Loop dominance analyses in system dynamics attempt to determine the effects of feedback loops on system behaviour. I have identified five methods

• Eigenvalue Elasticity Analysis (EEA)
• Loop Deactivation Method (LDM)
• Pathway Participation Method/Metric (PPM)
• Loop Impact / Newtonian Interpretive Framework (LI/NIF)
• Loops That Matter (LTM)

All the methods deal with dominant pathways except EEA, which deals with dominant modes. These differences are explained in Hayward & Roach (2024). Loop impact and sociomechanics are the subjects of this website.

I have given the references in date order and annotated them with the method(s) discussed where appropriate. Some papers are relevant to loop dominance but do not indicate any particular method. For some papers, I describe their influence on me while developing loop impact and the Newtonian Interpretive Framework.

References

Richardson GP. 1986. [1976]. Problems with causal loop diagrams, System Dynamics Review, 2:2, 158–170.
The original paper dates from 1976 as report D-3312 at MIT and is available on the Creative Learning Exchange Road Maps. The paper highlights the difficulties in relating structure to behaviour in causal loop diagrams because information about which elements are stocks is suppressed. Richardson points out that because stocks accumulate, plus and minus polarities mean “add to” and “subtract from” respectively. This meaning contrasts with links to other elements where the polarities mean “same way” and “opposite way”. This distinction is essential to the pathway methods PPM, LI/NIF and LTM.

Graham  AK. 1977. Principles on the relationship between structure and behavior of dynamic systems. Doctoral Thesis. Massachusetts Institute of Technology 1977.
Graham relates feedback structure to behaviour using the concept of phase-lag subsystems. He also relates his work to traditional phase plane analysis and the control theory approach to feedback. His identification of the roles of minor loops and higher-order loops closely matches the energy analogy in the Newtonian Interpretive Framework.

Forrester NB. 1982. A dynamic synthesis of basic macroeconomic theory: Implications for stabilization policy analysis. PhD thesis, Sloan School of Management, MIT, Cambridge, MA. [EEA]
Forrester’s thesis introduced the use of eigenvalue elasticities in policy analysis and their role in identifying dominant feedback loops. His work became the foundation of Eigenvalue Elasticity Analysis.

Forrester NB. 1983. Eigenvalue analysis of dominant feedback loops. 1st International Conference of the System Dynamics Society, Chestnut Hill, United States. [EEA]

Richardson GP. 1995. [1984]. Loop polarity, loop dominance, and the concept of dominant polarity, System Dynamics Review, 11:1, 67–88. First appeared in Proceedings of the 2nd International Conference of the System Dynamics Society, Oslo, Norway, 1984.
Richardson rigorously defined loop polarity and dominant polarity. I found this paper especially useful because of its mathematical approach to feedback and stock behaviour. The PPM method was built on this work.

Kim D. 1995. A new approach for finding dominant feedback loops: Loop by loop simulation for tracking feedback loop gains, Proceedings of the 13th International Conference of the System Dynamics Society, Tokyo, Japan.
Kim computes feedback gains by computing the change in an initial small disturbance in stock value. I am not aware that this method was ever followed up, but it has potential.

Mojtahedzadeh M, Richardson GP. 1995. Confusion in the polarity of major loops. Proceedings of the 13th International Conference of the System Dynamics Society, Tokyo, Japan. Albany, NY.
The authors deal with subtleties in the polarity of loops with two or more stocks. They note that balancing loops of three stocks or more are unstable. A negative loop polarity does not guarantee stability. The work builds on Grahan (1977) and is used in PPM.

Kampmann CE. 2012. [1996]. Feedback loop gains and system behaviour (1996). System Dynamics Review 28(4): 370–395. First appeared in Proceedings of the 14th International Conference of the System Dynamics Society, Cambridge, MA, 1996. [EEA]
Kampmann rigorously analyzes feedback loop structure and how eigenvalue elasticities relate to feedback loop strength. He defines feedback loop gain and dependent and independent loop sets. He applies the method to Sterman’s Economic Long-Wave model, giving the comparative influence of loops. I relied heavily on this paper when developing the Loop Impact Method using the same model, even though my pathway approach was very different. It was originally produced in 1996.

Richardson GP. 1997. Problems in causal loop diagrams revisited. System Dynamics Review 13(3): 247–252.
This paper describes the problems with Causal Loop Diagrams that use S (same way) and O (opposite way) links. He recommends using + and – and indicating which elements are stocks where the links mean “add to” and “subtract from”.

Ford DN. 1999. A behavioral approach to feedback loop dominance analysis. System Dynamics Review 15(1): 3–36. [LDM]
Ford’s paper describes how dominant loops may be identified by deactivating loops and comparing results. The method can be carried out in any SD simulator without programming. His approach inspired my approach to Loop Impact, where results could be produced without programming or black-box add-ons. I discovered Ford’s paper in 2004 when my students asked me how they could measure loop influence. I gave them this paper to investigate dominance.

Gonçalves P, Lerpattarapong C, Hines JH. 2000. Implementing formal model analysis. Proceedings of the 18th International System Dynamics Society Conference. Bergen, Norway. [EEA]
They build on the EEA work of Forrester (1982) and Kampmann [1996], deriving eigenvalue elasticities using Mason’s rule.

Myrtveit M, Saleh M. 2000. Superimposing dynamic behavior on causal loop diagrams of system dynamic models. Proceedings of the 18th International Conference of the System Dynamics Society, Bergen, Norway. [EEA]
They build on the EEA work of Forrester (1982) and Kampmann [1996], relating feedback gains to behaviour modes.

Saleh MM, Davidsen PI. 2000. An eigenvalue approach to feedback loop dominance analysis in non-linear dynamic models. Proceedings of the 18th International Conference of the System Dynamics Society, Bergen, Norway. [EEA]
They build on the EEA work of Forrester (1982) and Kampmann [1996], adding the transient behaviour to the steady-state behaviour of the original method.

Saleh MM. 2002. The Characterization of Model Behavior and its Causal Foundation. PhD Dissertation, University of Bergen, Bergen, Norway. [EEA].

Mojtahedzadeh M, Anderson D, Richardson GP. 2004. Using Digest to implement the pathway participation method for detecting influential system structure. System Dynamics Review 20(1): 1–20.[PPM]
The authors present a new method of loop dominance using the “Pathway Participation Metric” to measure the proportional influence of each loop on a given stock. This paper came out as I was faced with questions about measuring loop influence from my students. This was the breakthrough paper for me when I realised the stock behaviour being measured was its acceleration, that is, changes in its net flow. The appendix of this paper helped me develop the loop impact method.

Oliva R. 2004. Model structure analysis through graph theory: Partition heuristics and feedback structure decomposition. System Dynamics Review 20(4): 313–336.
Work on the shortest independent loop set used in EEA.

Güneralp  B. 2004. A principle on structure-behavior relations in system dynamics models. Proceedings of the 22nd International System Dynamics Conference, Oxford, UK. [EEA]
Builds on the EEA work of Forrester (1982) and Kampmann [1996].

Tseng YT, Tu YM. 2004. From Loop Dominance Analysis to System Behaviors, Proceedings of the 22nd International System Dynamics Conference, Oxford, UK. [PPM, LDM].
Their methods are similar to PPM but focus on links rather than loops and incorporate loop deactivation. Had they continued this approach, they would have ended up with something like the Loop Impact Method.

Güneralp B, Sterman J, Repenning  N, Langer R, Rowe J, Yani J. 2005. Progress in eigenvalue elasticity analysis as a coherent loop dominance analysis tool. Proceedings of the 23rd International Conference of The System Dynamics Society, Boston, MA. [EEA].
An application of EEA, extending the method to examine a variable of interest.

Phaff HWG, Slinger JH, Güneralp B, van Daalen CE. 2006. Investigating model behavioural analysis: a critical examination of two methods. Proceedings of the 24th International Conference of the System Dynamics Society, Nijmegen, The Netherlands. [LDM, EEA].
They compare the loop deactivation method with eigenvalue elasiticity, finding the former easier to use.

Güneralp B. 2006. Towards coherent loop dominance analysis: Progress in eigenvalue elasticity analysis. System Dynamics Review 22(3): 263–289. [EEA].
An application of EEA, extending the method to examine a variable of interest. Journal version of 2005 conference paper.

Kampmann CE, Oliva R. 2006. Loop eigenvalue elasticity analysis: Three case studies. System Dynamics Review 22(2): 141–162. [EEA, PPM].
EEA is applied to three models of different complexity. One has 2 stocks with 6 loops. Its results are compared with PPM. One is chaotic. The third has 8 stocks and 32 loops. Automation is used to generate results. They introduce a new measure loop influence as a more useful measure than eigenvalue elasticity.

Mojtahedzadeh M. 2007. Do the Parallel Lines Meet? A Comparison between Pathway Participation Metrics and Eigenvalue Analysis. Proceedings of the 26th International Conference of The System Dynamics Society. Athens, Greece. [PPM, EEA].
Mojtahedzadeh describes the relationship between the pathway participation metric and EEA metrics. He extends PPM to include stability and frequency metrics, which better explain oscillations.

Phaff HWG. 2008. Generalized loop deactivation method. Proceedings of the 26th International Conference of the System Dynamics Society, Athens, Greece. [LDM].
Phaff presents an automated version of Ford’s Loop Deactivation Method. He enhances the method using the shortest independent loop set to identify edges to be deactivated. Unlike the original method, results include the influence of more than just the dominant loop.

Kampmann CE, Oliva R. 2008. Structural dominance analysis and theory building in system dynamics. Systems Research and Behavioral Science 25(4): 505–519. [EEA, PPM].
They compare Control Theory, EEA and PPM. They also introduce dynamic weights into EEA to help determine the relative presence of behaviour modes in a variable of interest. This method is related to the computation of eigenvectors in conventional differential equation theory.

Mojtahedzadeh M. 2009. Do parallel lines meet? How can pathway participation metrics and eigenvalue analysis produce similar results? System Dynamics Review 24(4): 451–478. [PPM, EEA].
The journal version of Mojtahedzadeh (2007) above.

Huang J, Howley E, Duggan J. 2009. The Ford method: a sensitivity analysis approach. Proceedings of the 27th International Conference of the System Dynamics Society. Albuquerque, NM. [LDM].
They improve the Ford Loop Deactivation Method using sensitivity analysis and the identification of the shortest independent loop set.

Gonçalves P. 2009. Behavior modes, pathways and overall trajectories: Eigenvector and eigenvalue analysis of dynamic systems. System Dynamics Review 25(1): 35–62. [EEA].
Gonçalves applies EEA to a model to draw out the whole trajectory of a state variable, that is, the transients and behaviour modes. I found his analytical approach very helpful in elucidating the EEA method. I replicated his work in my 2024 paper.

Mojtahedzadeh M. 2009. Objective analysis of subjective feedback structures: The problem of consistency in explaining model behavior, Proceedings of the 27th International Conference of the System Dynamics Society, Albuquerque, USA. [PPM].
Mojtahedzadeh applies PPM to some idiosyncratic feedback structures for which existing analysis methods give misleading results. Such structures include self-cancelling loops and figure-of-8 loops. He includes more work on stability and frequency metrics.

Saleh M, Oliva R, Kampmann CE, Davidsen PI. 2010. A comprehensive analytical approach for policy analysis of system dynamics models. European Journal of Operational Research 203(3): 673–683. [EEA].
EEA is used to examine model policy analysis with the help of dynamic weights. They further define the eigenvalue loop influence as a more natural measure of the effect of a feedback loop on behaviour modes compared with the original eigenvalue elasticity.

Mojtahedzadeh M. 2011. Consistency in explaining model behavior based on its feedback structure, System Dynamics Review, 27:4, 358–373. [PPM].
The journal version of Mojtahedzadeh (2009) above.

Huang J, Howley E, Duggan J. 2012. Observations on the shortest independent loop set algorithm. System Dynamics Review 28(3): 276–280.
They develop an improved algorithm for identifying the shortest independent loop set.

Hayward J. 2012. Model Behavior and the Strengths of Causal Loops: Mathematical Insights and a Practical Method. Proceedings of the 30th International System Dynamics Conference, St Gallen, Switzerland. [LI].
My students kept asking me how we could measure the influence of a feedback loop on a stock using mathematics. Using Mojtahedzadeh et al. (2004), I developed a measure, which in 2014 came to be known as loop impact. This paper presents the measure before that name was chosen. My students had already been answering examination questions on this work. In addition, I developed a way of automating loop comparisons in Stella, inspired by Ford’s (1999) approach of only needing an SD simulator. That is, there are no add-on tools as in PPM and EEA.

Hayward J, Boswell GP. 2014. Model behaviour and the concept of loop impact: a practical method. System Dynamics Review 30(1): 29–57. [LI].
The journal version of the 2012 conference paper. Here, I describe the algorithm for choosing the dominant loop or loops. I also compare loop impact analysis with PPM, EEA and LDM for some well-known models. It was this paper that named the ratio of acceleration to stock rate of change as “loop impact”, the term being suggested by the sub-editor overseeing publication. It is fair to say that this work has caused some controversy, especially my emphasis on the analytical approach. I recruited the help of a mathematical colleague to help get the paper published.

Hayward J. 2015. Newton’s Laws of System Dynamics. Proceedings of the 33^rd International Conference of the System Dynamics Society, Cambridge, MA. [LI/NIF].
When constructing the loop impact method, I was struck with the analogy between a stock’s acceleration equation and Newton’s laws of motion. At this conference, I presented the 3 laws of stock dynamics and described the stock-to-stock causal link as the force of one stock on another. I developed an analytical notation to capture all the causal information of an SD model in differential equation form and defined “pathway differentiation”, a form of partial differentiation along causal links.

Oliva R. 2015. Linking structure to behavior using eigenvalue elasticity analysis. In Analytical Methods for Dynamics Modelers, Rahmandad H, Oliva R, Osgood ND (eds). MIT Press: Cambridge, MA; 207–239.[EAA].
A review of EEA, independent loop sets, dynamic weights and eigenvalue loop influence.

Oliva R. 2016. Structural dominance analysis of large and stochastic models. System Dynamics Review 32(1): 26–51. [EEA].
Further applications of EEA.

Moxnes E, Davidsen PI. 2016. Intuitive understanding of steady-state and transient behaviors. System Dynamics Review 32(2): 130–155.
A very helpful paper examining the pitfalls in failing to take into account transients and initial stock value when understanding system behaviour.

Sato JB. 2016. State space analysis of dominant structures in dynamic social systems. PhD thesis. Washington University, St Louis, MO.
In his thesis, Sato reviews the state of play of loop dominance methods. He then uses loop impact to develop a force decomposition approach to dominance. He uses the actual stock-to-stock force, the absolute measure, rather than impact, the ratio measure. Drawing analogies with engineering, he proves some useful theorems.

Sato JB. 2017. Dominance Analysis Using Pathway Force Decomposition,  Proceedings of the 35th International Conference of the System Dynamics Society, Cambridge MA.
A conference presentation of work in his thesis (2016).

Hayward J, Roach PA. 2017. Newton’s laws as an interpretive framework in system  dynamics. System Dynamics Review 33(3–4): 183–218. [LI/NIF],
The journal version of the 2015 conference paper and the definitive presentation of the Newtonian Interpretive Framework. This framework is more than an analysis tool. It also presents the theory underlying a model by describing the dynamical hypothesis in terms of force. As with loop impact, this work caused some controversy. I recruited the help of another colleague and was pleased we could present the work in the context of quantifying social theory.

Naumov S, Oliva R. 2018. Refinements to eigenvalue elasticity analysis: Interpretation of parameter elasticities. System Dynamics Review 34(3): 426–437. [EEA].
Refinements to EEA to eliminate inconsistencies in dynamic decomposition weight analysis. An updated toolset for EEA has been published, see below.

Naumov S, Oliva R 2018. Structural Dominance Analysis Toolset. Available from http://people.tamu.edu/$roliva/research/sd/ sda/. It requires the software Mathematica. [EEA].

Hayward J, Roach PA. 2019. The concept of force in population dynamics. Physica A: Statistical Mechanics and its Applications 531: 121736. [LI/NIF].
This paper presents loop impact and the Newtonian Interpretive Framework to a mathematics and physics audience. This strategy was a tough call, as these communities know little about system dynamics, and those who do know are often hostile. The paper tightens the mathematical foundations of loop impact and system dynamics.

Oliva R. 2020. On structural dominance analysis. System Dynamics Review 36(1): 8–28. [EEA].
A desciption of the evolution of EEA, its benefits and answers to criticisms.

Kampmann CE, Oliva R. 2020. Analytical methods for structural dominance analysis in system dynamics. In System Dynamics: Encyclopedia of Complexity and Systems Science Series, Dangerfield B (ed). Springer: New York; 153–176. [EEA, PPM, LDM, LI].
A review of loop dominance methods. This article is a revision of earlier versions of the encyclopedia and only effectively deals with EEA and PPM.

Schoenberg W. 2020. Loops that matter. PhD thesis, Department of Geography, University of Bergen, Norway. [LTM].
Loops That Matter (LTM) is a new method for assigning a measure of loop influence for the whole system not just a stock. It is a pathway method like PPM and Loop Impact. It achieves a single measure for the system by combining dimensionless measures of the loop’s influence on each stock in the loop. Schoenberg also describes how the method is implemented in the Stella software, including algorithms for independent loop sets and the Stella visualisation.

Schoenberg W, Davidsen P, Eberlein R. 2020. Understanding model behavior using the loops that matter method. System Dynamics Review 36(2): 158–190. [LTM].
A description of LTM in journal form.

Schoenberg W, Hayward J, Eberlein R. 2021. Improving loops that matter. Proceedings of the 37th International Conference of the System Dynamics Society. Chicago, IL. [LTM].
A change LTM was required to ensure consistency in the treatment of flows. Analysis shows its connection to Loop Impact and PPM.

Hayward J, Roach PA. 2022. The Concept of Energy in the Analysis of System Dynamics Models, System Dynamics Review, 38(1), 5-40. [LI/NIF].
This paper extends the Newtonian Interpretive Framework to include Energy and Power. This approach enables a systemwide connection between loop structure and behaviour. The explanatory narrative goes beyond a dominance analysis to more informal descriptions of a loop’s role in the dynamics. Energy is cumulative and explains stock and system behaviour over a time interval, not just at specific points in time. We tried to set this paper in the context of social theory, like our 2017 paper, but one referee would not let it through.

Schoenberg W, Hayward J, Eberlein R. 2023. Improving Loops That Matter, System Dynamics Review, 39(2).
The paper version of the 2021 conference paper.

Hayward J, Roach PA. 2023. Comparison of Loop Dominance Methods: Measures and Meaning. Proceedings of the 41st International Conference of the System Dynamics Society, Chicago July 2023. [LI/NIF, PPM, LTM, EEA].
In this work, we compare all the quantitative measures of loop dominance, concentrating on their conceptual meaning in a stock-flow model. This conference paper follows my talk at the conference and aims to lead people through each method in a range of models. The journal paper below focuses on one single model.

Hayward J, Roach PA. 2024. Comparison of Loop Dominance Methods: Measures and Meaning. System Dynamics Review 2024 Vol. 40(2), 1757. [LI/NIF, PPM, LTM, EEA].
The journal version of the 2023 conference paper, above. In the appendix, we provide analytical results for the measures used in each method. Our aim with both papers is to take the conceptual and analytical mystery out of the methods.