Modeling and simulation of information dissemination model considering user’s awareness behavior in mobile social networks

Physica A 537 (2020) 122639

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

Modeling and simulation of information dissemination model considering user’s awareness behavior in mobile social networks ∗

Chun-Yan Sang a , , Shi-Gen Liao b a b

School of Software Engineering, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China College of Mechanical Engineering, Chongqing University, Chongqing, 400030, China

article

info

Article history: Received 8 May 2019 Available online 14 September 2019 Keywords: Information propagation Mobile environment Dynamic system Stability analysis Equilibrium

a b s t r a c t With the rapid development of mobile sensing devices and Internet technology, any user can publish, comment, or forward information through social platforms. Meanwhile, while speeding up the speed and scope of information dissemination, it is difficult to distinguish the authenticity and reliability of the information. Users’ ability to discriminate information depends on their conscious behavior and knowledge level. According to the level of user awareness, the population can be split into five different states when information is generated. In order to know the evolution trend of information over time, a novel information dissemination model in mobile social networks based on the traditional SIR model is proposed in this work, which can be called SEIRD. From the theoretical point of view, the basic reproduction number R0 and equilibrium of the proposed model are given. We also analyze the local and global stability of the rumor-free equilibrium. Furthermore, to verify the effectiveness of the proposed model, a series of numerical simulations experiments and results analysis are given. Moreover, the impacts of different parameters of the proposed model are carried out. © 2019 Elsevier B.V. All rights reserved.

1. Introduction With the development of mobile networks and applications, the mobile Internet has greatly promoted the speed and scope of information propagation in Online Social Networks (OSNs) [1], which enable people to express their opinions, share their experiences, comment, and forward information, et al. However, whether the information is true or false, it can spread by posting on the wall or directly sending message to social neighbors in OSNs [2]. The dynamics of information dissemination in OSNs has attracted much attention and became a hot topic in complex networks and social science [3,4]. Diffusion models are utilized to explain and reproduce the spread of information in the network. Researchers proposed some dynamics models from different perspectives to explore the process of information on social platforms [3,5–8]. Epidemic models are the way infectious diseases are spread among the population, which can be divided into four basic types as SI, SIS, SIR, SIRS [9–11]. To take an epidemic modeling approach to the study of viral videos, Rahil Sachak-Patwa et al. proposed SEIRS DDE epidemic model for video popularity, which incorporates time-delay to accurately describe the virtual contact process between individuals and the temporary immunity of individuals to videos [8]. A dynamical model by taking into account the impact of social media applications and the rumor-refuting actions of rumor-deniers was ∗ Corresponding author. E-mail address: [email protected] (C.-Y. Sang). https://doi.org/10.1016/j.physa.2019.122639 0378-4371/© 2019 Elsevier B.V. All rights reserved.

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C.-Y. Sang and S.-G. Liao / Physica A 537 (2020) 122639

proposed [4]. User’s interest also plays an important role in the process of information diffusion, Forum-LDA was proposed to learn more coherent topics and serious interests and identify unserious users who publish many irrelevant posts [12]. Furthermore, an authoritative information propagation model which considered the super-spreading mechanism was proposed [13]. The impacts of perceived message features and network characteristics on size and structural virility of information diffusion on Twitter was proposed [14]. In mobile Internet, the dynamics of information propagation is influenced by the characteristics of individuals and temporal [15]. To explore the effect of awareness, a dynamic system of SIR with Unaware–Aware process is developed by using the compartment model through an analytical approach with the assumption of an infinite and well-mixed population [5]. In the process of awareness diffusion, the unaware individuals will be aware of the epidemics if the ratio between their awareness neighbors and their degrees reaches the specified ratio [16]. The information awareness about the contagious disease has an influential effect on an individual’s decision to suppress the diffusion of infections. Ariful Kabir et al. proposed a mathematical framework for a vaccination game combined with SIR and Unaware–Aware situation [7], which consider that the information spreading effect might be represented the situation of self-protection. Based on the classical SIR model and the awareness spread, a two-layered network model is proposed for modeling the epidemic spreading and its related awareness diffusion among the population [17]. SEIRS-QV model was proposed to consider the impacts of user awareness, network delay and diverse configuration of nodes to study the dynamical behavior of malware propagation [18]. Sometimes, people get the rumor messages posted by different users and forward them to the other individuals or groups, the rumor spreading model is proposed in mobile social networks based on SIR model, which investigate the impact of group propagation on the dynamics of rumor spreading process [3]. Based on the investigated of the topological structure via Facebook network datasets, an information model was proposed for simulating true and fake information diffusion [19]. To explore the effect of the heterogeneous population in social networks, a social contagion model is proposed on multiplex networks with a heterogeneous population [6]. Because anyone can publish information through social platforms, users need to identify the reliability and authenticity of information by themselves. Although most of the previous models contribute to the study of information dissemination from different perspectives, the process of information dissemination on social platforms is complicated with time. The heterogeneity of users widely exists in social networks, for example, each user may have different social relationship, education background, interest preference et al. In fact, all comments and forwards depending on the user’s ability to judge the authenticity of information and conscious behavior. User awareness plays a very important role in the process of information dissemination. Motivated by the above analysis and previous studies [9], we introduce a novel information dissemination model to incorporate the heterogeneous characteristics and user awareness, in which, the population can be divided into five different parts: Susceptible, Exposed, Infected, Recovered, and Defended. The influence of user awareness on different states of information dissemination will be considered. In theory, we derive the basic reproductive number and equilibrium of the proposed model. The stable analysis of equilibrium is also given. The rest of the paper is organized as follows. In Section 2, we give preliminaries and model information propagation model in heterogeneous networks. In Section 3, we analyze the dynamics of the SEIRD model and give the stability of the rumor-free equilibrium. In Section 4, we provide a set of numerical simulations to verify the theoretical analysis and study the influence of effective factors and parameters in the SEIRD model. In Section 5, we conclude the whole work in this paper and give some future directions. 2. Modeling for information diffusion in heterogeneous networks 2.1. Problem statement According to the information spreading process in heterogeneous networks, we introduce the information dissemination model in this section, which can be called SEIRD model. When information is generated, it is assumed that users are impossible to immediately diffuse the message in social networks. The state of the user may change based his/her awareness, social relationship, knowledge level et al. We assume that each user is represented by a node in a heterogeneous networks and the total population of social network nodes are divided into five different groups: S-nodes (Susceptible nodes), E-nodes (Exposed nodes), I-nodes (Infected nodes), R-nodes (Recovered nodes) and D-nodes (Defended nodes). S-nodes: users on social platforms are vulnerable when the information is generated. S(t) denotes the quantity of S-nodes at time. E-nodes: users have already received the information, but they hesitate whether to spread it or not. E(t) denotes the quantity of E-nodes at time t. I-nodes: users choose to believe the information and spread it to their friends by social platforms. I(t) denotes the quantity of I-nodes at time t. R-nodes: after reading the information, they choose to deny it and also tell others that it is a rumor. R(t) denotes the quantity of R-nodes at time t. D-nodes: users are proactively controlled and will block the information. D(t) means the quantity of D-nodes at time t. Next, we will present the hypotheses between state transitions and parameters descriptions

C.-Y. Sang and S.-G. Liao / Physica A 537 (2020) 122639

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Fig. 1. The state transition diagram of the SEIRD model.

H1: It is assumed that all nodes are S-nodes first. After reading the information, a user will usually make a choice like believe it, deny it, or spread it. This process may be affected by many factors such as users’ educational level, user activities, sources of information, et al. If a user is well educated and has strong analytical ability, he/she can evaluate the reliability of information more accurately. Then, the user may convert to D-node with probability γ . On the contrary, if the user cannot make a correct judgment on the information, he/she may convert to be E-node or I-node with probabilities

α or β , respectively. H2: When a user hesitates to disseminate information, he is in the exposed state. He or she may be infected by the external environment or social relations to spread the information or not. If most of the friends on the user’s social platform are forwarding the same message, the user may become an I-node with probability δ . Conversely, there are few people pay attention on this information, the user may become a R-node with probability λ. H3: Once the information is spreading widely, the government will take effective measures to restrain the spread of the information. We assume that the users who are vaccinated may convert to R-nodes with probability χ . H4: The coming rate of new users is b. With time passes, the leaving rate of users in five different states is µ. Based on the above hypotheses and analysis, the state transition diagram of the SEIRD model is shown in Fig. 1. Next, we discuss how information can be disseminated in heterogeneous networks. On average, a node of the degree k has k neighbors. p (k) is the degree distribution, and p(k) ∼ k−r , k = 1, . . . ∆. The mean degree ⟨k⟩ is ⟨k⟩ =

∑

k

kp(k).

The infected node can not only infect its susceptible neighbor with the infection probability β , but also infect its exposed neighbor with the infection probability nodes will be able to infect their susceptible neighbors ∑ δ . At time t, newly infected ∑ n

kp(k)I (t)

n

kp(k)I (t)

and exposed neighbors, thus kβ Sk (t) k=m ⟨k⟩ k and kδ Ek (t) k=m ⟨k⟩ k denote the probability that a connection from a node in degree k to an infected node in degree k exists to an infected node in k (m ≤ k ≤ n).

2.2. Model formulation

When an information occurs in the network, the density of the susceptible nodes transmitting to the exposed state will be reduced with information propagation rate. As stated earlier, information propagation rate is formulated by user awareness, users’ educational level, the source of information et al. By further getting immunity through vaccinations, the infected nodes go into state R with rate χ . Furthermore, the recovered nodes can transfer into the defended state with rate η. For convenience, several notations and quantities are introduced as follows: (1) ˜ Sk (t): the number of S-nodes with degree k at time t. (2) ˜ Ek (t): the number of E-nodes with degree k at time t. (3) ˜ Ik (t): the number of I-nodes with degree k at time t. (4) ˜ Rk (t): the number of R-nodes with degree k at time t. (5) ˜ Dk (t): the number of D-nodes with degree k at time t.

(6) Sk (t): the relative density of S-nodes with degree k at time t, Sk (t) = ˜ Sk (t)/Nk (t)

(7) Ek (t): the relative density of E-nodes with degree k at time t, Ek (t) = ˜ Ek (t)/Nk (t). (8) Ik (t): the relative density of I-nodes with degree k at time t, Ik (t) = ˜ Ik (t)/Nk (t).

(9) Rk (t): the relative density of R-nodes with degree k at time t, Rk (t) = ˜ Rk (t)/Nk (t).

(10) Dk (t): the relative density of D-nodes with degree k at time t, Dk (t) = ˜ Dk (t)/Nk (t).

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C.-Y. Sang and S.-G. Liao / Physica A 537 (2020) 122639

Based on the above considerations, information diffusion leads to dynamic transitions among different states, shown in Fig. 1. We also denote N(t) by the number of all nodes at time t. The dynamic differential model is shown in Eq. (1)

∑n ⎧ dSk (t) k=m kp (k) Ik (t) ⎪ ⎪ − γ Sk (t) − µSk (t) = b + ϕ R (t) − α S (t) − k β S (t) k k k ⎪ ⎪ ⟨k⟩ dt ⎪ ⎪ ∑ ⎪ n ⎪ kp (k) Ik (t) dEk (t) ⎪ ⎪ ⎪ − λEk (t) − µEk (t) = α Sk (t) − kδ Ek (t) k=m ⎪ ⎪ ⟨k⟩ dt ⎪ ⎨ ∑n ∑n dIk (t) k=m kp (k) Ik (t) k=m kp (k) Ik (t) = k β S (t) + k δ E (t) − χ Ik (t) − µIk (t) k k ⎪ ⎪ ⟨k ⟩ ⟨k⟩ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dRk (t) = λEk + χ Ik (t) − ϕ Rk (t) − ηRk (t) − µRk (t) ⎪ ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎩ dDk (t) = γ S (t) + ηR (t) − µD (t) k k k

(1)

dt

The initial conditions for SEIRD model of Eq. (1) are as follows: 0 < Sk ≤ 1, 0 ≤ Ek < 1, 0 ≤ Ik < 1, 0 ≤ Rk < 1, and 0 ≤ Dk < 1 for any m ≤ k ≤ n. Summing up the five equations of SEIRD model of Eq. (1), we can get Eq. (2): dNk (t) dt

= b − µNk (t)

(2)

Since Sk (t) + Ek (t) + Ik (t) + Rk (t) + Dk (t) = 1, then the fifth equation in the system (1) can be ignored and system (1) be reduced as follows:

⎧ ∑n kp (k) Ik (t) dSk (t) ⎪ ⎪ ⎪ = b + ϕ Rk (t) − α Sk (t) − kβ Sk (t) k=m − γ Sk (t) − µSk (t) ⎪ ⎪ ⟨k⟩ dt ⎪ ⎪ ∑n ⎪ ⎪ kp (k) Ik (t) dEk (t) ⎪ ⎪ ⎨ = α Sk (t) − kδ Ek (t) k=m − λEk (t) − µEk (t) ⟨k⟩ dt (3) ∑n ∑n ⎪ dIk (t) ⎪ k=m kp (k) Ik (t) k=m kp (k) Ik (t) ⎪ ⎪ = k β S (t) + k δ E (t) − χ I (t) − µ I (t) k k k k ⎪ ⎪ ⟨k ⟩ ⟨k⟩ dt ⎪ ⎪ ⎪ ⎪ dRk (t) ⎪ ⎩ = λEk + χ Ik (t) − ϕ Rk (t) − ηRk (t) − µRk (t) dt { } The feasible region of system (3) is Ω = (Sk , Ek , Ik , Rk ) ∈ Γ+4 : 0 < Sk + Ek + Ik + Rk ≤ 1, k = m, . . . , n , which can be confirmed as a positively invariant set for the system (3), and it is sufficient to study the dynamics of the system (1) in Ω. For the SEIRD model of Eq. (3), R0 is the asymptotic per generation growth factor of the infected state and it determines the global dynamics of the SEIRD model. According to the concept of the next-generation matrix [20,21], the basic reproductive number R0 is proportional to R0 = ρ (FV −1 ), where ρ (G) is the spectral radius of the matrix G. Then, R0 can be obtained as Eq. (4):

∑n

k=m

R0 =

kp (k)

bk (ϕ + η + µ) [β (λ + µ) + αδ ] [(α + γ + µ) (λ + µ) (ϕ + η + µ) − αϕλ] (χ + µ)

⟨k⟩

(4)

3. Dynamical analysis of the information spreading model In this section, the equilibrium and dynamical behavior of the proposed model in this work are given in this section. 3.1. Basic reproduction number and equilibrium Lemma 1. Consider system (3). The following assertions hold: (1) There always exists a rumor-free equilibrium E 0 = (Sk0 , Ek0 , Ik0 , R0k ), where b (λ + µ) (ϕ + η + µ)

Sk0 = Ek0

(α + γ + µ) (λ + µ) (ϕ + η + µ) − αϕλ bα (ϕ + η + µ) = , (α + γ + µ) (λ + µ) (ϕ + η + µ) − αϕλ

Ik0 = 0, and R0k =

bαλ . (α + γ + µ) (λ + µ) (ϕ + η + µ) − αϕλ

C.-Y. Sang and S.-G. Liao / Physica A 537 (2020) 122639

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(2) If R0 > 1, there exists a unique endemic equilibrium E ∗ = (S ∗ , E ∗ , I ∗ , R∗ ) of the system (3), where

] [ (kδ A∗ + λ + µ) b (ϕ + η + µ) + ϕχ Ik∗ (t) S = (kβ A∗ + α + γ + µ) (kδ A∗ + λ + µ) (ϕ + η + µ) − ϕλα [ ] λ bα (ϕ + η + µ) + ϕχα Ik∗ (t) + χ Ik∗ (t) (kβ A∗ + α + γ + µ) (kδ A∗ + λ + µ) (ϕ + η + µ) − ϕλα ∗ R = ϕ+η+µ b α + η + µ) + ϕχα Ik∗ (t) (ϕ E∗ = , and (kβ A∗ + α + γ + µ) (kδ A∗ + λ + µ) (ϕ + η + µ) − ϕλα A∗ ck b (ϕ + η + µ) [β (kδ A∗ + λ + µ) + αδ ] I∗ = , (χ + µ) [(kβ A∗ + α + γ + µ) (kδ A∗ + λ + µ) (ϕ + η + µ) − ϕλα ] − A∗ ck [β (kδ A∗ + λ + µ) + αδ ] ϕχ ∗

where A∗ is the unique positive zero of the following function: kp (k) kb (ϕ + η + µ) [β (kδ x + λ + µ) + αδ ]

(

∑n

)

(χ + µ) [(kβ x + α + γ + µ) (kδ x + λ + µ) (ϕ + η + µ) − ϕλα ] − xk [β (kδ x + λ + µ) + αδ ] ϕχ ⟨k⟩

k=m

f (x) =

Proof. For finding equilibrium points, let

dSk dt

= 0,

dEk (t) dt

= 0,

dIk (t) dt

dRk (t) dt

= 0, then we have ∑n ⎧ kp (k) Ik (t) ⎪ ⎪ − γ Sk (t) − µSk (t) = 0 ⎪b + ϕ Rk (t) − α Sk (t) − kβ Sk (t) k=m ⎪ ⎪ ⟨k⟩ ⎪ ∑ ⎪ n ⎪ ⎪ kp (k) Ik (t) ⎨ α Sk (t) − kδ Ek (t) k=m − λEk (t) − µEk (t) = 0 ⟨k⟩ ∑ ∑n ⎪ n ⎪ ⎪ ⎪ ⎪kβ Sk (t) k=m kp (k) Ik (t) + kδ Ek (t) k=m kp (k) Ik (t) − χ Ik (t) − µIk (t) = 0 ⎪ ⎪ ⎪ ⟨k⟩ ⟨k⟩ ⎪ ⎩ λEk + χ Ik (t) − ϕ Rk (t) − ηRk (t) − µRk (t) = 0

By letting Ik = 0, k = 1, . . . , ∆, then we can easily get Sk = Rk = E0.

bαλ

(α+γ +µ)(λ+µ)(ϕ+η+µ)−αϕλ

= 0,

− 1 (5)

b(λ+µ)(ϕ+η+µ) , (α+γ +µ)(λ+µ)(ϕ+η+µ)−αϕλ

Ek =

bα(ϕ+η+µ) , (α+γ +µ)(λ+µ)(ϕ+η+µ)−αϕλ

(6)

and

for each k = 1, . . . , ∆. Therefore, system (3) always has a unique rumor-free equilibrium

⎛ k=1,...∆ ⎞ k=1,...∆ k=1,...∆ k=1,...∆             Next, let E ∗ = ⎝S1∗ , . . . , Sk∗ , E1∗ , . . . , Ek∗ , I1∗ , . . . , Ik∗ , R∗1 , . . . , R∗k ⎠ represent an endemic equilibrium of system (3). Then,

it follows by the system (3), we can obtain Eq. (7):

⎧ b + ϕ R∗k (t) − α Sk∗ (t) − kβ Sk∗ (t)A∗ − γ Sk∗ (t) − µSk∗ (t) = 0 ⎪ ⎪ ⎨α S ∗ (t) − kδ E ∗ (t)A∗ − λE ∗ (t) − µE ∗ (t) = 0 k

k

k

k

⎪ kβ S ∗ (t)A∗ + kδ Ek∗ (t)A∗ − χ Ik∗ (t) − µIk∗ (t) = 0 ⎪ ⎩ ∗k λEk + χ Ik∗ (t) − ϕ R∗k (t) − ηR∗k (t) − µR∗k (t) = 0 where A∗ =

(7)

∑n

∗ k=m kp(k)Ik

⟨k⟩

. From the above analysis, we can get

[ ] (kδ A∗ + λ + µ) b (ϕ + η + µ) + ϕχ Ik∗ (t) S = , (kβ A∗ + α + γ + µ) (kδ A∗ + λ + µ) (ϕ + η + µ) − ϕλα [ ] λ bα (ϕ + η + µ) + ϕχα Ik∗,c (t) ∗

+ χ Ik∗ (t) (kβ A∗ + α + γ + µ) (kδ A∗ + λ + µ) (ϕ + η + µ) − ϕλα R = , ϕ+η+µ bα (ϕ + η + µ) + ϕχα Ik∗ (t) E∗ = ∗ (kβ A + α + γ + µ) (kδ A∗ + λ + µ) (ϕ + η + µ) − ϕλα A∗ kb (ϕ + η + µ) [β (kδ A∗ + λ + µ) + αδ ] I∗ = , (χ + µ) [(kβ A∗ + α + γ + µ) (kδ A∗ + λ + µ) (ϕ + η + µ) − ϕλα ] − A∗ k [β (kδ A∗ + λ + µ) + αδ ] ϕχ ∗

Substituting I ∗ into the expression of A∗ , we can get

∑n

k=m

A∗ =

kp (k) A∗ kb (ϕ + η + µ) [β (kδ A∗ + λ + µ) + αδ ]

(χ +

µ) [(kβ A∗

+ α + γ + µ) (kδ A∗ + λ + µ) (ϕ + η + µ) − ϕλα ] − A∗ k [β (kδ A∗ + λ + µ) + αδ ] ϕχ ⟨k ⟩

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C.-Y. Sang and S.-G. Liao / Physica A 537 (2020) 122639

Thus, A∗ is a positive root of the equation f (x) = 0 shown in. It is obvious that the function f (x) is monotonically decreasing in the interval [0, ∞), i.e. for all x ≥ 0, we can have f ′ (x) < 0 [22]. Moreover,

∑n

kp (k) kb (ϕ + η + µ) [β (λ + µ) + αδ ]

(

)

(χ + µ) [(α + γ + µ) (λ + µ) (ϕ + η + µ) − ϕλα ] ⟨k⟩

k=m

f (0) =

− 1 = R0 − 1 and f (+∞) = −1

f (+∞) = −1. Thus, if R0 > 1, f (x) = 0 has a unique positive root. The proof is complete. □ 3.2. Global and local stability analysis of E0 equilibrium In this section, the stability of model (3) will be discussed, which include the local and global stability of the rumor-free equilibrium. It is obvious that model (3) always possesses a unique rumor-free equilibrium E 0 . Theorem 1. If R0 < 1, then the rumor-free equilibrium E 0 is locally asymptotically stable. Proof. The corresponding Jacobin matrix at the rumor-free equilibrium E 0 is

⎡ ⎢−α − γ − µ ⎢ ⎢ ⎢ α ⎢ 0 J(E ) = ⎢ ⎢ ⎢ ⎢ 0 ⎣

−kβ S0

0

−λ − µ

−kδ E0

∑n 0

k=m

kp (k)

⟨k⟩

∑n

k=m

⎤ 0

⟨k⟩

∑n

k=m

kp (k)

0

⟨k⟩

(kβ S0 + kδ E0 ) − χ − µ

λ

0

kp (k)

0

χ

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(8)

−ϕ − η − µ

Then,

⏐ ⏐ ⏐−α − γ − µ − λ0 ⏐ ⏐ ⏐ ( ) ⏐⏐ α 0 det J − λ = ⏐ ⏐ ⏐ ⏐ 0 ⏐ ⏐ ⏐ 0

−kβ

0

−λ − µ − λ0

−kδ

∑n

k=m

0

kp (k)

⟨k⟩

λ

S0

∑n

E0

∑n

k=m

kp (k)

⟨k⟩ k=m

⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ 0 ⏐ ⏐ ⏐ ⏐ ⏐ 0 ⏐ ⏐ 0⏐ −ϕ − η − µ − λ 0

kp (k)

⟨k⟩

(kβ S0 + kδ E0 ) − χ − µ − λ0 χ

(9)

[ We can get λ01 = −α − γ − µ, λ02 = −λ − µ, λ03 = (χ + µ)

∑n kp(k) bk k=⟨m (ϕ+η+µ)[β (k)(λ+µ)+δ (k)α ] k⟩ [(α+γ +µ)(λ+µ)(ϕ+η+µ)−αϕλ](χ+µ)

] − 1 , λ04 = −ϕ − η − µ < 0.

If R0 < 1, then λ03 < 0. Based on the method from Hurwitz criterion [23], if R0 < 1, the rumor-free equilibrium E 0 of the system (3) is locally asymptotically stable. Next, we will present a theorem to address the global stability of the rumor-free equilibrium of the system (3). Theorem 2. If R0 < 1, then the rumor-free equilibrium E 0 of the model (3) is globally asymptotically stable. Proof. To investigate the global asymptotically stability of E 0 in the model (3), we can construct a Lyapunov function L(t) for E 0 as Eq. (10): L(t) =

1

χ +µ

A∗

(10)

Next, we calculate the time derivative of L(t) along with the equilibrium solution of Eq. (10):

[ ∑n

kp (k)

] L (t) = A k (β Sk (t) + δ Ek (t)) − χ − µ ⟨k⟩ χ +µ [ ∑n ] ) kp (k) ( 0 1 ≤ A∗ k k=m β Sk (t) + δ Ek0 (t) − χ − µ ⟨k⟩ χ +µ [ ∑n ( ) ] ( ) kp k ( ) bk (ϕ + η + µ) [β (λ + µ) + δα ] k=m ≤ A∗ − 1 ≤ A ∗ R0 − 1 ⟨k⟩ (χ + µ) [(α + γ + µ) (λ + µ) (ϕ + η + µ) − αϕλ] ′

1

∗

k=m

C.-Y. Sang and S.-G. Liao / Physica A 537 (2020) 122639

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Fig. 2. The global asymptotically stability of the SEIRD model when R0 < 1 and R0 > 1, respectively.

Fig. 3. The global asymptotically stability of the SEIRD model with k = 1, 2, . . . , 100, respectively.

Thus, based on LaSalle’s invariance principle [24], E 0 is global asymptotically stability if R0 < 1. □ 4. Numerical simulation To illustrate the above theoretical results, a series of numerical simulation examples are given, which are performed by MATLAB. The degree distribution of nodes in the online social networks follows a power-law distribution p(k) = k−v . 4.1. The equilibrium of SEIRD model To validate the equilibrium of the model (3), the parameter settings are shown in Table 1. As shown in Fig. 2, the model (3) has a rumor-free equilibrium point and is globally asymptotically stable when R0 < 1. We also find that when I increase and reach the peak, then descend and reach to 0. As shown in Fig. 2, the model (3) has an endemic equilibrium point when R0 = 4.4884 > 1. Thus, this is consistent with Lemma 1 and Theorem 1 in Section 3. When R0 > 1, the model (3) has a rumor-free equilibrium point and an endemic equilibrium. To validate the influence of k, parameter settings are the same as Example 1, except for k. As shown in Fig. 3, the model (3) is always has a rumor-free equilibrium point and is globally asymptotically stable when k = (1, 2, . . . , 100). This is because R0 < 1 when k varied from 1 to 100. We also find that the greater the degree of individuals, the bigger the impact on information spreading.

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C.-Y. Sang and S.-G. Liao / Physica A 537 (2020) 122639

Table 1 Parameter setting of the SEIRD model. Example 1 Example 2

k

b

ϕ

α

β

γ

µ

δ

λ

χ

η

R0

100 50

0.01 0.05

0.000013 0.00013

0.77 0.5

0.7 0.7

0.0133 0.133

0.01 0.05

0.6 0.7

0.43 0.33

0.25 0.15

0.05 0.15

0.5206 4.4884

Fig. 4. The global asymptotically stability of the SEIRD model with different initial values, respectively.

Fig. 5. The impact of the parameter β varied from 0.3 to 0.9 for the proposed model SEIRD.

To validate the influence of different initial values, the initial conditions are set as C 1 = (0.8, 0.05, 0.1, 0.02, 0.03), C 2 = (0.5, 0.15, 0.1, 0.15, 0.1), and C 3 = (0.4, 0.2, 0.15, 0.1, 0.15), respectively. As shown in Fig. 4, we can find that the stability of E 0 is not influenced by initial conditions. We also find that five different states can converge to the same rumor-free equilibrium point. 4.2. The effects of user awareness In the real world, user behavior plays a key impact of information propagation in online social networks, which is related to restrain strategy such as improving user defense awareness with rate γ , and taking immunity through vaccinations with rate χ , also quarantining nodes with rate λ.

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Fig. 6. The impact of the parameter γ varied from 0.02 to 0.32 for the proposed model SEIRD.

Fig. 7. The impact of the parameter χ varied from 0.25 to 0.4 for the proposed model SEIRD.

To verify the impact of the infected user with probability β , we set the other parameter settings are the same as Example 1, except β . As shown in Fig. 5, we can find that all states are impacted by β . With the increasing of β , the density of I nodes increasing obviously. And they all can converge to the same rumor-free equilibrium point. This is because if R0 < 1, the model (3) just has a rumor-free equilibrium. To discuss the impact of the rate γ , the parameter settings are similar to Example 1 except γ . From Fig. 6, we can find that the rate γ plays an important role for information dissemination. The densities of S, E, I, and R nodes are impacted by γ and have the same evolution trend. In reality, we can increase the treatment costs to restrain information diffusion. Fig. 7 shows us that the simulation results with different values of the parameter χ . We can find that with the increasing of χ , the density of I converges to 0 faster. The density of S is less affected by χ . The densities of E and D converge to rumor-free equilibrium faster and have the same evolution trend. Finally, we also discuss the impact of the parameter λ. As shown in Fig. 8, the density of S is not affected by λ. With the increasing of λ, the densities of E and I converge to equilibrium faster. We also validate the impact of parameters α and δ . As can be seen in Fig. 9, we can find that α plays an important role for each state. With the increasing of α from 0.1 to 0.7, the density of I converges to zero faster. As can be seen in Fig. 10, we can find that with the increasing of δ , the density of I increasing dramatically. Based on the above analysis, we can get different outbreak results by adjusting parameter β , γ , χ , and λ. We can observe the user awareness effect in the density of infected nodes when the rumor outbreak. We also can conclude that we can increase the cost of immune to restrain information diffusion.

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Fig. 8. The impact of parameter λ varied from 0.2 to 0. 8 for the proposed model SEIRD.

Fig. 9. The impact of parameter α varied from 0.1 to 0.7 for the proposed model SEIRD.

5. Conclusion

Motivated by the consideration that rumor spreading through social networks has some differences to the traditional rumor spreading by way of mouth-to-mouth, we attempt to develop a novel mathematical model to depict rumor spreading by incorporating its special characteristics. In this work, we investigate the process of information diffusion and introduce a novel model based on the traditional epidemic model. The basic reproduction number and stability of rumorfree equilibrium are given. To validate the theorem results, a series of numerical simulation experiments are designed with different parameter settings. Based on the theoretical and numerical simulation, we can find that user awareness plays an important role in information dissemination. In reality, we can improve user awareness or improve the cost immune to restrain the rumor spreading. On the contrary, we also can spread positive information by the high degree of nodes or authoritative users in online social networks. In future work, we will further study the impact evaluation methods of users and the potential influencing factors of information spreading in online social networks.

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Fig. 10. The impact of parameter δ varied from 0.1 to 0.7 for the proposed model SEIRD.

Acknowledgments The authors gratefully acknowledge the supports provided for this research by the China Scholarship Council, China (CSC No. 201707845011), the National Natural Science Foundation, China (Grant No. 61672117), and the research project of Chongqing CSTC, China (cstc2019jcyj-msxm1277, cstc2015jcyjA30001). References [1] A. Kuhnle, M.A. Alim, X. Li, et al., Multiplex influence maximization in online social networks with heterogeneous diffusion models, IEEE Trans. Comput. Soc. Syst. (2018). [2] S. Wen, M.S. Haghighi, C. Chen, et al., A sword with two edges: Propagation studies on both positive and negative information in online social networks, IEEE Trans. Comput. 64 (3) (2015) 640–653. [3] E. Sahafizadeh, B.T. Ladani, The impact of group propagation on rumor spreading in mobile social networks, Physica A 506 (2018) 412–423. [4] W.P. Liu, X. Wu, W. Yang, et al., Modeling cyber rumor spreading over mobile social networks: A compartment approach, Appl. Math. Comput. 343 (2019) 214–229. [5] K.M.A. Kabir, K. Kuga, J. Tanimoto, Analysis of SIR epidemic model with information spreading of awareness, Chaos Solitons Fractals 119 (2019) 118–125. [6] S.-S. Zhu, X.-Z. Zhu, J.-Q. Wang, et al., Social contagions on multiplex networks with heterogeneous population, Physica A 516 (2019) 105–113. [7] K.M.A. Kabir, K. Kuga, J. Tanimoto, Effect of information spreading to suppress the disease contagion on the epidemic vaccination game, Chaos Solitons Fractals 119 (2019) 180–187. [8] R. Sachak-Patwa, N.T. Fadai, R.A. Van Gorder, Understanding viral video dynamics through an epidemic modelling approach, Physica A 502 (2018) 416–435. [9] M.A. Khan, Y. Khan, S. Islam, Complex dynamics of an SEIR epidemic model with saturated incidence rate and treatment, Physica A 493 (2018) 210–227. [10] S. Jana, S.K. Nandi, T. Kar, Complex dynamics of an SIR epidemic model with saturated incidence rate and treatment, Acta Biotheor. 64 (1) (2016) 65–84. [11] W.-M. Liu, S.A. Levin, Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol. 23 (2) (1986) 187–204. [12] C. Chen, J. Ren, Forum latent Dirichlet allocation for user interest discovery, Knowl.-Based Syst. 126 (2017) 1–7. [13] Y. Zhang, Y. Su, L. Weigang, et al., Rumor and authoritative information propagation model considering super spreading in complex social networks, Physica A 506 (2018) 395–411. [14] J. Meng, W. Peng, P.-N. Tan, et al., Diffusion size and structural virality: The effects of message and network features on spreading health information on twitter, Comput. Hum. Behav. 89 (2018) 111–120. [15] H. Zhu, J. Ma, Knowledge diffusion in complex networks by considering time-varying information channels, Physica A 494 (2018) 225–235. [16] Z. Wang, Q. Guo, S. Sun, et al., The impact of awareness diffusion on SIR-like epidemics in multiplex networks, Appl. Math. Comput. 349 (2019) 134–147. [17] C. Xia, Z. Wang, C. Zheng, et al., A new coupled disease-awareness spreading model with mass media on multiplex networks, Inf. Sci. 471 (2019) 185–200. [18] S. Hosseini, M.A. Azgomi, The dynamics of an SEIRS-QV malware propagation model in heterogeneous networks, Physica A 512 (2018) 803–817. [19] D. Yang, T.W. Chow, L. Zhong, et al., True and fake information spreading over the Facebook, Physica A 505 (2018) 984–994. [20] J. Heffernan, R. Smith, L. Wahl, Perspectives on the basic reproductive ratio, J. R. Soc. Interface 2 (4) (2005) 281–293. [21] W.P. Liu, S.M. Zhong, A novel dynamic model for web malware spreading over scale-free networks, Physica A 505 (2018) 848–863. [22] W. Liu, S. Zhong, A novel dynamic model for web malware spreading over scale-free networks, Physica A 505 (2018) 848–863. [23] R.C. Robinson, An Introduction to Dynamical Systems: Continuous and Discrete, American Mathematical Soc., 2012. [24] J.P. Lasalle, The Stability of Dynamical Systems, SIAM, 1976.