, µ = 0.2, γ = 0.2, m = 1, β = 0.45, δ = 0.1, α = 0.35, x = 0.8, λ = 0.2, thus the basic reproduction number R1 = 1.33. The figure shows that when R1 < 1,the sharer will decrease gradually and disappear, when R1 > 1, the number of sharers gradually reach a high stable level. Fig. 7 describes the density of S100 in initial model, fixed weight and adaptive weight. In Fig. 7(a), we choose the same parameters f (k) = k1.2 , m = 1, x = 0.8, η = 0.2, µ = 0.2, γ = 0.55, β = 0.2, δ = 0.1, α = 0.25 when R < 1.This figure shows that weight can weaken the level of information spreading, especially with adaptive weight. Similarly, when R > 1, we choose η = 0.3, µ = 0.2, γ = 0.25,β = 0.45, δ = 0.15, α = 0.3 in Fig. 7(b). It is observed that the sharing behavior rapidly increase first and experience a drop then reach a positive constant. Due to the ability of people’s self-discrimination and self-protection, the spreading intensity will be weaker with adaptive weight, it more close to the actual situation. 7. Conclusions In this paper, a ICST online social information spreading model on scale-free networks has been presented. By mean-field theory, we obtain the basic reproduction number R0 and equilibriums. If R0 < 1, the information-elimination equilibrium is globally stability, which means the spread of information will eventually disappear regardless of the initial values of the shares individuals. If R0 > 1,the information spreading is permanent and information-prevailing equilibrium globally asymptotically stable, which means the information spreading will persist and converge to a positive level. Besides, the effect of sharing rate and effective comment rate have been analyzed in details. Moreover, we study the model with weight and analyze corresponding dynamics behavior, we get the conclusion that the adaptive weight cannot change the spreading threshold, but it can weaken information spreading level. The study has valuable guiding significance in effectively managing and controlling online social information spreading on scale-free networks. Acknowledgments This work is supported by the National Natural Science Foundation of China under Grant 61672112, 61873287 and Project in Hubei Provincial Department of Education, China under Grant B2016036. References [1] J.Y. Wei, C.D. Liu, S.Y. Park, et al., Network control and management for the next generation internet, IEICE Trans. Commun. 83 (10) (2000) 2191–2209. [2] G. Miritello, E. Moro, R. Lara, Dynamical strength of social ties in information spreading, Phys. Rev. E 83 (2011) 045102.

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