`Journal of Applied MathematicsVolume 2014 (2014), Article ID 940623, 12 pageshttp://dx.doi.org/10.1155/2014/940623`
Research Article

## Modeling and Analysis of New Products Diffusion on Heterogeneous Networks

Shuping Li1,2 and Zhen Jin2,3

1School of Mechatronic Engineering, North University of China, Taiyuan, Shan’xi 030051, China

2Department of Mathematics, North University of China, Taiyuan, Shan’xi 030051, China

3Complex Systems Research Center, Shanxi University, Taiyuan, Shan’xi 030006, China

Received 5 February 2014; Accepted 22 April 2014; Published 28 May 2014

Copyright © 2014 Shuping Li and Zhen Jin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 [13]. The following equation illustrates the classical Bass model [4]: where is the number of cumulative adopters, is the coefficient of external influence, is the coefficient of internal influence, and 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 [57]. 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 [1117]; 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 distinct groups of sizes () such that each individual in group has exactly contacts per day. If the whole population size is (), then the probability that a uniformly chosen individual has contacts is , which is called the degree distributions of the network. Let denote the number of those individuals who have not been aware of the product within group at time , let denote the number of those individuals who have been aware of the information about the product but have not yet adopted it within group at time , and let denote the number of those individuals who have adopted the product at time . 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 ordinary differential equations representing the product diffusion dynamics on the scale-free network: where All parameters are positive constants and the meaning of parameters is summarized in Table 1. When 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 , , and at time by , , and , respectively; then, system (2) can be rewritten as with the normalization condition , and , . Using this condition, system (4) is reduced to Prior to discussing the stability of system (5), we first study its variant set.

Let , ,  , ,  . Denoting that , we study system (5) for .

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

Proof. We will show that all the solutions starting from any initial value of system (5) satisfy . Denote that for some , for some ,where . Let the “outer normals” be denoted by and .

For arbitrary compact set , Nagumo had proved that is invariant for , if, at each point in (the boundary of ), the vector is tangent or pointing into the set [21, 22]. We can easily apply the result here, since is an -dimensional rectangle. Through Nagumo's result, it is not difficult to obtain that 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 ,(2)persuasive advertisements are considered in the decision-making stage, which means .

Case I (). When , 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 We can get the equivalent system of system (7) as follows: and it is obvious that satisfies the second formula in (8); substituting into the first formula in (8), we obtain and a self-consistency equation about as follows: Through some calculation, we have that According to formula (11), we can know that (10) has at least a positive solution in ; formula (12) shows that the function is a rigorous monotone increasing function, and formula (13) means that the function is convex, so (10) must has a unique positive solution in , and system (5) has a trivial solution , where The above results can be summarized in the following theorem.

Theorem 2. When , system (5) has a unique trivial equilibrium .

Furthermore, one will prove the local stability and global stability of the equilibrium . The Jacobin matrix of system (5) at is wherehere

Next, for studying the local stability of the equilibrium , we first estimate eigenvalues by carrying out the similarity transformation about the matrix . Letting , , , then Obviously, and have same eigenvalues. We can obtain using formula (14), so , ; furthermore, , so 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 has negative real part; that is to say, every eigenvalue of the matrix has negative real part.

To find eigenvalues of the matrix , we carry out similarity transformation to the matrix ; namely, the column multiplied by is added to the column, and the row multiplied by is added to the row, . Then we obtain the similarity matrix as follows: It is straightforward that has negative characteristic roots. When , is locally asymptotically stable; otherwise, it is not stable. Letting , we have the following theorem.

Theorem 3. When , if , is locally asymptotically stable, and if , is unstable.

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

Theorem 4. When , if , is globally asymptotically stable.

Proof. Considering system (5) with , by the second formula in system (5), we have and we consider the following auxiliary system: Multiplying formula (22) by and summing over , we obtain and the solution of (23) tends to 0 when , since system (23) is a quasi-monotone system; by the comparison theorem, in system (5), we consider the limit equation of the first equation of system (5): the above formula has a unique equilibrium and it is locally asymptotically stable. So , . is globally attractive when ; this, combined with Theorem 3, implies that is globally asymptotically stable when .

For proving the existence of the positive equilibrium of system (5) with , we discuss the existence of the positive solution of (8); by (8), we obtain the following equations about , : In consideration of , , from formula (25), we can know that . By the identical transformation of the second equation in formula (25), we have where and is continuous in . It is easy to testify that when , , and is continuous about in , so there must exist some sufficient small numbers and fully close to 1 that satisfy and , and is continuous about in ; similarly, , where is fully close to 1; thus there is at least one point satisfying , where . According to the implicit function theorem, equation can establish only a continuous function in the neighbor domain of : Substituting (28) into the first equation in formula (25), we can get the following equation: Noting that , , by the continuity of function , there at least exists satisfying formula (29), so formula (25) at least has a positive solution   , which means that system (5) at least has a positive equilibrium , where

We will subsequently show that system (5) has a unique positive equilibrium. We denote that , , and assume that and are two constant solutions of system (5). If , then there exists at least one , , such that , where is the component of the vector . Without loss of generality, we assume that and moreover that for all . Since and are constant solutions of system (5), and if , we obtain that or if , we have where After equivalent deformation, it follows that or But , for all , and ; thus from the above equalities we get This is a contradiction. Therefore, system (5) has a unique positive equilibrium. that is .

Hence, the conclusion follows.

Theorem 5. When , if , system (5) has a unique positive equilibrium .

Case II (). When , we can get equations that the steady state of system (5) satisfies and self-consistency equations about , are obtained such that From formula (38), we can obtain , . By the identical transformation of the second equation in formula (38), we can obtain where and since and for any , and is continuous about , then there is at least satisfying ; substituting and to (37), we have a positive equilibrium of system (5) , where We summarize these results in the following theorem.

Theorem 6. When , system (5) has at least a positive equilibrium .

#### 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 (, , ). Parameters in system (5) are chosen as , , , , and .

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 , , and , Figures 1(a)1(c) show that system (5) has a stable trivial equilibrium; namely, new products cannot diffuse on the market; when , , and , Figures 1(d)1(f) show that system (5) has a positive equilibrium; new products can diffuse on the market. Letting , Figure 2 gives time series of , , and , and it indicates that system (5) has a positive equilibrium when .

Figure 1: Time series when . (a)–(c) time series with different degrees, respectively, when ; (d)–(f) time series with different degrees, respectively, when .
Figure 2: Time series with different degrees when .

Figure 3: (a)–(d) time series of , , , and , respectively, when changes from to (); (e) time series of when .

In addition, fixing the degree of nodes in (a)–(d) of Figure 3, it is found that the peak of increases with increment of ; namely, the peak of adopters will increase as the advertisement is enhanced. In Figure 3(e), by fixing , 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 , 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 versus times, and we find that the mean density of networks increases with the increment of ; 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 when the change of 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 [2429]; 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).

#### References

1. F. M. Bass, “A new product growth for model consumer durables,” Management Science, vol. 50, no. 12, pp. 1825–1832, 2004.
2. V. Mahajan, E. Muller, and R. A. Kerin, “Introduction strategy for a new products with positive and negative word-of-mouth,” Management Science, vol. 30, pp. 1389–1404, 1984.
3. V. Mahajan and R. Peterson, Models for Innovation Diffusion, Sage Publication, Beverly Hills, Calif, USA, 1985.
4. V. Mahajan, E. Muller, and F. Bass, “New Product Diffusion Models in Marketing: a Review and Directions for Research,” Journal of Marketing, vol. 54, pp. 1–26., 1990.
5. Y. Yu, W. Wang, and Y. Zhang, “An innovation diffusion model for three competitive products,” Computers & Mathematics with Applications. An International Journal, vol. 46, no. 10-11, pp. 1473–1481, 2003.
6. Y. Yu and W. Wang, “Global stability of an innovation diffusion model for $n$ products,” Applied Mathematics Letters, vol. 19, no. 11, pp. 1198–1201, 2006.
7. H. S. Yan and K. P. Ma, “Competitive diffusion process of repurchased products in knowledgeable manufacturing,” European Journal of Operational Research, vol. 208, no. 3, pp. 243–252, 2011.
8. W. Wang, P. Fergola, and C. Tenneriello, “Innovation diffusion model in patch environment,” Applied Mathematics and Computation, vol. 134, no. 1, pp. 51–67, 2003.
9. W. Wang, P. Fergola, S. Lombardo, and G. Mulone, “Mathematical models of innovation diffusion with stage structure,” Applied Mathematical Modelling, vol. 30, no. 1, pp. 129–146, 2006.
10. S. Li and Z. Jin, “Global dynamics analysis of homogeneous new products diffusion model,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 158901, 6 pages, 2013.
11. R. Pastor-Satorras and A. Vespignani, “Epidemic spreading in scale-free networks,” Physical Review Letters, vol. 86, no. 14, pp. 3200–3203, 2001.
12. R. Pastor-Satorras and A. Vespignani, “Epidemic dynamics in finite size scale-free networks,” Physical Review E, vol. 65, no. 3, Article ID 035108, 2002.
13. X. Fu, M. Small, D. M. Walker, and H. Zhang, “Epidemic dynamics on scale-free networks with piecewise linear infectivity and immunization,” Physical Review E, vol. 77, no. 3, Article ID 036113, 2008.
14. J. Z. Wang, Z. R. Liu, and J. Xu, “Epidemic spreading on uncorrelated heterogenous networks with non-uniform transmission,” Physica A, vol. 382, no. 2, pp. 715–721, 2007.
15. T. Zhou, J. G. Liu, W. J. Bai, et al., “Behaviors of susceptible-infected epidemics on scale-free networks with identical infectivity,” Physical Review E, vol. 74, Article ID 056109, 2006.
16. J. Joo and J. L. Lebowitz, “Behavior of susceptible-infected-susceptible epidemics on heterogeneous networks with saturation,” Physical Review E, vol. 69, no. 6, Article ID 066105, 2004.
17. H. Zhang and X. Fu, “Spreading of epidemics on scale-free networks with nonlinear infectivity,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3273–3278, 2009.
18. M. Boguna, R. Pastor-satorras, and A. Vespignani, “Epidemic spreading in complex networks with degree correlations,” in Statistical Mechanics of Complex Networks, vol. 625 of Lecture Notes in Physics, pp. 127–147, 2003.
19. D. López-Pintado, “Diffusion in complex social networks,” Games and Economic Behavior, vol. 62, no. 2, pp. 573–590, 2008.
20. S. Kim, K. Lee, J. K. Cho, and C. O. Kim, “Agent-based diffusion model for an automobile market with fuzzy TOPSIS-based product adoption process,” Expert Systems with Applications, vol. 38, no. 6, pp. 7270–7276, 2011.
21. J. A. Yorke, “Invariance for ordinary differential equations,” Mathematical Systems Theory, vol. 1, pp. 353–372, 1967.
22. A. Lajmanovich and J. A. Yorke, “A deterministic model for gonorrhea in a nonhomogeneous population,” Mathematical Biosciences, vol. 28, pp. 221–236, 1976.
23. L. Li, “Some applications of the criteria for generalized diagonally dominant matrices,” International Journal of Pure and Applied Mathematics, vol. 7, no. 4, pp. 389–398, 2003.
24. S. N. Dorogovtsev and J. F. F. Mendes, “Evolution of networks,” Advances in Physics, vol. 51, pp. 1079–1187, 2002.
25. M. E. J. Newman, “The structure and function of complex networks,” SIAM Review, vol. 45, no. 2, pp. 167–256, 2003.
26. The Structure and Dynamics of Networks, Princeton University Press, Princeton, NJ, USA, 2006.
27. M. Molloy and B. Reed, “A critical point for random graphs with a given degree sequence,” Random Structures and Algorithms, vol. 6, pp. 161–179, 1995.
28. M. Molloy and B. Reed, “The size of the giant component of a random graph with a given degree sequence,” Combinatorics, Probability and Computing, vol. 7, no. 3, pp. 295–305, 1998.
29. M. E. J. Newman, S. H. Strogatz, and D. J. Watts, “Random graphs with arbitrary degree distributions and their applications,” Physical Review E, vol. 64, no. 2, Article ID 026118, 2001.