Dominant structure RESEARCH PROBLEMS. The concept of dominant structure. The need. George P. Richardson

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1 RESEARCH PROBLEMS Dominant structure George P. Richardson In this section the System Dynamics Review presents problems having the potential to stimulate system dynamics research. Articles may address realworld dynamic problems that could be approached fruitfully from the system dynamics perspective, or methodological problems affecting the field. A paper submitted to the Research Problems section should concisely motivate and define a problem and start a process of conceptualization or formulation that can open the way for further studies. Manuscripts not exceeding 2,000 words should be sent to George P. Richardson, Department of Mathematics and Computer Science, Wheaton College, Norton, MA 02766, U.S.A. The concept of dominant structure Underlying the formal, quantitative methods of system dynamics is the goal of understanding how the feedback structure of a system contributes to its dynamic behavior. Understandings are captured and communicated in terms of stocks and flows, the polarities of feedback loops interconnecting them, and shifts in the significance or dominance of various loops. However, there is a conspicuous gap in our literature between intuitive statements about loop dominance and precise statements about how to define and detect dominant structure. Intuitively, the term dominant structure refers to some subset of the stock-and-flow1 feedback structure of a model that is principally responsible for a particular pattern of model behavior. To move beyond intuition, two questions need to be answered: What do we mean by a particular pattern of behavior? and What do we mean by principally responsible? Precise answers would raise exciting possibilities. If we can define these terms formally and unambiguously, we might then be able to devise means of detecting, rapidly and with certainty, the dominant structures underlying the patterns of behavior exhibited by a model. A long-term goal of research into these questions, with potentially great significance for simulation modeling, is the widespread availability of simulation languages that would, as a matter of course, identify dominant structure. The need System dynamicists make use of the idea of dominant feedback structure in at least seven ways: I. Philosophy. Most fundamentally, the concept of dominant structure underlies the premise that is often repeated as a cornerstone of the system dynamics perspective: Behavior is a consequence of structure. In this sentence, behavior refers to an identifiable dynamic pattern; structure is the dominant interconnected stock-andflowlfeedback structure in the system. 2. Understanding. In analyzing model behavior modelers search for variables, information links, and causal loops that significantly influence the patterns observed. They describe their conclusions at this stage in terms of particularly important stocks and flows and feedback loops, along with significant nonlinearities that produce observed shifts from one set of dominant loops to another. 3. Evaluation. Sensitivity testing, for example, involves the hunt for parameters that have unusually strong influences on model behavior. One would infer that sensitive parameters are those that appear in dominant or influential structures. 4. Policy analysis. The search for sensitive parameters and equations also character- The author is indebted to Nathan Forrester for his help in preparing this research problem statement. System Dynamics Review 2 (no. 1. Winter 1986): ISSN by the System Dynamics Society. 68

2 Richardson: Dominant Structure 69 izes model-based policy analyses. Here the modeler and the client search for leverage points, which by definition must be located in portions of system structure that have significant influence in particular problematic behavior modes. 5. Communication. The concept of dominant structure plays the major role in the communication of model-based insights. Explanations of model behavior and policy simulations are phrased for the client in simplified terms that emphasize dominant structure. The concepts of loop dominance, dominant structure, and shifts in loop dominance are important bridges between complex interactions in a quantitative model and the understandings of people that a modeler tries to influence. 6. Theory. Nonlinear dynamic systems can endogenously generate bifurcations (sudden shifts in goal states) and chaotic behavior. Systems must also be nonlinear to show the phenomenon of shifting loop dominance. It is natural to conjecture that endogenous bifurcations in continuous systems are consequences of shifts in loop dominance (Richardson 1984). Thus the concept of loop dominance provides a potential bridge between research in the theory of bifurcating, chaotic systems and applied, policy-oriented system dynamics work. 7. Generic structures. Finally, the concept of transferability of structure and the related idea of generic feedback structures rest on perceptions of dominant structure. Generic structures can be thought of as especially vivid structurehehavior pairs that are perceived to support particular but widely applicable insights. They usually appear first in applied modeling work as subsets of the structure and behavior of a complex system. They must be teased out of larger mental or formal models by processes that discern dominant structure and behavior. In each of these areas, investigations would be considerably facilitated by the availability of powerful means of detecting loop dominance in higher-order systems. Approaches to the problem Determining dominant structure involves two sets of choices: how to characterize behavior, and how to define principal structure. We shall view the different possible approaches in terms of the twelve cells shown in Figure 1. Represented in that figure are three ways of characterizing behavior: in terms of graphs over time, eigenvalues, and frequency response. On the vertical dimension are two ways of thinking about dominant structure: in terms of the marginal contribution of a loop to a given behavior, and in terms of a model reduction or simplification that preserves a given behavior. The third dimension represents the necessary distinction between linear and nonlinear systems. We shall consider the problem of detecting dominant structure from the point of view of each of these cells. The marginal contribution of a feedback loop, viewed in terms of graphs over time One of the traditional simulation approaches in both linear and nonlinear systems is represented by the two cells on the lower left edge of Figure 1. The approach they

3 70 System Dynamics Review Volume 2 Number 1 Winter 1986 Fig. 1. Classification scheme for approaches to the problem of determining dominant model structure - d --J 2 - x \lode1 Reduction Loop ( ontrihution / / Time ~igen\alue ~~~~~~~~ Behavior characterize is a visual, experimental approach, involving repeated simulations. Parameters are varied to weaken or strengthen a feedback loop, and the effect of the change in the plotted output of the model is observed. This approach, which is reflected in almost all system dynamics studies, is outlined in N. Forrester (1983) and described with examples in Richardson and Pugh (1981, ). Model reduction, viewed in terms of graphs over time Another traditional simulation approach strives to identify dominant structure in a complex model by formulating a simplified model designed to reproduce some particular behavior of the larger model. In Figure 1 the cells that characterize this approach lie along the upper left edge. By repeated simulations, a hypothesis about dominant structure is formed. A simplified model embodying that dynamic hypothesis is then formulated and tested. If the simple model qualitatively reproduces the behavior of interest in the more complex model, as judged visually or statistically, the modeler concludes that a structure underlying that behavior in the large model has been located. This largely intuitive approach produced, for example, the simple model of the economic long wave reported in Sterman (1985), which was developed out of the large national socioeconomic model developed under J. W. Forrester at M.I.T. The simple model provides strong evidence that one dominant influence underlying the long wave in the large model is the feedback structure associated with capital self-ordering. The approaches described in these four cells on the left side of Figure 1 consume much time and effort. Indeed, it may be argued that it is impossible to do a complete analysis in very large models using these traditional approaches. In their defense, however, we must observe that they have generated our most insightful statements about the relationships between feedback structure and dynamic behavior. It is these insightful, repeated simulation approaches that we hope to facilitate by technical aids embedded in simulation software, which are represented by the other cells in Figure 1.

4 Richardson: Dominant Structure 71 The marginal contribution of a feedback loop, viewed in terms of eigenvalues In a linear model the eigenvalues are a summary of the behavior modes of the system. They capture some, but not all, of the information contained in the graphs of the variables over time. Suppose a system dynamics model is thought of as a nonlinear system of levels x, parameters p, and perhaps time t. Then x = f(x, p, t) (1) Linearizing the system about some point xo yields x = Ax, where A = - The eigenvalues of the linearized system are the numbers s, perhaps complex, that satisfy the characteristic equation Is1 - A1 = 0. Any level in the system can then be expressed as a linear combination of complex exponential functions of time. Its behavior over time in the original system is thus approximated as a sum of sinusoids and exponentials. In this way, the eigenvalues become a substitute for time graphs as a description of the behavior modes of the system. Each eigenvalue s depends on the structure and parameters of the system f. Thus, in the linearized system it is possible to compute the change in an eigenvalue that would result from a given change in a parameter. An eigenvalue elasticity is defined as the ratio of the fractional change in an eigenvalue to a fractional change in a given parameter. For a given eigenvalue s and parameter p it is computed as If s is a complex number, its eigenvalue elasticity is complex. The magnitude of the elasticity measures the change in a behavior mode that a given parameter change would produce. Comparing the magnitudes of all the eigenvalue elasticities with respect to all the parameters of the system should identify which parameters have strong influences on which behavior modes. Loops containing parameters that have relatively high elasticities for a given eigenvalue can then be inferred to be the dominant loops for that mode. This sort of approach to the determination of dominant structure is described in detail and applied in N. Forrester (1982). N. Forrester (1983) contains an extension of the methods, which enables the direct computation of eigenvalue elasticities with respect to the gain of individual loops. This improvement rests upon the fact that the elasticity of an eigenvalue with respect to a given link is equal to the sum of elasticities of that eigenvalue with respect to all the loops containing that link (Forrester 1983, ). Software embodying this approach would require a means of linearizing a nonlinear model, computing eigenvalues of the linear system, and identifying all feedback loops

5 72 System Dynamics Review Volume 2 Number 1 Winter 1986 in the model. It would then have to compute eigenvalue elasticities with respect to elements in the A matrix. (Forrester (1982; 1983) shows how to do this using left and right eigenvectors.) Finally, it would have to solve a system of linear equations for the eigenvalue elasticities with respect to the loop gains. All of this is very possible with today s technology. Linearizing, for example, can be done by symbolic differentiation or by approximation (Forrester 1983, 192). The mathematics of the remaining manipulations has been worked out and used. What remains is to make the tools widely available in user-friendly software that automatically produces summary loop dominance information. Model reduction, viewed in terms of eigenvalues The two cells in the middle of the top layer in Figure 1 represent the use of eigenvalue analyses to locate a subset of a large model that by itself can exhibit some behavior of the large model. The small structure identified in this way is then inferred to be a dominant influence responsible for that behavior in the large model. Like the eigenvalue techniques described previously, the theory here works for linear models and may or may not apply in nonlinear cases. There are several model simplification techniques that fall in or near this category. Eberlein (1984, 31-79) reviews the literature associated with aggregation based on eigenvalues, singular perturbation, weak coupling, and selective modal analysis. In each approach the basic idea is to start with a linear model as in Eq. 2 and to decompose it as representing a set of states (levels) x1 that we wish to keep and a set x2 that we wish to eliminate. The reduced model would be and we wish to approximate it without the levels xz by for some suitably chosen M. It should be noted that such a reduced model preserves some, but not all, of the identifiable structure of the original model. In addition, the matrix M may represent some new feedback connections among the reduced set of levels xl. We presume, however, that for virtually all system dynamics applications the reduced system must

6 Richardson: Dominant Structure 73 be recognizable in terms of the original model and the real system it purports to model. To try to use such techniques to identify dominant structure, we are interested in how to choose the levels to keep. The most promising approach for system dynamicists appears to be selective modal analysis. In this approach, the tool for selecting the levels to keep is the notion of a participation factor, Pik. The Pik is a dimensionless measure of the interaction of the ith level and the kth behavior mode or eigenvalue. It serves much the same purpose in locating dominant model structure as the eigenvalue elasticities discussed previously. It can be computed easily from the ith components of the kth left and right eigenvectors as plk = (Ik)i(rk)l In selective modal analysis a behavior mode in a large model is selected and represented in terms of one or more eigenvalues of the system. The model states (levels) that are important in that mode are then identified as those whose participation factors for the selected modes have relatively large magnitudes. It is then possible to compute the matrix of constants M in Eq. 6 and to formulate a reduced or simplified model that approximately reproduces the desired behavior modes. Eberlein (1984, 70-82) shows the necessary details in terms of linear difference equations. (7) The marginal contribution of a feedback loop, viewed in terms of frequency response In models that oscillate, the notion of frequency response gives a different way of characterizing dynamic behavior. Frequency response is defined in terms of a pair of variables: an input variable u and an output variable y. When u is disturbed by a sine wave of frequency w, and y comes to oscillate at that frequency in steady state, then the amplitude and phase shift of the oscillations of y relative to u define the frequency response for that pair of variables at the frequency w. In particular, the ratio of the output amplitude over the input amplitude is the gain for that input-output pair at the frequency w. Finding the gains between u and y over a whole range of frequencies produces the gain curve of the frequency response of u to y. A peak in the gain curve for some frequency wo corresponds to a tendency for the model structure to reinforce or resonate to an oscillation of that frequency that involves u and y. Suppose a model oscillates with an observed frequency w and we wish to identify the loops that are influential in that oscillation. Intuitively input-output pairs of variables that are important in that oscillatory behavior should show peaks in their gain curves at that frequency. Plotting all the gain curves for all the likely inputoutput pairs in the model should show which pairs have such peaks. Locating those pairs in the model and finding the loops that link them should identify the model structure that is influential in that oscillatory behavior. In practice, the analysis can be done without simulation, because the frequency response for all input-output pairs in a linear model can be computed from the matrices that define the system. Suppose k=ax+bu (8)

7 74 System Dynamics Review Volume 2 Number 1 Winter 1986 and y = Cx + Du where u and y represent input and output variables. Then the frequency response of y, stimulated by a sinusoid of frequency w at input uk is given for all (k, m) pairs by the matrix of complex numbers TOW) = C [jwi -A ]-'B + D (10) where j' = -1 (Ogata 1970, 689). The frequency response gains are given by the matrix of the magnitudes of these complex numbers. The details are nicely worked out and applied in Forrester (1982, 48-52). As with eigenvalues, it is also possible to compute the change in frequency response with respect to a parameter change and to represent it as an elasticity: (9) The matrix computations are worked out in Forrester (1982, 59). One ought to be able to use frequency response elasticities to identify parameters that are influential in particular oscillatory frequencies. The loops in which such parameters appear ought to be the loops that are responsible for oscillations appearing in model behavior at those frequencies. While the mathematics of this frequency response approach is well worked out, it has not yet, to my knowledge, been applied to identify dominant structure. And, like the eigenvalue approach, the frequency response approach applies to linear systems. Its applicability to nonlinear systems is totally untested. Literature associated with three of the four cells at the right in Figure 1 is essentially nonexistent. Promising directions for research Mathematical approaches to determining dominant structure apparently have promise in linear models. In nonlinear models, however, behavior cannot be concisely summarized in terms of eigenvalues or frequency response. In nonlinear systems eigenvalues can change continuously over time. We should expect that, of course, since nonlinear models have the capability endogenously to shift loop dominance and change the structures that are active in the system's observed behavior. Nonetheless, using linear approaches in nonlinear models raises a number of problems and unanswered questions. Would applications of linear mathematical techniques in nonlinear models give us the insights we desire between feedback structure and dynamic behavior? Can linearization be performed about successive operating points quickly enough to make eigenvalue or frequency response methods feasible in large models? Under what conditions would such linear techniques in a nonlinear model be seriously misleading? Do the approaches in terms of graphs over time, eigenvalues, and frequency response ever produce different conclusions about dominant structure? Under what conditions can the formal linear techniques be trusted in nonlinear systems?

8 Richardson: Dominant Structure 75 These questions are important for us to answer. Paradoxically, we need the widespread availability of these formal tools to enable us to determine if they are the tools we want to help us understand nonlinear systems. Thus, the first necessary step is the creation and dissemination of simulation software that can report appropriate eigenvalue and frequency response information and use it to identify dominant structure. There are movements in this direction (Graham and Pugh 1983), but we must have more. The second step is widespread use of such formal tools and organized reporting of the results. We should also not ignore a potential third step. There remains the possibility that totally different lines of research in the detection of dominant structure may be developed, not out of linear mathematics but rather directly out of investigations of nonlinear structures. There are clearly recognizable nonlinear behavior modes, such as logistic growth. We may be able to find nonlinear tools for characterizing nonlinear behavior modes and identifying the dominant shifting loop structure responsible for them (see Richardson 1984 for a modest attempt). Unfortunately, how to proceed in this direction is far from clear. However achieved, improvements in our abilities to detect dominant structure and shifts in loop dominance would greatly facilitate applied policy-oriented research in system dynamics. References Eberlein, R. L Simplifying Dynamic Models by Retaining Selected Behavior Modes. Ph.D. Dissertation, Sloan School of Management, M.I.T., Cambridge, MA Forrester, J. W Market Growth as Influenced by Capital Investment. Sloan Management Review 9(2): Forrester, N. B A Dynamic Synthesis of Basic Macroeconomic Theory: Implications for Stabilization Policy Analysis. Ph.D. Dissertation, Sloan School of Management, M.I.T., Cambridge, MA Eigenvalue Analysis of Dominant Feedback Loops. Proceedings of the 1983 International System Dynamics Conference, Plenary Session Papers. Pp Graham, A. K., and A. L. Pugh Behavior Analysis Software for Large DYNAMO Models. Proceedings of the 1983 International System Dynamics Conference 1: Ogata, K Modern Control Engineering. Englewood Cliffs, N.J.: Prentice-Hall. Perez, J. I Selective Modal Analysis With Applications to Electric Power Systems. Ph.D. Dissertation, Sloan School of Management, M.I.T., Cambridge, MA Perez, J. I., F. C. Schweppe, and G. C. Verghese Selective Modal Analysis With Applications to Electric Power Systems. IEEE Transactions on Electric Power Systems POS-101: Porter, B., and R. Crossley Modal Control: Theory and Applications. London: Taylor and Francis. Richardson, G. P Loop Polarity, Loop Dominance, and the Concept of Dominant Polarity. Proceedings of the 1984 International System Dynamics Conference, Oslo, Norway. Pp Richardson, G. P., and A. L. Pugh Introduction to System Dynamics Modeling With DYNAMO. Cambridge, Mass.: MIT Press. Sterman, J. D A Behavioral Model of the Economic Long Wave. Journal of Economic Behavior and Organization 6:17-53.