Modeling and Analysis of New Products Diffusion on Heterogeneous Networks

Abstract

We present a heterogeneous networks model with the awareness stage and the decision-making stage to explain the process of new products diffusion. If mass media is neglected in the decision-making stage, there is a threshold whether the innovation diffusion is successful or not, or else it is proved that the network model has at least one positive equilibrium. For networks with the power-law degree distribution, numerical simulations confirm analytical results, and also at the same time, by numerical analysis of the influence of the network structure and persuasive advertisements on the density of adopters, we give two different products propagation strategies for two classes of nodes in scale-free networks.

1. Introduction

The Bass model has become an important exemplar in marketing science. For over three decades, this model has been the main impetus underlying diffusion research and has been widely used to understand the diffusion of new products [1–3]. The following equation illustrates the classical Bass model [4]: 𝑑𝑁(𝑡)=𝑞𝑑𝑡𝑚𝑁(𝑡)(𝑚−𝑁(𝑡))+𝑝(𝑚−𝑁(𝑡)), (1) where 𝑁(𝑡) is the number of cumulative adopters, 𝑝 is the coefficient of external influence, 𝑞 is the coefficient of internal influence, and 𝑚 is the market potential or potential number of ultimate adopters. On the basis of the Bass model, assuming that populations on the market are homogeneous and well mixed, many dynamics models are used to study new products diffusion; Yu et al. proposed mathematical models to describe the dynamics of competitive products in the market and analyzed the stability of equilibria of the model [5–7]. Wang et al. proposed describing the dynamics of users of one product in two different patches; periodic advertisements are also incorporated and the existence and uniqueness of positive periodic solutions are investigated [8]. Considering that an observation process cannot be neglected for the consumers of many products, especially for the products with high values, Wang et al. established a new production diffusion model with two stages (the awareness stage and the decision-making stage) and obtained a threshold above which innovation diffusion is successful [9, 10]. All these models incorporate the imitation among the population but without the explicit network structure. In fact, the consumer's decision-making process on the new product adoption involves a complex interaction of various external and internal factors like mass media, advertising, word of mouth, personal preferences, and experience. Without a doubt, a friend's opinion or advice often can be a decisive argument for a purchase. So, network models considering social contacts and the individual heterogeneity about new products diffusion are relatively reasonable.

The products information spread in the diffusion process is similar to the typical virus spread model such as SIS model and SIR model. The spread mechanism of diseases on networks has been widely studied by many researchers. Many researchers have given different interpretations for the propagation of diseases on uncorrelated complex networks [11–17]; Boguna et al. discuss the epidemic spreading on complex networks in which there are explicit correlations among the degrees of connected vertices [18]. With the development of the disease model on networks, researches about new products diffusion on complex networks have raised a growing interest [19, 20]; however, they still do not have specific network dynamics models or dynamics analysis of the model. So, we intend to model and analyze the process of new products diffusion on heterogeneous networks in this paper. Considering two stages, the awareness stage and the decision-making stage in the product diffusion process, and assuming that the product information is spread due to mouth-to-mouth method among people's direct contacts and mass media channels (in the awareness stage, mass media is informative, and in the decision-making stage, mass media is persuasive), we propose the heterogeneous networks model of new products diffusion with two stages.

The organization of this paper is as follows. In the next section, we set up a heterogeneous network model about new products diffusion and discuss its invariant set. In Section 3, when mass media is neglected in the decision-making stage, we discuss equilibria and their stabilities and obtain conditions that the diffusion is successful; when mass media is considered, we obtain that the system at least has a positive equilibrium. The theoretical results are also confirmed by numerical simulations in Section 4; furthermore, we also obtain two different propagation strategies by simulations analysis. Finally, we give a brief conclusion and discussion in Section 5.

2. Dynamics Model of New Products Diffusion

By incorporating the impact of social neighborhood in the new products diffusion process, we establish networks model describing the new products diffusion procedure. Here, we consider the whole population and their contacts in networks. Each individual in the region under consideration can be regarded as a vertex in the network, and each contact between two individuals is represented as an edge connecting the vertices. The number of edges emanating from a vertex, that is, the number of contacts a person has, is called the degree of the vertex. Therefore, we divide the population into 𝑛 distinct groups of sizes 𝑁_𝑘 (𝑘=1,2,…,𝑛) such that each individual in group 𝑘 has exactly 𝑘 contacts per day. If the whole population size is 𝑁 (𝑁=𝑁₁+𝑁₂+⋯+𝑁_𝑛), then the probability that a uniformly chosen individual has 𝑘 contacts is 𝑝(𝑘)=𝑁_𝑘/𝑁, which is called the degree distributions of the network. Let 𝑈_𝑘(𝑡) denote the number of those individuals who have not been aware of the product within group 𝑘 at time 𝑡, let 𝐼_𝑘(𝑡) denote the number of those individuals who have been aware of the information about the product but have not yet adopted it within group 𝑘 at time 𝑡, and let 𝐴_𝑘(𝑡) denote the number of those individuals who have adopted the product at time 𝑡. We consider two stage: in the first stage, enterprises transfer new production information to consumers through informative advertisements, or individuals obtain new product information from their neighbors, so individuals become aware of information; in the decision-making stage, enterprises change consumers' preference through persuasive advertisements, or a friend's opinion or advice also influences individuals' preferences, so individuals try or adopt the product. We give a system of 3𝑛 ordinary differential equations representing the product diffusion dynamics on the scale-free network: 𝑑𝑈_𝑘(𝑡)𝑑𝑡=−𝑝𝑈_𝑘(𝑡)−𝛽₁𝑘𝑈_𝑘(𝑡)Θ₁(𝑡)−𝛽₂𝑘𝑈_𝑘(𝑡)Θ₂(𝑡)+𝛾𝐼_𝑘(𝑡)+𝜇𝐴_𝑘(𝑡),𝑑𝐼_𝑘(𝑡)𝑑𝑡=𝑝𝑈_𝑘(𝑡)+𝛽₁𝑘𝑈_𝑘(𝑡)Θ₁(𝑡)+𝛽₂𝑘𝑈_𝑘(𝑡)Θ₂(𝑡)−𝛼𝐼_𝑘(𝑡)−𝛾𝐼_𝑘(𝑡)−𝛽₃𝑘𝐼_𝑘(𝑡)Θ₂(𝑡),𝑑𝐴_𝑘(𝑡)𝑑𝑡=𝛼𝐼_𝑘(𝑡)+𝛽₃𝑘𝐼_𝑘(𝑡)Θ₂(𝑡)−𝜇𝐴_𝑘(𝑡), (2) where Θ₁(𝑡)=∑^𝑛_𝑘=1𝑘𝐼_𝑘∑^𝑛_𝑘=1𝑘𝑁_𝑘,Θ₂(𝑡)=∑^𝑛_𝑘=1𝑘𝐴_𝑘∑^𝑛_𝑘=1𝑘𝑁_𝑘. (3) All parameters are positive constants and the meaning of parameters is summarized in Table 1. When 𝛽₁=0 and population contact is assumed to be homogeneous, system (2) becomes the model in [9].

Table 1 

Parameters of model 1.

Denote the relative densities of 𝑈_𝑘(𝑡), 𝐼_𝑘(𝑡), and 𝐴_𝑘(𝑡) at time 𝑡 by 𝑢_𝑘(𝑡), 𝑖_𝑘(𝑡), and 𝑎_𝑘(𝑡), respectively; then, system (2) can be rewritten as 𝑑𝑢_𝑘(𝑡)𝑑𝑡=−𝑝𝑢_𝑘(𝑡)−𝛽₁𝑘𝑢_𝑘(𝑡)Θ₁(𝑡)−𝛽₂𝑘𝑢_𝑘(𝑡)Θ₂(𝑡)+𝛾𝑖_𝑘(𝑡)+𝜇𝑎_𝑘(𝑡),𝑑𝑖_𝑘(𝑡)𝑑𝑡=𝑝𝑢_𝑘(𝑡)+𝛽₁𝑘𝑢_𝑘(𝑡)Θ₁(𝑡)+𝛽₂𝑘𝑢_𝑘(𝑡)Θ₂(𝑡)−𝛼𝑖_𝑘(𝑡)−𝛾𝑖_𝑘(𝑡)−𝛽₃𝑘𝑖_𝑘(𝑡)Θ₂(𝑡),𝑑𝑎_𝑘(𝑡)𝑑𝑡=𝛼𝑖_𝑘(𝑡)+𝛽₃𝑘𝑖_𝑘(𝑡)Θ₂(𝑡)−𝜇𝑎_𝑘(𝑡), (4) with the normalization condition 𝑢_𝑘(𝑡)+𝑖_𝑘(𝑡)+𝑎_𝑘(𝑡)=1, and Θ₁(𝑡)=(1/⟨𝑘⟩)∑^𝑛_𝑘=1𝑘𝑝(𝑘)𝑖_𝑘, Θ₂(𝑡)=(1/⟨𝑘⟩)∑^𝑛_𝑘=1𝑘𝑝(𝑘)𝑎_𝑘. Using this condition, system (4) is reduced to 𝑑𝑖_𝑘(𝑡)=𝑑𝑡𝑝+𝛽₁𝑘Θ₁(𝑡)+𝛽₂𝑘Θ₂(𝑡)1−𝑖_𝑘(𝑡)−𝑎_𝑘(𝑡)−𝛾𝑖_𝑘(𝑡)−𝛼+𝛽₃𝑘Θ₂(𝑡)𝑖_𝑘(𝑡),𝑑𝑎_𝑘(𝑡)=𝑑𝑡𝛼+𝛽₃𝑘Θ₂(𝑡)𝑖_𝑘(𝑡)−𝜇𝑎_𝑘(𝑡). (5) Prior to discussing the stability of system (5), we first study its variant set.

Let 𝑖_𝑘=𝑦_𝑘, 𝑘=1,2,…,𝑛,  𝑎_𝑘=𝑦_𝑛+𝑘, 𝑘=1,2,…,𝑛,  𝑦=(𝑦₁,𝑦₂,…,𝑦_2𝑛)=(𝑖₁,𝑖₂,…,𝑖_𝑛,𝑎₁,𝑎₂,…,𝑎_𝑛). Denoting that Δ_2𝑛=∑^2𝑛_𝑙=1[0,1], we study system (5) for 𝑦(𝑡)∈Δ_2𝑛.

Lemma 1. The set Δ_2𝑛 is positively invariant set of system (5).

Proof. We will show that all the solutions starting from any initial value 𝑦(0)∈Δ_2𝑛 of system (5) satisfy 𝑦(𝑡)∈Δ_2𝑛. Denote that 𝜕Δ¹_2𝑛={𝑦∈Δ_2𝑛∣𝑦_𝑖=0 for some 𝑖}, 𝜕Δ²_2𝑛={𝑦∈Δ_2𝑛∣𝑦_𝑖=1 for some 𝑖},where 𝑖=1,2,…,2𝑛. Let the “outer normals” be denoted by 𝜂¹_𝑖=(0,…,−1,…,0) and 𝜂²_𝑖=(0,…,1,…,0).
For arbitrary compact set Γ, Nagumo had proved that Γ is invariant for 𝑑𝑥/𝑑𝑡=𝑓(𝑥), if, at each point 𝑦 in 𝜕Γ (the boundary of Γ), the vector 𝑓(𝑥) is tangent or pointing into the set [21, 22]. We can easily apply the result here, since Γ is an 2𝑛-dimensional rectangle. Through Nagumo's result, it is not difficult to obtain that 𝑑𝑦𝑑𝑡_𝑦𝑖=0⋅𝜂¹_𝑖𝛽=−𝑝+¹𝑖⟨𝑘⟩_𝑘≠𝑖𝑘𝑝(𝑘)𝑦_𝑘+𝑖𝛽₂Θ₂1−𝑦_𝑖+𝑛≤0,𝑖=1,2,…,𝑛,𝑑𝑦𝑑𝑡_{𝑦𝑛+𝑖=0}⋅𝜂¹_𝑖𝛽=−𝛼+³𝑖⟨𝑘⟩_𝑘≠𝑖𝑘𝑝(𝑘)𝑦_𝑛+𝑘≤0,𝑖=1,2,…,𝑛,𝑑𝑦𝑑𝑡_𝑦𝑖=1⋅𝜂²_𝑖≤0,𝑖=1,2,…,2𝑛. (6) Hence, Lemma 1 is proved.

3. Model Analysis

Next, for investigating the effect of mass media in the decision-making stage, we will analyze the dynamics of system (5) under two kinds of assumptions as followed:(1)persuasive advertisements are neglected in the decision-making stage, which means 𝛼=0,(2)persuasive advertisements are considered in the decision-making stage, which means 𝛼≠0.

Case I (𝛼=0). When 𝛼=0, equilibria for system (5) can be found by setting the right sides of two differential equations of system (5) equal to zero, giving the algebraic system 𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂1−𝑖_𝑘−𝑎_𝑘−𝛾𝑖_𝑘−𝛽₃𝑘𝑖_𝑘Θ₂𝛽=0,₃𝑘𝑖_𝑘Θ₂−𝜇𝑎_𝑘=0. (7) We can get the equivalent system of system (7) as follows: 𝑖_𝑘=𝜇𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂𝜇+𝛽₃𝑘Θ₂𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂+𝜇𝛾+𝛽₃𝑘Θ₂,𝑎_𝑘=𝛽₃𝑘Θ₂𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂𝜇+𝛽₃𝑘Θ₂𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂+𝜇𝛾+𝛽₃𝑘Θ₂, (8) and it is obvious that 𝑎_𝑘=0 satisfies the second formula in (8); substituting 𝑎_𝑘=0 into the first formula in (8), we obtain 𝑖_𝑘=𝑝+𝛽₁𝑘Θ₁𝑝+𝛾+𝛽₁𝑘Θ₁ (9) and a self-consistency equation about Θ₁ as follows: Θ₁=1⟨𝑘⟩^𝑛_𝑘=1𝑘𝑝(𝑘)𝑝+𝛽₁𝑘Θ₁𝑝+𝛾+𝛽₁𝑘Θ₁≜𝑓Θ₁. (10) Through some calculation, we have that (0)=𝑝0<𝑓(1)=1𝑝+𝛾<1,0<𝑓⟨𝑘⟩^𝑛_𝑘=1𝑘𝑝(𝑘)𝑝+𝛽₁𝑘𝑝+𝛾+𝛽₁𝑘<1,𝑓^′Θ₁=1⟨𝑘⟩^𝑛_𝑘=1(𝑘)𝛽𝑘𝑝¹𝛾𝑘𝑝+𝛾+𝛽₁𝑘Θ₁²>0,𝑓^′′Θ₁=1⟨𝑘⟩^𝑛_𝑘=1𝑘𝑝(𝑘)−2𝛽²₁𝑘²𝛾𝑝+𝛾+𝛽₁𝑘Θ₁³<0. (11)(12)(13) According to formula (11), we can know that (10) has at least a positive solution in (0,1); formula (12) shows that the function 𝑓(Θ₁) is a rigorous monotone increasing function, and formula (13) means that the function 𝑓(Θ₁) is convex, so (10) must has a unique positive solution Θ⁰₁ in (0,1), and system (5) has a trivial solution 𝐸⁰=(𝑖⁰₁,𝑖⁰₂,…,𝑖⁰_𝑛,0,0,…,0), where 𝑖⁰_𝑘=𝑝+𝛽₁𝑘Θ⁰₁𝑝+𝛾+𝛽₁𝑘Θ⁰₁,𝑘=1,2,…,𝑛,Θ⁰₁=1⟨𝑘⟩^𝑛_𝑘=1𝑘𝑝(𝑘)𝑖⁰_𝑘. (14)(15) The above results can be summarized in the following theorem.

Theorem 2. When 𝛼=0, system (5) has a unique trivial equilibrium 𝐸⁰.

Furthermore, one will prove the local stability and global stability of the equilibrium 𝐸⁰. The Jacobin matrix of system (5) at 𝐸⁰ is 𝐽|_𝐸⁰=,𝐵𝐶0𝐷 (16) where𝑏𝐵=¹¹𝛽₁2𝑝(2)𝑢⟨𝑘⟩⁰¹⋯𝛽₁𝑛𝑝(𝑛)𝑢⟨𝑘⟩⁰¹2𝛽₁𝑝(1)𝑢⟨𝑘⟩⁰²𝑏₂₂⋯2𝛽₁𝑛𝑝(𝑛)𝑢⟨𝑘⟩⁰²⋮⋮⋱⋮𝑛𝛽₁𝑝(1)𝑢⟨𝑘⟩^0𝑛𝑛𝛽₁2𝑝(2)𝑢⟨𝑘⟩^0𝑛⋯𝑏_𝑛𝑛,𝑐𝐶=¹¹2𝑝(2)𝛽⟨𝑘⟩²𝑢⁰₁−𝛽₃𝑖⁰₁⋯𝑛𝑝(𝑛)𝛽⟨𝑘⟩²𝑢⁰₁−𝛽₃𝑖⁰₁2𝑝(1)𝛽⟨𝑘⟩²𝑢⁰₂−𝛽₃𝑖⁰₂𝑐₂₂(𝑛)⋯2𝑛𝑝𝛽⟨𝑘⟩²𝑢⁰₂−𝛽₃𝑖⁰₂𝑛𝑝(1)⋮⋮⋱⋮⟨𝑘⟩𝛽₂𝑢⁰_𝑛−𝛽₃𝑖⁰_𝑛𝑛2𝑝(2)𝛽⟨𝑘⟩²𝑢⁰_𝑛−𝛽₃𝑖⁰_𝑛⋯𝑐_𝑛𝑛,𝛽𝐷=³𝑝(1)𝑖⟨𝑘⟩⁰¹−𝜇𝛽₃2𝑝(2)𝑖⟨𝑘⟩⁰¹⋯𝛽₃𝑛𝑝(𝑛)𝑖⟨𝑘⟩⁰¹2𝛽₃𝑝(1)𝑖⟨𝑘⟩⁰²2𝛽₃2𝑝(2)𝑖⟨𝑘⟩⁰²−𝜇⋯2𝛽₃𝑛𝑝(𝑛)𝑖⟨𝑘⟩⁰²⋮⋮⋱⋮𝑛𝛽₃𝑝(1)𝑖⟨𝑘⟩^0𝑛𝑛𝛽₃2𝑝(2)𝑖⟨𝑘⟩^0𝑛⋯𝑛𝛽₃𝑛𝑝(𝑛)𝑖⟨𝑘⟩^0𝑛;−𝜇 (17)here 𝑏_𝑙𝑙=−𝑝+𝑙𝛽₁Θ⁰₁+𝛾+𝑙²𝛽₁𝑝(𝑙)⟨𝑘⟩𝑢⁰_𝑙𝑐,𝑙=1,…,𝑛,_𝑙𝑙=𝑙²𝑝(𝑙)⟨𝑘⟩𝛽₂𝑢⁰_𝑙−𝛽₃𝑖⁰_𝑙−𝑝+𝑙𝛽₁Θ⁰₁𝑢,𝑙=1,…,𝑛,_0𝑙=1−𝑖⁰_𝑙,𝑙=1,…,𝑛. (18)

Next, for studying the local stability of the equilibrium 𝐸⁰, we first estimate eigenvalues by carrying out the similarity transformation about the matrix 𝐵. Letting 𝑇=diag(𝛿₁,𝛿₂,…,𝛿_𝑛), 𝛿_𝑖=𝑖𝑢⁰_𝑖, 𝑖=1,2…,𝑛, then 𝑇⁻¹𝑏𝐵𝑇=¹¹𝛽₁2²𝑝(2)𝑢⟨𝑘⟩⁰²⋯𝛽₁𝑛²𝑝(𝑛)𝑢⟨𝑘⟩^0𝑛𝛽₁𝑝(1)𝑢⟨𝑘⟩⁰¹𝑏₂₂⋯𝛽₁𝑛²𝑝(𝑛)⟨𝑘⟩𝑢⁰_𝑛𝛽⋮⋮⋱⋮₁𝑝(1)𝑢⟨𝑘⟩⁰¹𝛽₁2²𝑝(2)⟨𝑘⟩𝑢⁰₂⋯𝑏_𝑛𝑛≜𝐵^∗. (19) Obviously, 𝐵 and 𝐵^∗ have same eigenvalues. We can obtain 𝛽₁(⟨𝑘²𝑢⁰_𝑘⟩/⟨𝑘⟩)=𝑝+𝛾−𝑝/Θ⁰₁ using formula (14), so 𝑏_𝑙𝑙=−(𝑝+𝑙𝛽₁Θ⁰₁+𝛾)+𝑙²𝛽₁(𝑝(𝑙)/⟨𝑘⟩)𝑢⁰_𝑙<−(𝑝+𝑙𝛽₁Θ⁰₁+𝛾)+𝛽₁(⟨𝑘²𝑢⁰_𝑘⟩/⟨𝑘⟩)=−𝑙𝛽₁Θ⁰₁−𝑝/𝜃⁰₁<0, 𝑙=1,2,…,𝑛; furthermore, |𝑏_𝑙𝑙|−∑^𝑛_{𝑗≠𝑙,𝑗=1}|𝐵^∗_𝑙𝑗|=(𝑝+𝑙𝛽₁Θ⁰₁+𝛾)−𝛽₁(⟨𝑘²𝑢⁰_𝑘⟩/⟨𝑘⟩)>0, so 𝐵^∗ is strictly diagonally dominant and the principle diagonal elements are negative, according to the results of estimating distribution of eigenvalues for generalized diagonally dominant matrices [23], and then every eigenvalue of the matrix 𝐵^∗ has negative real part; that is to say, every eigenvalue of the matrix 𝐵 has negative real part.

To find eigenvalues of the matrix 𝐷, we carry out similarity transformation to the matrix 𝐷; namely, the 𝑗 column multiplied by −𝑝(𝑗−1)/𝑗𝑝(𝑗) is added to the 𝑗−1 column, and the 𝑗−1 row multiplied by 𝑝(𝑗−1)/𝑗𝑝(𝑗) is added to the 𝑗 row, 𝑗=2,3,…,𝑛. Then we obtain the similarity matrix 𝐷^∗ as follows: 𝐷^∗=−𝜇0⋯𝛽₃𝑛𝑝(𝑛)𝑖⟨𝑘⟩⁰¹0−𝜇⋯𝛽₃𝑛𝑝(𝑛)⟨𝑘⟩𝑝(1)(2)𝑖2𝑝⁰¹+2𝑖⁰₂⋮⋮⋱⋮00⋯𝛽₃⟨𝑘²𝑖⁰_𝑘⟩.⟨𝑘⟩−𝜇 (20) It is straightforward that 𝐽|_𝐸⁰ has 2𝑛−1 negative characteristic roots. When 𝛽₃(⟨𝑘²𝑖⁰_𝑘⟩/⟨𝑘⟩)−𝜇<0, 𝐸₀ is locally asymptotically stable; otherwise, it is not stable. Letting 𝑅≜(𝛽₃/𝜇)(⟨𝑘²𝑖⁰_𝑘⟩/⟨𝑘⟩), we have the following theorem.

Theorem 3. When 𝛼=0, if 𝑅<1, 𝐸₀ is locally asymptotically stable, and if 𝑅>1, 𝐸₀ is unstable.

Actually, one can further obtain the global stability of 𝐸₀ under stronger parameter conditions.

Theorem 4. When 𝛼=0, if 𝑅<(𝛽₃⟨𝑘²⟩/𝜇⟨𝑘⟩)<1, 𝐸₀ is globally asymptotically stable.

Proof. Considering system (5) with 𝛼=0, by the second formula in system (5), we have 𝑑𝑎_𝑘(𝑡)𝑑𝑡≤𝛽₃𝑘Θ₂(𝑡)−𝜇𝑎_𝑘(𝑡), (21) and we consider the following auxiliary system: 𝑑𝑎_𝑘(𝑡)𝑑𝑡=𝛽₃𝑘Θ₂(𝑡)−𝜇𝑎_𝑘(𝑡). (22) Multiplying formula (22) by 𝑘𝑝(𝑘)/⟨𝑘⟩ and summing over 𝑘, we obtain 𝑑Θ₂(𝑡)𝑑𝑡=𝜇Θ₂𝛽₃𝜇𝑘²⟨𝑘⟩−1, (23) and the solution of (23) tends to 0 when 𝛽₃⟨𝑘²⟩/𝜇⟨𝑘⟩<1, since system (23) is a quasi-monotone system; by the comparison theorem, lim_𝑡→∞𝑎_𝑘=0(𝑘=1,2,…,𝑛) in system (5), we consider the limit equation of the first equation of system (5): 𝑑𝑖_𝑘(𝑡)=𝑑𝑡𝑝+𝛽₁𝑘Θ₁(𝑡)1−𝑖_𝑘(𝑡)−𝛾𝑖_𝑘(𝑡); (24) the above formula has a unique equilibrium (𝑖⁰₁,𝑖⁰₂,…,𝑖⁰_𝑛) and it is locally asymptotically stable. So lim_𝑡→∞𝑖_𝑘=𝑖⁰_𝑘, 𝑘=1,2,…,𝑛. 𝐸₀ is globally attractive when 𝛽₃⟨𝑘²⟩/𝜇⟨𝑘⟩<1; this, combined with Theorem 3, implies that 𝐸₀ is globally asymptotically stable when 𝑅<𝛽₃⟨𝑘²⟩/𝜇⟨𝑘⟩<1.

For proving the existence of the positive equilibrium of system (5) with 𝛼=0, we discuss the existence of the positive solution of (8); by (8), we obtain the following equations about Θ₁, Θ₂: Θ₁=1⟨𝑘⟩^𝑛_𝑘=1(𝑘)×𝜇𝑘𝑝𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂×𝜇+𝛽₃𝑘Θ₂𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂+𝜇𝛾+𝛽₃𝑘Θ₂⁻¹,Θ₂=1⟨𝑘⟩^𝑛_𝑘=1(𝑘)×𝛽𝑘𝑝₃𝑘Θ₂𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂×𝜇+𝛽₃𝑘Θ₂𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂+𝜇𝛾+𝛽₃𝑘Θ₂⁻¹. (25) In consideration of 𝑎_𝑘≠0, Θ₂≠0, from formula (25), we can know that (Θ₁,Θ₂)∈Ω₁≜{0<Θ₁<1,0<Θ₂<1}. By the identical transformation of the second equation in formula (25), we have 𝑓Θ₁,Θ₂=0, (26) where 𝑓Θ₁,Θ₂≜1⟨𝑘⟩^𝑛_𝑘=1(𝑘)×𝑘𝑝𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂𝛽₃𝑘1−Θ₂−𝜇−𝜇𝛾+𝛽₃𝑘Θ₂×𝜇+𝛽₃𝑘Θ₂𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂+𝜇𝛾+𝛽₃𝑘Θ₂⁻¹, (27) and 𝑓(Θ₁,Θ₂) is continuous in Ω≜[0,1]×[0,1]. It is easy to testify that 𝑓(Θ⁰₁,0)=𝑅−1>0 when 𝑅>1, 𝑓(1,1)<0, and 𝑓(Θ₁,Θ₂) is continuous about Θ₂ in [0,1], so there must exist some sufficient small numbers 𝜀₁(0<𝜀₁<1) and 𝜀₂(0<𝜀₂<1) fully close to 1 that satisfy 𝑓(Θ⁰₁,𝜀₁)>0 and 𝑓(1,𝜀₂)<0, and 𝑓(Θ₁,Θ₂) is continuous about Θ₁ in [0,1]; similarly, 𝑓(𝜀₃,𝜀₂)<0, where 𝜀₃ (Θ⁰₁<𝜀₃<1) is fully close to 1; thus there is at least one point 𝑃(Θ₁,Θ₂) satisfying 𝑓(Θ₁,Θ₂)=0, where 𝑃∈[Θ⁰₁,𝜀₃]×[𝜀₁,𝜀₂]. According to the implicit function theorem, equation 𝑓(Θ₁,Θ₂)=0 can establish only a continuous function in the neighbor domain Ω₂(⊂Ω₁) of 𝑃: Θ₂=𝑔Θ₁,Θ₁,Θ₂∈Ω₂. (28) Substituting (28) into the first equation in formula (25), we can get the following equation: Θ₁=1⟨𝑘⟩^𝑛_𝑘=1(𝑘)×𝜇𝑘𝑝𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘𝑔Θ₁×𝜇+𝛽₃𝑘𝑔Θ₁𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘𝑔Θ₁+𝜇𝛾+𝛽₃𝑘𝑔Θ₁⁻¹≜𝑔Θ₁. (29) Noting that 𝑔(0)−0>0, 𝑔(1)−1<0, by the continuity of function 𝑔(Θ₁), there at least exists Θ^∗₁∈(0,1) satisfying formula (29), so formula (25) at least has a positive solution (Θ^∗₁,Θ^∗₂)  (Θ^∗₂=𝑔(Θ^∗₁)), which means that system (5) at least has a positive equilibrium 𝐸₁(𝑖^∗₁,𝑖^∗₂,…,𝑖^∗_𝑛,𝑎^∗₁,𝑎^∗₂,…,𝑎^∗_𝑛), where 𝑖^∗_𝑘=𝜇𝑝+𝛽₁𝑘Θ^∗₁+𝛽₂𝑘Θ^∗₂𝜇+𝛽₃𝑘Θ^∗₂𝑝+𝛽₁𝑘Θ^∗₁+𝛽₂𝑘Θ^∗₂+𝜇𝛾+𝛽₃𝑘Θ^∗₂,𝑎𝑘=1,…,𝑛,_∗𝑘=𝛽₃𝑘Θ^∗₂𝑝+𝛽₁𝑘Θ^∗₁+𝛽₂𝑘Θ^∗₂𝜇+𝛽₃𝑘Θ^∗₂𝑝+𝛽₁𝑘Θ^∗₁+𝛽₂𝑘Θ^∗₂+𝜇𝛾+𝛽₃𝑘Θ^∗₂,𝑘=1,…,𝑛. (30)

We will subsequently show that system (5) has a unique positive equilibrium. We denote that 𝑖^∗_𝑘=𝑦^∗_𝑘, 𝑎^∗_𝑘=𝑦^∗_𝑛+𝑘, 𝑘=1,…,𝑛 and assume that 𝑦=𝑦^∗>0 and 𝑦=𝑧^∗>0 are two constant solutions of system (5). If 𝑦^∗≠𝑧^∗, then there exists at least one 𝑖₀, 𝑖₀=1,2,…,2𝑛, such that 𝑦^∗_𝑖0≠𝑧^∗_𝑖0, where 𝑦^∗_𝑖0 is the 𝑖^th₀ component of the vector 𝑦^∗. Without loss of generality, we assume that 𝑦^∗_𝑖0>𝑧^∗_𝑖0 and moreover that 𝑦^∗_𝑖0/𝑧^∗_𝑖0≥𝑦^∗_𝑖/𝑧^∗_𝑖 for all 𝑖=1,…,2𝑛. Since 𝑦^∗ and 𝑧^∗ are constant solutions of system (5), and if 1≤𝑖₀≤𝑛, we obtain that 𝑝+𝛽₁𝑖₀Θ₁𝑦^∗+𝛽₂𝑖₀Θ₂𝑦^∗1−𝑦^∗_𝑖0−𝑦^∗_𝑛+𝑖0−𝛾𝑦^∗_𝑖0−𝛽₃𝑘Θ₂𝑦^∗𝑦^∗_𝑖0=𝑝+𝛽₁𝑖₀Θ₁𝑧^∗+𝛽₂𝑖₀Θ₂𝑧^∗1−𝑧^∗_𝑖0−𝑧^∗_𝑛+𝑖0−𝛾𝑧^∗_𝑖0−𝛽₃𝑘Θ₂𝑧^∗𝑧^∗_𝑖0=0, (31) or if 𝑛<𝑖₀≤2𝑛, we have 𝛽₃𝑖₀Θ₂𝑦^∗𝑦^∗_𝑖0−𝑛−𝜇𝑦^∗_𝑖0=𝛽₃𝑖₀Θ₂𝑧^∗𝑧^∗_𝑖0−𝑛−𝜇𝑧^∗_𝑖0=0, (32) where Θ₁𝑦^∗=1⟨𝑘⟩^𝑛_𝑘=1𝑘𝑝(𝑘)𝑦^∗_𝑘,Θ₂𝑦^∗=1⟨𝑘⟩^𝑛_𝑘=1𝑘𝑝(𝑘)𝑦^∗_𝑛+𝑘,Θ₁𝑧^∗=1⟨𝑘⟩^𝑛_𝑘=1𝑘𝑝(𝑘)𝑧^∗_𝑘,Θ₂𝑧^∗=1⟨𝑘⟩^𝑛_𝑘=1𝑘𝑝(𝑘)𝑧^∗_𝑛+𝑘. (33) After equivalent deformation, it follows that 𝑧^∗_𝑖0𝑦^∗_𝑖0𝑝+𝛽₁𝑖₀Θ₁𝑦^∗+𝛽₂𝑖₀Θ₂𝑦^∗1−𝑦^∗_𝑖0−𝑧^∗_𝑛+𝑖0−𝛾𝑧^∗_𝑖0−𝛽₃𝑘Θ₂𝑧^∗𝑧^∗_𝑖0=𝑝+𝛽₁𝑖₀Θ₁𝑧^∗+𝛽₂𝑖₀Θ₂𝑧^∗1−𝑧^∗_𝑖0−𝑧^∗_𝑛+𝑖0−𝛾𝑧^∗_𝑖0−𝛽₃𝑘Θ₂𝑧^∗𝑧^∗_𝑖0=0, (34) or 𝑧^∗_𝑖0𝑦^∗_𝑖0𝛽₃𝑖₀Θ₂𝑦^∗𝑧^∗_𝑖0−𝑛−𝜇𝑧^∗_𝑖0=𝛽₃𝑖₀Θ₂𝑧^∗𝑧^∗_𝑖0−𝑛−𝜇𝑧^∗_𝑖0=0. (35) But 𝑦^∗_𝑖0>𝑧^∗_𝑖0, (𝑧^∗_𝑖0/𝑦^∗_𝑖0)𝑦^∗_𝑖≤𝑧^∗_𝑖 for all 𝑖, and 1−𝑦^∗_𝑖0<1−𝑧^∗_𝑖0; thus from the above equalities we get 𝑧^∗_𝑖0𝑦^∗_𝑖0𝑝+𝛽₁𝑖₀Θ₁𝑦^∗+𝛽₂𝑖₀Θ₂𝑦^∗1−𝑦^∗_𝑖0−𝑧^∗_𝑛+𝑖0−𝛾𝑧^∗_𝑖0−𝛽₃𝑘Θ₂𝑧^∗𝑧^∗_𝑖0<𝑝+𝛽₁𝑖₀Θ₁𝑧^∗+𝛽₂𝑖₀Θ₂𝑧^∗1−𝑧^∗_𝑖0−𝑧^∗_𝑛+𝑖0−𝛾𝑧^∗_𝑖0−𝛽₃𝑘Θ₂𝑧^∗𝑧^∗_𝑖0,𝑧^∗_𝑖0𝑦^∗_𝑖0𝛽₃𝑖₀Θ₂𝑦^∗𝑧^∗_𝑖0−𝑛−𝜇𝑧^∗_𝑖0<𝛽₃𝑖₀Θ₂𝑧^∗𝑧^∗_𝑖0−𝑛−𝜇𝑧^∗_𝑖0. (36) This is a contradiction. Therefore, system (5) has a unique positive equilibrium. 𝐸₁(𝑖^∗₁,𝑖^∗₂,…,𝑖^∗_𝑛,𝑎^∗₁,𝑎^∗₂,…,𝑎^∗_𝑛) that is 𝑦^∗=(𝑦^∗₁,𝑦^∗₂,…,𝑦^∗_𝑛+1)∈Δ_2𝑛.

Hence, the conclusion follows.

Theorem 5. When 𝛼=0, if 𝑅>1, system (5) has a unique positive equilibrium 𝐸₁.

Case II (𝛼≠0). When 𝛼≠0, we can get equations that the steady state of system (5) satisfies 𝑖_𝑘=𝜇𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂×𝛼+𝜇+𝛽₃𝑘Θ₂𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂+𝜇𝛼+𝛾+𝛽₃𝑘Θ₂⁻¹,𝑎_𝑘=𝛼+𝛽₃𝑘Θ₂𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂×𝛼+𝜇+𝛽₃𝑘Θ₂𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂+𝜇𝛼+𝛾+𝛽₃𝑘Θ₂⁻¹, (37) and self-consistency equations about Θ₁, Θ₂ are obtained such that Θ₁=1⟨𝑘⟩^𝑛_𝑘=1(𝑘)×𝜇𝑘𝑝𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂×𝛼+𝜇+𝛽₃𝑘Θ₂𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂+𝜇𝛼+𝛾+𝛽₃𝑘Θ₂⁻¹,Θ₂=1⟨𝑘⟩^𝑛_𝑘=1(𝑘)×𝑘𝑝𝛼+𝛽₃𝑘Θ₂𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂×𝛼+𝜇+𝛽₃𝑘Θ₂𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂+𝜇𝛼+𝛾+𝛽₃𝑘Θ₂⁻¹. (38) From formula (38), we can obtain 0<Θ₁<1, 0<Θ₂<1. By the identical transformation of the second equation in formula (38), we can obtain 𝑓Θ₁,Θ₂=0, (39) where 𝑓Θ₁,Θ₂≜1⟨𝑘⟩^𝑛_𝑘=1(𝑘)×𝑘𝑝𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂𝛼+𝛽₃𝑘Θ₂1−Θ₂−𝜇Θ₂𝑝+𝛼+𝛾+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂+𝛽₃𝑘Θ₂×𝛼+𝜇+𝛽₃𝑘Θ₂𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂+𝜇𝛼+𝛾+𝛽₃𝑘Θ₂⁻¹, (40) and since 𝑓Θ₁=1,0⟨𝑘⟩^𝑛_𝑘=1𝑘𝑝(𝑘)𝑝+𝛽₁𝑘Θ₁𝛼𝛼+𝜇𝑝+𝛽₁𝑘Θ₁𝑓Θ+𝜇𝛼+𝛾>0,₁=1,1⟨𝑘⟩^𝑛_𝑘=1(𝑘)×𝑘𝑝−𝜇𝑝+𝛼+𝛾+𝛽₁𝑘Θ₁+𝛽₂𝑘+𝛽₃𝑘×𝛼+𝜇+𝛽₃𝑘𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘+𝜇𝛼+𝛾+𝛽₃𝑘⁻¹<0, (41)𝑓(Θ₁,0)>0 and 𝑓(Θ₁,1)<0 for any Θ₁∈(0,1), and 𝑓(Θ₁,Θ₂) is continuous about Θ₂, then there is at least Θ₂(Θ₂∈(0,1)) satisfying 𝑓(Θ₁,Θ₂)=0; substituting Θ₁ and Θ₂ to (37), we have a positive equilibrium of system (5) 𝐸₂(𝑖₁,𝑖₂,…,𝑖_𝑛,𝑎₁,𝑎₂,…,𝑎_𝑛), where 𝑖_𝑘=𝜇𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂×𝛼+𝜇+𝛽₃𝑘Θ₂𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂+𝜇𝛼+𝛾+𝛽₃𝑘Θ₂⁻¹𝑎,𝑘=1,…,𝑛,_𝑘=𝛼+𝛽₃𝑘Θ₂𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂×𝛼+𝜇+𝛽₃𝑘Θ₂𝑝+𝛽₁𝑘Θ₁+𝛽₂𝑘Θ₂+𝜇𝛼+𝛾+𝛽₃𝑘Θ₂⁻¹,𝑘=1,…,𝑛. (42) We summarize these results in the following theorem.

Theorem 6. When 𝛼≠0, system (5) has at least a positive equilibrium 𝐸₂.

4. Numerical Simulation

In this section, we will perform a series of numerical simulations to verify the mathematical analysis on a scale-free network with power-law distribution (𝑝(𝑘)=8𝑘⁻⁴, 𝑘=2,3,…,150, ⟨𝑘⟩=9.1824). Parameters in system (5) are chosen as 𝑝=0.001, 𝛽₁=0.0001, 𝛽₂=0.0004, 𝛾=0.0001, and 𝜇=0.015.

We give numerical simulations about the network model (5) with different degrees and different initial value in Figures 1 and 2. In Figure 1, when 𝛼=0, 𝛽₃=0.0004, and 𝑅=0.8915, Figures 1(a)–1(c) show that system (5) has a stable trivial equilibrium; namely, new products cannot diffuse on the market; when 𝛼=0, 𝛽₃=0.01, and 𝑅=1.7310, Figures 1(d)–1(f) show that system (5) has a positive equilibrium; new products can diffuse on the market. Letting 𝛼=0.005, Figure 2 gives time series of 𝑎₅₀, 𝑎₁₀₀, and 𝑎₁₅₀, and it indicates that system (5) has a positive equilibrium when 𝛼=0.005.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1 

Time series when 𝛼=0. (a)–(c) time series with different degrees, respectively, when 𝑅=0.8915; (d)–(f) time series with different degrees, respectively, when 𝑅=1.7310.

(a)

(b)

(c)

Figure 2 

Time series with different degrees when 𝛼=0.005.

In Figure 3, (a) and (b) indicate that the relative density of adopters always increases with the enhancement of mass media' influence; furthermore, by comparing Figure 3(a) to Figure 3(b), we can find that the bigger the degree of adopters is, the smaller the increment of adopters with the enhancement of mass media' influence is; however, Figures 3(c) and 3(d) manifest that the enhancement of mass media' influence can increase adopters only in the early stage as the adopters' degree continuously increases; moreover, in Figure 3(d), we find that mass media is effective only in [0,𝜏]. These simulation results can be theoretically explained as follows: 𝑖_𝑘 exponentially decreases with the increment of 𝛼 and 𝑘 in system (5), so (𝛼+𝛽₃𝑘Θ₂(𝑡))𝑖_𝑘(𝑡) decreases with the increment of 𝛼 and 𝑘; namely, 𝑑𝑎_𝑘(𝑡)/𝑑𝑡 decreases when 𝛼 or 𝑘 increases. In other words, the larger 𝛼 is (or the larger 𝑘), the smaller 𝑖_𝑘 is, and the smaller 𝑑𝑎_𝑘(𝑡)/𝑑𝑡 is, the faster 𝑎_𝑘 decrease is if 𝑑𝑎_𝑘(𝑡)/𝑑𝑡<0 and the slower 𝑎_𝑘 increase is if 𝑑𝑎_𝑘(𝑡)/𝑑𝑡>0. In view of economics, Figures 3(a)–3(d) imply two different propagation strategies in the decision-making stage of new products diffusion process; for those majority of nodes having small degree in scale-free networks, it is very effective and permanent to persuade them to adopt new products by mass media channels, despite the fact that the effect in the late period is less significant than the one in the early stage, so businesses should propagate chronically new products by mass media among them. For those few hubs’ nodes with larger degree, mass media' effect is only very temporary; if businesses propagate and persuade people to adopt new products only in [0,𝜏], it will be useful to save resources for businesses; when 𝑡>𝜏, businesses have to change advertisement strategies.

(a)

(b)

(c)

(d)

(e)

Figure 3 

(a)–(d) time series of 𝑎5, 𝑎10, 𝑎15, and 𝑎25, respectively, when 𝛼 changes from 0 to 0.05 (𝛼=0,0.0005,0.005,0.05); (e) time series of 𝑎𝑘(𝑘=5,10,15,25) when 𝛼=0.0005.

In addition, fixing the degree of nodes in (a)–(d) of Figure 3, it is found that the peak of 𝑎_𝑘 increases with increment of 𝛼; namely, the peak of adopters will increase as the advertisement is enhanced. In Figure 3(e), by fixing 𝛼, we can find that the larger the degree and the larger the relative density of adopters, the smaller the diffusion time; Figure 3(e) means that hubs’ nodes are more easy to become adopters than nodes with the small degrees, so businesses should propagate timely and effectively new products among them only in some period of time 𝜏, and they should take different advertising strategies for hubs’ nodes and nodes with the small degrees.

In Figure 4, we keep track of the mean density 𝑎 versus times, and we find that the mean density of networks 𝑎 increases with the increment of 𝛼; that is to say, the number of adopters increases when mass media' influence is enhanced, and it is reasonable. Moreover, we can also see that the effect of mass media is more pronounced before the stable state of the adopters than after the stable state; this indicates that the enterprise should take persuasive advertisement strategy to tell consumers why to choose a particular brand; after adopters tend to stable states, namely, once the brand is popularized, the enterprise should take remindful advertisement to frequently remind consumer to have such a product, and it is useful to save advertising cost.

Figure 4 

Time series of the mean density 𝑎 when the change of 𝛼 is as shown in the figure.

5. Conclusion and Discussion

In this paper, we have proposed a complex networks model with the awareness stage and the decision-making stage to expound adoption processes. Unlike the classical diffusion model, where population contacts are homogeneous mixing, populations have complex and heterogeneous connectivity patterns. We study the existence and stability of equilibrium when the influence of mass media is neglected or considered in the decision-making stage. In simulations of networks with the power-law distribution, we analyze the effect of the network structure and mass media in the decision-making stage on the density of adopters and obtain different propagation strategies to persuade individuals to adopt the products. From the local point of view, businesses should take different propagation strategies for hub nodes and other nodes with the small degree. From the global point of view, because the effect of mass media is more significant before the arrival of the stable state of the adopters than after the arrival of the stable state, businesses should change propagation strategies after the average density of adopters in the network reaches the stable state.

It is worth noting here that we only consider the influence of nodes’ degree on diffusion; however, in recent years there has been considerable interest within the physics community in the network structures, such as clusters, path length, and centrality indices [24–29]; if other network structures are considered in the model, it could have a profound influence.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by a Grant from the National Natural Science Foundation of China (no. 11171314), National Natural Science Foundation for Youths (no. 11201434), Shanxi Province Natural Science Foundation (no. 2012011002-1), and Scientific Research Item for the Returned Overseas Chinese Scholars of Shanxi Province (2010-074).