- Research
- Open Access
Spreading dynamics of a preferential information model with hesitation psychology on scale-free networks
- Xiongding Liu^{1, 2},
- Tao Li^{1, 2}Email author,
- Xinming Cheng^{3},
- Wenjin Liu^{1, 2} and
- Hao Xu^{1, 2}
https://doi.org/10.1186/s13662-019-2221-2
© The Author(s) 2019
- Received: 26 January 2019
- Accepted: 27 June 2019
- Published: 9 July 2019
Abstract
Considering the influence of the preferential degree, the hesitation psychology of customers and the heterogeneity of underlying networks in preferential information spreading, we propose a novel model called AHFB (adherent–hesitator–forwarder–beneficiary) model to illustrate the dynamic behaviors of preferential information spreading on scale-free networks. The mean-field theory is adopted to describe the formulas of AHFB model. To begin with, we analyze the spreading dynamics of preferential information. Then we determine the basic reproductive number and equilibria by the next generation matrix method. The relationship among the basic reproductive number, preferential degree, and hesitation parameter is also analyzed. In addition, the globally asymptotical stability of information-eliminate equilibrium and the permanence of the preferential information spreading are proved in detail. Furthermore, a preferential information competition model is proposed, and the corresponding dynamic behaviors are studied. We have found that under certain conditions, a greater competitive advantage can be achieved by improving a certain range of preferential strengths. Numerical simulations are also presented to verify and extend theoretical results.
Keywords
- Attractivity parameter
- Preferential information spread model
- Heterogeneity
- Scale-free networks
- Hesitation psychology
1 Introduction
With the continuous prosperity of mobile Internet technology and the appearance of the 5G, e-commerce technology is rapidly developing and multifarious applications are emerging. As a kind of Internet networks, e-commerce networks not only play an important role in spreading information and promoting communication, but also penetrate people’s daily life, such as convenient shopping, mobile payment [1–3]. In most instances, e-commerce networks provide businesses with a platform to release preferential information and display merchandises [4, 5]. In order to obtain discounts or small gifts from the store during the shopping process, customers need to forward the corresponding preferential information [6]. Therefore, the preferential information will be spread widely.
In the field of complex networks, researchers have made some achievements in information spreading [7–14] and network control [15–19]. Based on the characteristics of network structure, the effects of different network topologies, such as small world networks, random networks, scale-free networks, community structure networks, and multiplex networks, on information spreading models are studied in detail. The authors of [20] described information spreading by ODE (ordinary differential equation) dynamic system and built a linear algebra model in small world networks. They studied the relationship between the ODE function groups and the graph topology, and also found the rule of the information’s distribution in networks. Lim et al. [21] studied the influence of clustering coefficients under the SIR model with heterogeneous contact rates and provided a novel iterative algorithm to estimate the conditions and sizes of global cascades. In order to study the influence of human subjective value and psychological status on information dissemination, Liu et al. [22] proposed a novel social network information dissemination with negative feedback NFSIR (negative feedback–susceptible–infected–removed) model on scale-free networks, which showed that the intensity of information feedback has a significant impact on the process of information dissemination. The authors of [23, 24] presented a privacy protection and emotional behavior information spreading model with a community structure on social networks. Zhang et al. [25, 26] proposed a node measurement augmented system model in multirate systems networks with dynamic quantization. Recently, scholars have begun to take serious consideration about the role of human behavior and the multiplex networks structure in information spreading [27–29]. The interplay between the epidemic spreading and the diffusion of awareness in multiplex networks was analyzed in [30, 31].
Another generalization of the initial simple determinacy is to focus on different models and spreading mechanisms. The typical information behavior in social networks is a kind of information sharing and interaction. Ally et al. [32] proposed two rewiring SIR (susceptible–infected–removed) models on information spreading in scale-free and small world networks. In order to study the effect of hesitation mechanism, the authors of [33, 34] proposed an SEIR (susceptible–exposed–infected–removed) model by introducing the exposed nodes between the ignorant nodes and the spread nodes in heterogeneity networks, they found that hesitant individuals have a very important influence on the spreading of information. To further investigate the influence of heterogeneity of the underlying complex networks and quarantine strategy, an SIQRS (susceptible–infected–quarantined–recovered–susceptible) epidemic model on the scale-free networks has been proposed [35, 36].
It has been found that most real networks are actually scale-free networks, so the scale-free property is a particularly important one in social networks [37, 38]. To further understand the information spreading dynamics in real world, the scale-free property of social networks has been taken into account by many information spreading models [39–41]. In networks, nodes represent individual and edges represent the relationship of people, as for preferential spreading dynamics, that relationship is contacted to forward the preferential information. With the rapid development of social networks and new media, increasingly many business groups choose social networks to promote their products by publishing preferential information. Hence, it is of great importance to study the spreading dynamics of preferential information on scale-free networks. Due to the influences of the merchandise’s preferential degree, personal economic level, psychological and other factors, some customers will not immediately forward the preferential information after they know it in the real social life, and they will become hesitant at first. However, in the literature on preferential information spreading, some researchers ignore the hesitating mechanism, which will affect people’s attitudes and behaviors on preferential information and further influence its spreading. People in the hesitating state will read some comments or opinions about the products because of the attractiveness of preferential information, which may make one enter the forwarder status to forward the preferential information with a certain probability. An FSFC (follower–super-forwarder–client) preferential information model was proposed by Fu et al. [42] in online social networks, who found that different infection rates have a great impact on large- and small-degree node forwarders. Wan et al. [43] proposed an SIB (susceptible–infected–beneficial) model based on scale-free networks, and they further studied the influence of the preferential degree and the heterogeneity of underlying networks on the spread of preferential e-commerce information. However, the influence of the parameters in the model on the basic reproductive number has not been analyzed in detail, and the hesitation psychology of people is also not considered. Motivated by the above, we establish a novel AHFB preferential information spreading model with hesitation mechanism on scale-free networks.
The remainder of this paper is organized as follows. Section 2 presents an AHFB preferential information spreading model. In Sect. 3, the basic reproductive number and equilibriums are obtained. Section 4 analyzes the globally asymptotic stability of equilibriums and the permanence of preferential information spreading. In Sect. 5, the modified preferential information spreading model with competitive mechanism is introduced and the corresponding dynamical behaviors are studied. In Sect. 6, numerical simulations are presented to illustrate our main results. Finally, we give the discussions and conclusions in Sect. 7.
2 Model formulation
3 The basic reproductive number and existence of equilibriums
3.1 The basic reproductive number \(R_{0}\)
3.2 Existence of equilibriums
Theorem 1
Consider system (2.1) and define \(R_{0} = \frac{ \langle \varphi (k)k \rangle }{ \langle k \rangle } \frac{(\alpha (\mu + \beta ) + \delta \beta )(\mu \rho _{1} + \gamma \rho _{2})}{\mu (\beta + \mu )(\mu + \gamma )}\). There always exists an information-elimination equilibrium \(E_{0}(\eta _{k}, 0, 0, 0)\) when \(R_{0} < 1\). When \(R_{0} > 1\), the system has an information-prevailing equilibrium \(E^{ *} (A_{k}^{ *}, H_{k}^{ *}, F _{k}^{ *}, B_{k}^{ *} )\).
Proof
So, a nontrivial solution exists if and only if \(R_{0} > 1\). Inserting the nontrivial solution of (3.7) into Eq. (3.6), we obtain \(F_{k}^{ *} \). Then by (3.7) and (3.8), we can easily get \(0 < A_{k}^{ *}, H_{k} ^{ *}, F_{k}^{ *}, B_{k}^{ *} < \eta _{k}\) for \(k = 1, 2, \ldots, n\). Thus, the equilibrium \(E^{ *} (A_{k}^{ *}, H_{k}^{ *}, F_{k}^{ *}, B _{k}^{ *} )\) is well-defined. Hence, when \(R_{0} > 1\), only one positive equilibrium \(E^{ *} (A_{k}^{ *}, H_{k}^{ *}, F_{k}^{ *}, B_{k}^{ *} )\) of system (2.1) exists. The proof is completed. □
Remark
- (1)
The basic reproductive number \(R_{0}\) depends on some model parameters and fluctuations of the degree distribution. It can be found that \(R_{0}\) has no correlation with the degree-dependent new immigration individuals \(b(k)\). It seems that the attraction parameters α and the infection rate \(\rho _{1}, \rho _{2}\) have the same effects, because \(R_{0}\) will increase when they increase, the effects will be explored by the detailed numerical calculation.
- (2)
If \(\varphi (k) = k, \beta = 0\), and \(b(k) = \mu \) (i.e., new immigration individuals are balanced by emigration rate), then the model can be simplified to the network based SIB model with \(R_{0} = \frac{ \langle k^{2} \rangle }{ \langle k \rangle } \frac{v(l\beta _{1} + \varepsilon \beta _{2})}{l(l + \varepsilon )}\), which is investigated in [43].
4 Global dynamics of the model
In this section, qualitative analysis of the model is presented. Firstly, we consider the local asymptotical stability of the information-elimination equilibrium \(E_{0}\).
Theorem 2
For system (2.5), the information-elimination equilibrium \(E_{0}\) is locally asymptotically stable if \(R_{0} < 1\).
Proof
In this section, the global attractivity of the information-prevailing equilibrium is discussed.
Theorem 3
If \(R_{0} > 1\), then \(\lim_{t \to \infty } \inf \{ A_{k}(t), H _{k}(t), F_{k}(t), B_{k}(t) \} = \{ A_{k}^{ *}, H_{k} ^{ *}, F_{k}^{ *}, B_{k}^{ *} \}\), where (\(A_{k}^{ *}, H_{k} ^{ *}, F_{k}^{ *}, B_{k}^{ *} \)) is the unique information-prevailing equilibrium of (2.1) for \(k = 1,2, \ldots,n\).
Proof
Thus, the information-prevailing equilibrium \(E^{ *} (A_{k}^{ *},H _{k}^{ *},F_{k}^{ *},B_{k}^{ *} )\) is globally asymptotically stable by LaSalle Invariance Principle [46]. This completes the proof. □
5 A preferential information spreading model with competitive mechanism
Theorem 4
- (1)
There always exists an information-elimination equilibrium \(E_{0}^{ *} = \{ (1,0,0,0,0) \}_{k}\) when \(R_{0}^{ *} < 1\).
- (2)
There is an information-prevailing equilibrium \(E_{1}^{ *} = \{ (A_{k}^{ *},H_{k}^{ *},F1_{k}^{ *},F2_{k}^{ *},B_{k}^{ *} ) \}_{k}\) when \(R_{0}^{ *} > 1\).
Proof
Obviously, \(\rho = 0\) satisfies (5.3). Hence, \(A_{k} = 1\) and \(H_{k} = F1_{k} = F2_{k} = B_{k} = 0\) is an equilibrium of (5.1), which is called the information-elimination equilibrium.
6 Numerical simulation and discussion
In this section, we conduct simulations to investigate the dynamics of the model parameters and network topology structures to support and explain our theoretical results. Here the degree distribution is \(P(k) = ck^{ - l}(2 < l \le 3)\) in which \(l = 3\) and c satisfies \(\sum_{k = 1}^{n} P(k) = 1\) [50], \(\sum_{k = 1}^{n} P(k) = 1, n = 4000, b(k) = b/n\); if we choose \(\varphi (k) = k^{m}\) with \(m = 1\), then some interesting phenomena can be observed in our simulations.
7 Conclusion
In this paper, we propose a novel AHFB model to describe the dynamic behavior of preferential information spreading on scale-free networks. We obtain the basic reproductive number \(R_{0}\) with the next generation matrix method, which is closely related to the topology of the underlying networks and some model parameters. Interestingly, the basic reproductive number \(R_{0}\) has no relationship with the degree-dependent new register \(b(k)\). More specifically, by using the comparison principle and Lyapunov function, we prove that the information-eliminate equilibrium \(E_{0}\) is globally asymptotically stable when \(R_{0} < 1\); the preferential information is uniformly persistent on the network when \(R_{0} > 1\). In all cases, the effects of attractiveness in preferential information and the hesitation psychology of customers on the information spreading dynamics have been discussed. It seems that increasing the attractiveness parameter α or decreasing the hesitation parameter δ can improve preferential information spreading. Moreover, we study the preferential information spreading model with competitive mechanism and analyze corresponding dynamics behavior, then we draw a conclusion that the degree of preferential for products in the process of commodity competition is a core competitiveness factor. We have found that the degree of merchandise preference is 0.001 or 0.01 more than that of competitors and the number of forwarders will increase significantly. At the same time, increasing the spreading rate of beneficiaries, that is, the acknowledged comment and positive feedback of the beneficiaries can help promote the sale of products. The study has valuable guiding significance in effectively managing and controlling preferential information spreading on scale-free networks.
Declarations
Acknowledgements
We thank the referees and the editor for their careful reading of the original manuscript and many valuable comments and suggestions that greatly improved the presentation of this paper.
Funding
This work is supported by the National Natural Science Foundation of China under Grant 61672112, 61873287, Project in Hubei Province Department of Education under Grant B2016036 and Yangtze University Excellent Doctoral and Master’s Thesis Cultivation Program Funding Project.
Authors’ contributions
The authors contributed equally to this work. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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