### Abstract

- Top of page
- Abstract
- Introduction
- Graph theory and dynamical systems
- Loop gains and eigenvalues
- Example: the long wave model
- Conclusion
- Acknowledgements
- References
- Biography
- Supporting Information

Kampmann C. E. (1996) Feedback loop gains and system behavior

Proceedings of the 1996 International System Dynamics Conference. System Dynamics Society, Boston, MA, USA

In the past two decades, there has been a steady growth of analytical tools that can help the system dynamicist understand and test the behavior of large models. Without such tools, we must resort to simulation experiments and intuition developed from simple, low-order, systems. While this traditional approach has served us well, there is always the danger that we overlook important mechanisms in larger systems or falsely attribute behavior patterns to particular structures.

Among these new analytical tools, eigenvalue elasticity analysis (EEA) has received the most attention, perhaps because it rests on a consistent and comprehensive theoretical foundation. The approach involves decomposing system outcomes into characteristic behavior modes, each characterized by an eigenvalue of the linearized system matrix, and then examining how each eigenvalue or behavior mode is affected by small changes in the parameters of the system. The eigenvalue elasticity of a parameter change is a dimensionless number, defined as the fractional change in the eigenvalue resulting from a fractional infinitesimal change in that parameter. Parameters or structural links or loops in the model that have large elasticities are then interpreted as playing a dominant role in that behavior mode.

EEA was first used by Nathan Forrester in his doctoral dissertation from 1982. However, it did not receive much attention in the field until the 1996 International System Dynamics Conference, where Christian Kampmann presented a rigorous analysis of the topology of feedback loop structures and how eigenvalue elasticities are related to the strength of individual feedback loops. Since then, an entire strand of research has been devoted to the development and application of EEA, where Kampmann's conference paper remains a seminal reference. Yet until now, the paper has never been published, and researchers have had to rely on private circulation of various drafts of the paper. With this archival publication, we wish to remedy this deficiency in the hope that the work will inspire others to continue developing EEA and further the high standards of analytical rigor that Kampmann's paper represents.

#### Abstract

Linking feedback loops and system behavior is part of the foundation of system dynamics, yet the lack of formal tools has so far prevented a systematic application of the concept, except for very simple systems. Having such tools at their disposal would be a great help to analysts in understanding large, complicated simulation models. The paper applies tools from graph theory formally linking individual feedback loop strengths to the system eigenvalues.

The significance of a link or a loop gain and an eigenvalue can be expressed in the eigenvalue elasticity, i.e., the relative change of an eigenvalue resulting from a relative change in the gain. The elasticities of individual links and loops may be found through simple matrix operations on the linearized system.

Even though the number of feedback loops can grow rapidly with system size, reaching astronomical proportions even for modest systems, a central result of the paper is that one may restrict attention to an *independent* subset which typically grows only linearly and at most as the square of system size. An algorithm for finding an independent loop set is presented, along with suggestions for how to augment it to select loops with large elasticities.

For illustration, the method is applied to a well-known system: the simple long-wave model. Because this model exhibits highly nonlinear behavior, it sheds light on the usefulness of linear methods to nonlinear system. The analysis leads to a more thorough and deeper understanding of the system and sheds new light on conventional wisdom regarding the role of many of the system's feedback loops. Copyright © 2012 System Dynamics Society