OPERATIONS RESEARCH Vol. 58, No. 5, September–October 2010, pp. 1450–1468 issn 0030-364X  eissn 1526-5463  10  5805  1450 inf orms ® doi 10.1287/opre.1100.0818 © 2010 INFORMS Aggregate Diffusion Dynamics in Agent-Based Models with a Spatial Structure Gadi Fibich, Ro’i Gibori Department of Applied Mathematics, Tel Aviv University, Tel Aviv 69978, Israel {fibich@tau.ac.il, gibori@platonix.com} We explicitly calculate the aggregate diffusion dynamics in one-dimensional agent-based models of adoption of new prod- ucts, without using the mean-field approximation. We then introduce a clusters-dynamics approach, and use it to derive an analytic approximation of the aggregate diffusion dynamics in multidimensional agent-based models. The clusters-dynamics approximation shows that the aggregate diffusion dynamics does not depend on the average distance between individuals, but rather on the expansion rate of clusters of adopters. Therefore, the grid dimension has a large effect on the aggregate adoption dynamics, but a small-world structure and heterogeneity among individuals have only a minor effect. Our results suggest that the one-dimensional model and the Bass model provide a lower bound and an upper bound, respectively, for the aggregate diffusion dynamics in agent-based models with “any” spatial structure. Subject classifications : agent-based model; cellular automata; Bass model; diffusion; new products; small-world; mean-field approximation; heterogeneity. Area of review : Marketing Science. History : Received October 2008; revisions received September 2009, November 2009; accepted December 2009. Published online in Articles in Advance July 14, 2010. 1. Introduction Diffusion of new products is a fundamental problem in marketing. This problem has been studied in diverse areas such as retail service, industrial technology, agriculture, and educational, pharmaceutical, and consumer-durables markets (Mahajan et al. 1993). Typically, the diffusion pro- cess begins when the product is introduced into the market, and progresses through a series of adoption events. The first quantitative model of diffusion of new products was the Bass model (Bass 1969). This model inspired a huge body of theoretical and empirical research, and was selected as one of the 10 most-cited papers in the 50-year history of Management Science (Hopp, ed., 2004). In the Bass model, the adoption rate is given by dnt dt = M - nt  p + q M nt   n0 = 0 (1) where nt is the number of individuals that adopted the product by time t , and M is the population size. The param- eters p and q describe the likelihood of an individual to adopt the product due to external influences such as mass media or commercials, and due to internal influences by other individuals who have already adopted the product, respectively. Because the hazard of adoption of each indi- vidual is p + qn/M, each individual is affected by both external and internal influences. Equation (1) can be solved explicitly, yielding n Bass t = M 1 - e -p+qt 1 + q/pe -p+qt  or, equivalently, f Bass t = 1 - e -p+qt 1 + q/pe -p+qt  (2) where f t = nt/M is the fraction of adopters at time t . Empirically, the Bass model was found to capture the S-shape of the adoption curve of various products. Typical values for the parameters were found to be p = 003/year and q = 038/year, with p often less than 001/year and q typically in the range 03–05/year (Mahajan et al. 1995). The Bass model is an aggregate model, i.e., it describes the diffusion in terms of the behavior of the entire popu- lation. Therefore, a considerable research effort has been devoted to modeling the individual adoption behavior, and to analyzing how it affects the aggregate diffusion process. Thus, Sinha and Chandrashekaran (1992) studied individ- ual adoption behavior using hazard modeling on empirical data. Subsequently, Van den Bulte and Lilien (2001) used hazard modeling to study social contagion with social net- work data. Bronnenberg and Mela (2004) and Bell and Song (2007) have done this with spatial data. Beginning with Goldenberg et al. (2000), this line of research has been carried out by using agent-based (cellular-automata) mod- els to compute numerically the aggregate adoption curve from the individual-based behaviors, which are based on external and internal effects. In the Bass model, the rate of new adoptions due to internal effects is equal to q/MM - nn. This expres- sion is based on the assumption that each of the M - n 1450 INFORMS holds copyright to this article and distributed this copy as a courtesy to the author(s). Additional information, including rights and permission policies, is available at http://journals.informs.org/.