Heterogeneity and Network Structure in the Dynamics of Diffusion: Comparing Agent-Based and Differential Equation Models
Hazhir Rahmandad hazhir@794f30c8e518964bce847c37
Virginia Tech, Falls Church, VA 22043
John Sterman jsterman@794f30c8e518964bce847c37
MIT Sloan School of Management, Cambridge MA 02142
Revision of August 2007
Forthcoming
Management Science
We thank Reka Albert, Joshua Epstein, Rosanna Garcia, Ed Kaplan, David Krackhardt, Marc Lipsitch, Nelson Repenning, Perwez Shahabuddin, Steve Strogatz, Duncan Watts, Larry Wein, members of the MIDAS Group and 2006 MIDAS workshop, the associate editor and referees, and seminar participants at MIT, the 2004 NAACSOS conference and 2004 International System Dynamics Conference for helpful comments. Ventana Systems and XJ Technologies generously provided their simulation software and technical support. Financial support provided by the Project on Innovation in Markets and Organizations at the MIT Sloan School.
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Heterogeneity and Network Structure in the Dynamics of Diffusion: Comparing Agent-Based and Differential Equation Models
Abstract
When is it better to use agent-based (AB) models, and when should differential equation (DE) models be used? Where DE models assume homogeneity and perfect mixing within compartments, AB models can capture heterogeneity across inpiduals and in the network of interactions among them. AB models relax aggregation assumptions but entail computational and cognitive costs that may limit sensitivity analysis and model scope. Because resources are limited, the costs and benefits of such disaggregation should guide the choice of models for policy analysis. Using contagious disease as an example, we contrast the dynamics of a stochastic AB model with those of the analogous deterministic compartment DE model. We examine the impact of inpidual heterogeneity and different network topologies, including fully connected, random, Watts-Strogatz small world, scale-free, and lattice networks. Obviously deterministic models yield a single trajectory for each parameter set, while stochastic models yield a distribution of outcomes. More interestingly, the DE and mean AB dynamics differ for several metrics relevant to public health, including diffusion speed, peak load on health services infrastructure and total disease burden. The response of the models to policies can also differ even when their base case behavior is similar. In some conditions, however, these differences in means are small compared to variability caused by stochastic events, parameter uncertainty and model boundary. We discuss implications for the choice among model types, focusing on policy design. The results apply beyond epidemiology: from innovation adoption to financial panics, many important social phenomena involve analogous processes of diffusion and social contagion.
Keywords: Agent Based Models, Networks, Scale free, Small world, Heterogeneity, Epidemiology, Simulation, System Dynamics, Complex Adaptive Systems, SEIR model
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Spurred by growing computational power, agent-based modeling (AB) is increasingly applied to physical, biological, social and economic problems previously modeled with nonlinear differential equations (DE). Both approaches have yielded important insights. In the social sciences, agent models explore phenomena from the emergence of segregation to organizational evolution to market dynamics (Schelling 1978; Levinthal and March 1981; Carley 1992; Axelrod 1997; Lomi and Larsen 2001; Axtell, Axelrod, Epstein and Cohen 2002; Epstein 2006; Tesfatsion 2002). Differential and difference equation models, also known as compartmental models, have an even longer history in social science, including innovation diffusion (Mahajan, Muller and Wind 2000) and epidemiology (Anderson and May 1991).
When should AB models be used, and when are DE models appropriate? Each method has strengths and weaknesses. The importance of each depends on the model purpose. Nonlinear DE models can easily encompass a wide range of feedback effects, but typically aggregate agents into a relatively small number of states (compartments). For example, innovation diffusion models may aggregate the population into categories including unaware, aware, in the market, adopters, and so on (Urban, Hauser and Roberts 1990; Mahajan et al. 2000). However, within each compartment people are assumed to be homogeneous and well mixed; the transitions among states are modeled as their expected value, possibly perturbed by random events. In contrast, AB models can readily include heterogeneity in inpidual attributes and in the network structure of their interactions; like DE models, AB models can be deterministic or stochastic and can capture feedback effects.
The granularity of AB models comes at some cost. First, the extra complexity significantly increases computational requirements, constraining the ability to conduct sensitivity analysis. A second cost of agent-level detail is the cognitive burden of understanding model behavior. Linking the behavior of a model to its structure becomes more difficult as model complexity grows. Finally, limited time and resources force modelers to trade off disaggregate detail and the breadth of the model boundary. Model boundary here stands for the richness of the feedback structure captured endogenously in the model (Meadows and Robinson 1985, Sterman 2000). For example, an agent-based demographic model may portray each inpidual separately but assume exogenous fertility and mortality; such a model has a narrow boundary. In contrast, an aggregate model may lump the entire population into a single compartment, but model fertility and mortality as functions
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of food per capita, health care, pollution, norms for family size, etc., each of which, in turn, are modeled endogenously; such a model has a broad boundary. DE and AB models may in principle fall anywhere on these dimensions of disaggregation and scope. In particular, there is no intrinsic limitation that prevents AB models from incorporating behavioral feedback effects or encompassing a broad model boundary. In practice, however, where time, budget, and computational resources are limited, modelers must trade off disaggregate detail and breadth of boundary. Choosing wisely is central in selecting appropriate methods for any problem.
The stakes are large. Consider potential bioterror attacks. Kaplan, Craft, and Wein (2002) used a deterministic nonlinear DE model to examine smallpox attack in a large city, comparing mass vaccination (MV), in which essentially all people are vaccinated after an attack, to targeted vaccination (TV), in which health officials trace and immunize those contacted by potentially infectious inpiduals. Capturing vaccination capacity and logistics explicitly, they conclude MV significantly reduces casualties relative to TV. In contrast, using different AB models, Eubank et al. (2004) and Halloran et al. (2002) conclude TV is superior, while Epstein et al. (2004) favor a hybrid strategy. The many differences among these models make it difficult to determine whether the conflicting conclusions arise from relaxing the perfect mixing and homogeneity assumptions of the DE (as argued by Halloran et al. 2002) or from other assumptions such as the size of the population (ranging from 10 million for the DE model to 2000 in Halloran et al. to 800 in Epstein et al.), other parameters, or boundary differences such as whether capacity constraints on immunization are included (Koopman 2002; Ferguson et al. 2003; Kaplan and Wein 2003). Kaplan and Wein (2003) and Kaplan, Craft and Wein (2003) show that their DE model closely replicates the Halloran et al. AB results when simulated with the Halloran et al. parameters, including vaccination rates, population and initial attack size, concluding that parameterization accounts for the different conclusions, not differences in mixing and homogeneity.
Here we carry out controlled experiments to compare AB and DE models in the context of contagious disease. We choose disease diffusion for four reasons. First, the dynamics of contagion involve important characteristics of complex systems, including positive and negative feedbacks, time delays, nonlinearities, stochastic events, and inpidual heterogeneity. Second, network topologies linking inpiduals are important in the diffusion process (Davis 1991; Watts and
