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A stochastic two-stage innovation diffusion model on a lattice

Part of: Special processes Markov processes

Published online by Cambridge University Press:  09 December 2016

Cristian F. Coletti ,
Karina B. E. de Oliveira  and
Pablo M. Rodriguez
Cristian F. Coletti*
Affiliation:
Universidade Federal do ABC
Karina B. E. de Oliveira*
Affiliation:
Universidade de São Paulo
Pablo M. Rodriguez*
Affiliation:
Universidade de São Paulo
*
* Postal address: Centro de Matemática, Computação e Cognição, Universidade Federal do ABC, Avenida dos Estados, 5001 Bangu, Santo André, São Paulo, Brasil, . Email address: cristian.coletti@ufabc.edu.br
** Postal address: Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Av. Trabalhador são-carlense 400, Centro, CEP 13560-970, São Carlos, SP, Brasil.
** Postal address: Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Av. Trabalhador são-carlense 400, Centro, CEP 13560-970, São Carlos, SP, Brasil.
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Abstract

We propose a stochastic model describing a process of awareness, evaluation, and decision making by agents on the d-dimensional integer lattice. Each agent may be in any of the three states belonging to the set {0, 1, 2. In this model 0 stands for ignorants, 1 for aware, and 2 for adopters. Aware and adopters inform its nearest ignorant neighbors about a new product innovation at rate λ. At rate α an agent in aware state becomes an adopter due to the influence of adopters' neighbors. Finally, aware and adopters forget the information about the new product, thus becoming ignorant, at rate 1. Our purpose is to analyze the influence of the parameters on the qualitative behavior of the process. We obtain sufficient conditions under which the innovation diffusion (and adoption) either becomes extinct or propagates through the population with positive probability.

Keywords

Interacting particle systeminnovation diffusionstochastic modelBass modelcontact processoriented percolation

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60K10: Applications (reliability, demand theory, etc.) 60J28: Applications of continuous-time Markov processes on discrete state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 4 , December 2016 , pp. 1019 - 1030
Copyright
Copyright © Applied Probability Trust 2016 

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References

[1] Agliari, E.,Burioni, R.,Cassi, D. and Neri, F. M. (2010).Word-of-mouth and dynamical inhomogeneous markets: an efficiency measure and optimal sampling policies for the pre-launch stage.IMA J. Manag. Math. 21,67–83.Google Scholar
[2] Agliari, E.,Burioni, R.,Cassi, D. and Neri, F. M. (2006).Efficiency of information spreading in a population of diffusing agents.Phys. Rev. E 73,046138.Google Scholar
[3] De Arruda, G. F.,Lebensztayn, E.,Rodrigues, F. A. and Rodríguez, P. M. (2015).A process of rumor scotching on finite populations.R. Soc. Open Sci. 2,150240.CrossRefGoogle Scholar
[4] Bass, F. M. (1969).A new product growth for model consumer durables.Manag. Sci. 15,215–227.Google Scholar
[5] Bass, F. M. (2004).Comments on `A new product growth for model consumer durables the Bass model'.Manag. Sci. 50,1833–1840.Google Scholar
[6] Bezuidenhout, C. and Grimmett, G. (1990).The critical contact process dies out.Ann. Appl. Prob. 18,1462–1482.Google Scholar
[7] Coletti, C. F.,Rodríguez, P. M. and Schinazi, R. B. (2012).A spatial stochastic model for rumor transmission.J. Stat. Phys. 147,375–381.Google Scholar
[8] Comets, F.,Delarue, F. and Schott, R. (2014).Information transmission under random emission constraints.Combin. Prob. Comput. 23,973–1009.Google Scholar
[9] Comets, F.,Gallesco, C.,Popov, S. and Vachkovskaia, M. (2016).Constrained information transmission on Erdös‒Rényi graphs.Markov Proc. Relat. Fields 22,111–138.Google Scholar
[10] Durrett, R. (1995).Ten lectures on particle systems.In Lectures on Probability Theory.Springer,Berlin.Google Scholar
[11] Garber, T.,Goldenberg, J.,Libai, B. and Muller, E. (2004).From density to destiny: using spatial dimension of sales data for early prediction of new product success.Marketing Sci. 23,419–428.CrossRefGoogle Scholar
[12] Goldenberg, J.,Libai, B. and Muller, E. (2001).Talk of the network: a complex systems look at the underlying process of word-of-mouth.Marketing Lett. 12,211–223.Google Scholar
[13] Goldenberg, J.,Libai, B. and Muller, E. (2002).Riding the saddle: how cross-market communications can create a major slump in sales.J. Marketing 66,1–16.Google Scholar
[14] Grimmett, G. (2010).Probability on Graphs: Random Processes on Graphs and Lattices.Cambridge University Press.Google Scholar
[15] Harris, T. E. (1972).Nearest-neighbor Markov interaction processes on multidimensional lattices.Adv. Math. 9,66–89.Google Scholar
[16] Isham, V.,Harden, S. and Nekovee, M. (2010).Stochastic epidemics and rumours on finite random networks.Phys. A 389,561–576.Google Scholar
[17] Konno, N.,Schinazi, R. B. and Tanemura, H. (2004).Coexistence results for a spatial stochastic epidemic model.Markov Proc. Relat. Fields 10,367–376.Google Scholar
[18] Lebensztayn, E.,Machado, F. P. and Rodríguez, P. M. (2011).On the behaviour of a rumour process with random stifling.Environ. Modell. Softw. 26,517–522.Google Scholar
[19] Lebensztayn, E.,Machado, F. and Rodríguez, P. M. (2011).Limit theorems for a general stochastic rumour model.SIAM J. Appl. Math. 71,1476–1486.Google Scholar
[20] Liggett, T. M. (1995).Improved upper bounds for the contact process critical value.Ann. Prob. 23,697–723.Google Scholar
[21] Liggett, T. M. (1999).Stochastic Interacting Systems: Contact, Voter, and Exclusion Processes.Springer,Berlin.Google Scholar
[22] McCullen, N. J. et al. (2013).Multiparameter models of innovation diffusion on complex networks.SIAM J. Appl. Dyn. Syst. 12,515–532.Google Scholar
[23] Mountford, T. S. (1993).A mestastable result for the finite multidimensional contact process.Canad. Math. Bull. 36,216–226.CrossRefGoogle Scholar
[24] Niu, S. C. (2002).A stochastic formulation of the Bass model of new-product diffusion.Math. Probl. Eng. 8,249–263.Google Scholar
[25] Rogers, E. M. (1962).Diffusion of Innovations.The Free Press,New York.Google Scholar
[26] Van den Berg, J.,Grimmett, G. R. and Schinazi, R. B. (1998).Dependent random graphs and spatial epidemics.Ann. Appl. Prob. 8,317–336.Google Scholar
[27] Wang, W.,Fergola, P.,Lombardo, S. and Mulone, G. (2006).Mathematical models of innovation diffusion with stage structure.Appl. Math. Model. 30,129–146.Google Scholar