## Introduction

The link between system structure and dynamic behavior is one of the defining elements of system dynamics (SD). In a sense, a simulation model can be viewed as an explicit and consistent theory of the behavior it exhibits (Kampmann and Oliva, 2009; Oliva, 2003). Although this point of view has certain merits, not least of which is the fact that it lifts the discussion from outcomes to causes and from events to underlying structure (Forrester, 1961; Sterman, 2000), system dynamicists often need more compact explanations of the system's behavior. In fact, most dynamic modeling projects report their results with simple explanations, typically in terms of dominant feedback loops and, occasionally, external driving forces to the system that produce the salient features of the behavior.

For simple systems with relatively few variables, it is usually easy to use intuition and trial-and-error simulation experiments to explain the dynamic behavior as resulting from particular feedback loops. In larger systems, this method becomes increasingly difficult and the risk of incorrect explanations rises accordingly. There is a need, therefore, for analytical methods that provide consistency and rigor to this process.

Eigenvalue elasticity analysis (EEA) is a set of methods to assess the effect of structure on behavior in dynamic models (Kampmann, 2012; Kampmann and Oliva, 2006; Oliva, 2015). It works by considering model behavior as a combination of characteristic behavior modes and assessing the relative importance of particular elements of system structure in influencing these behavior modes. Elements of the model structure that have a large influence on particular behaviors can provide important clues to identify areas for further testing and policy design. EEA uses linear systems theory to (i) decompose the observed behavior into its constituent *behavior modes*, such as oscillation, growth and exponential adjustment, and (ii) outline how a particular behavior mode and its appearance in a given system variable depend upon particular parameters and structural elements (links and loops) in the system. In this manner, it provides a precise account of the relationship between structure and behavior through mathematical rigor absent in the experimental methods normally used in the field (Duggan and Oliva, 2013).

The last decade and a half has seen significant efforts to develop and automate methods for identifying structural dominance in SD models (see Duggan and Oliva, 2013, for an overview of this literature). To date, however, the interpretation and testing of these methods have been with small deterministic models (fewer than five stocks) that show smooth behavioral transitions (e.g. Gonçalves, 2009; Güneralp, 2006; Kampmann and Oliva, 2006; Mojtahedzadeh, 2011; Mojtahedzadeh *et al.*, 2004; Saleh *et al.*, 2010). While the analysis of simple and stable models is an obvious first step in providing proof of concept, the methods have become stable enough to be tested in a wider range of models. In this paper I report the findings from expanding the application domain of these methods in two dimensions: increasing model size and incorporating stochastic variance in some model variables. These expansions are important tests of the applicability of EEA, as most realistic applications of SD require higher-order models than the demonstration models used until now. Furthermore, an intrinsic part of testing SD models and assessing the robustness of proposed policies is to subject these models to expected random noise (Sterman, 2000, section 4.3.2 and Appendix B). As noise can have a significant effect on model performance and hide otherwise easy-to-observe core behavior modes (thus making the use of exploratory methods for structural analysis more difficult), it is important to assess whether these methods can work under noisy conditions.

While I only show the results of the analysis of one large and stochastic model, the results are promising. I find that the methods work as predicted, that they generate insights that are consistent with the existing explanations for the behavior of the tested model, and that they do so in an efficient way.

The next section provides an overview of the eigenvalue elasticity methods as well as the most common strategies to interpret the results from the methods. The overview is followed by a brief presentation of the model used for our analysis, as well as a description of the modifications that had to be performed for the methods to work. The following section describes the strategy followed to select operating points 1 to perform the model linearization and analysis. EEA results and a comparison with previous analyses, narratives and policy recommendations are presented in the next section. The paper concludes by summarizing findings and implications.