loop eigenvalue elasticity analysis: three case studies

C. E. Kampmann and R. Oliva: Loop Eigenvalue Elasticity Analysis 141

Published online in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/sdr

Loop eigenvalue elasticity analysis:three case studies

Christian Erik Kampmanna* and Rogelio Olivab

Abstract

We explore the application of loop eigenvalue elasticity analysis (LEEA) to three different models

in order to assess the potential of the method for generating insights about model behavior and

to uncover issues in developing the method further. The results indicate that the utility of themethod depends upon the character of the model and dynamics involved. In models where the

transient behavior is of interest, the method yields insights on a par with the pathway participa-

tion method, though better tools to link the method to time paths of particular variables areneeded. In quasi-linear models, LEEA shows the most promise, quickly revealing the different

behavior modes and the associated dominant structures. Finally, analysis of a nonlinear chaotic

model reveals that the eigenvalues may change rapidly even during phases when the mode ofbehavior appears constant, limiting the insights gained from LEEA analysis. The paper concludes

with our thoughts on the strengths and weaknesses of the LEEA and suggestions for future work.

Copyright © 2006 John Wiley & Sons, Ltd.

Syst. Dyn. Rev. 22, 141–162, (2006)

Introduction

There is a long-standing desire in the system dynamics scientific communityfor tools for analyzing the behavior of large-scale models. The field has reliedheavily on intuitive approaches, based on the notions of system archetypes(Senge, 1990), simple models linking feedback loops to system behavior (Ford,1999; Richardson, 1995), and a structured approach to model testing andsimulation experiments (Barlas, 1989; Forrester and Senge, 1980).

A more formal approach, developed by Mojtahedzadeh et al. (2004), uses theso-called pathway participation metric (PPM) to identify the structure that ismost influential in affecting the qualitative time path of a given variable. Themain strength of PPM is that it takes place in the time domain and does notrequire calculating eigenvalues and other highly computation-intensive tasks.It has, however, some limitations. First, the method is not particularly wellsuited for systems that oscillate: in an oscillating system, PPM would indicateshifting loop dominance over the course of each cycle, even though the relativestrength of each loop may not change. Second, the current implementationof PPM uses a depth-first search for the single most influential pathway for avariable. This strategy does not capture the situation where more than one

System Dynamics Review Vol. 22, No. 2, (Summer 2006): 141–162Published online in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/sdr.333Copyright © 2006 John Wiley & Sons, Ltd.

141

a Copenhagen Business School, Birk Centerpark 40, 7400 Herning, Denmark. E-mail: [email protected] Mays Business School, Texas A&M University, College Station, TX 77843-4217, U.S.A. E-mail: [email protected]

* Correspondence to: Christian E. Kampmann.

Received September 2005; Accepted February 2006

Christian Erik

Kampmann is

Associate Professorat the Copenhagen

Business School

(CBS), Center forApplied Market

Science. He holds an

MSc in engineeringfrom the Technical

University of Denmark

(DTU) and a PhD fromthe System Dynamics

Group at MIT. Since

graduating from MIT,he worked as a

postdoctoral fellow

at the Department ofPhysics, DTU, focusing

on nonlinear dynamicsand chaos theory,

and subsequently in

the universityadministration, where

he was responsible for

establishing a Masterin Management of

Technology program.

After a 6-year stint inprivate business, he

returned to CBS,

where he teachesindustrial marketing

and system dynamics

and is programdirector for the MSc

program in Industrial

Marketing andPurchasing. His

research interests

include methodologydevelopment,

industrial organization,

non-traditionaleconomic theory, and

energy economics.

142 System Dynamics Review Volume 22 Number 2 Summer 2006


structure may contribute significantly to the model behavior and, since PPM iscontingent upon the selected variable of interest, the method could lead to“local” explanations of the variable’s behavior—as opposed to a system-wideanalysis of the structure. More generally, the focus in PPM on identifying asingle dominant pathway may be an oversimplification.

The approach that underlies this paper is eigenvalue elasticity analysis, firstintroduced by Forrester (1982), which uses linear systems theory to decomposethe behavior into simple modes, each of which corresponds to an eigenvalue.Measuring how much a given eigenvalue changes with a small change in aparameter or link in the model then gives an indication of how much that linkcontributes to that behavior mode.

Kampmann (1996) further refined the approach by considering feedbackloops as the structural unit of analysis. Since system dynamics models typi-cally contain a very large number of feedback loops, Kampmann used graphtheory to develop the independent loop set (ILS), a much smaller set of loops,within which it is valid to speak of the individual loop elasticities. Oneproblem with Kampmann’s approach, as pointed out by Oliva (2004), is thatthe elasticity measures depend upon the choice of the independent loop set,which is not unique. Oliva proposed the notion of a shortest independent loopset (SILS), consisting entirely of geodetic loops (a geodetic loop between twovariables x and y consists of a shortest path from x to y and a shortest path fromy to x). Under some conditions the SILS is unique and the shorter loopsfrequently lead to more intuitive interpretations. Furthermore, preliminarytesting of the PPM method suggests that it identifies loops within the SILS asbeing dominant (Oliva and Mojtahedzadeh, 2004).

Although still in its infancy, the loop eigenvalue elasticity analysis (LEEA)method has been employed in a variety of models and has proven to be a usefultool for gaining deeper insight into model behavior (Gonçalves, 2003; Gonçalveset al., 2000; Güneralp, 2005; Hines, 2005; Kampmann, 1996; Saleh et al., 2005).While these initial results are encouraging, it remains an open question howapplicable a linear approach like LEEA is to system dynamics models, whichare usually nonlinear. The motivation for the present paper is to address thisissue. We thus explore the range of insights that the LEEA method may provideby applying it to three very different models, each of which exhibits a particulartype of behavior pattern. The first model, Industrial Structures, is (essentially)the same as the one used in Mojtahedzadeh et al. (2004) to demonstrate thePPM method. The model shows a single transient behavior where the growthof industry is limited by a non-renewable resource (water) and allows forcomparisons with the PPM method.

The second model, first proposed by Lorenz (1963), is a classic model exhib-iting deterministic chaos. The model cycles around two unstable equilibriumpoints, tracing out a characteristic butterfly-shaped (strange) attractor.Nonlinearities play a key role in chaotic dynamics, and the model was chosento explore whether eigenvalue analysis could yield an understanding of the

Rogelio Oliva is

Associate Professor

in the Informationand Operations

Management department

at Mays BusinessSchool at Texas A&M

University. He holds

a BS in industrial andsystems engineering

from ITESM (Mexico),

an MA in systems inmanagement from

Lancaster University

(UK), and a PhD inoperations management

and system dynamics

from MIT. Hisresearch explores how

behavioral and social

aspects of anorganization interact

with its technicalcomponents to

determine the firm’s

operationalperformance. His

current research

interests includeservice operations,

behavioral decision

making in supplychains, and

the transition that

product manufacturersare making to become

service providers.



nature of the behavior phases where the model cycles around a given equilib-rium point and the phases where it suddenly shifts to cycle around the otherequilibrium point.

The third model is a medium-sized model developed by Mass (1975) toanalyze the causes of different types of economic cycles. Mass identifies dis-tinct structures responsible for the short-term business cycles. The model waschosen both because it is of a fairly large size and because it exhibits oscilla-tions around an equilibrium point, an area where the eigenvalue method oughtto have the greatest potential because of the quasi-linearity of the model. Thequestion is both whether the eigenvalue method for such a large model willyield insights, given that there is bound to be a very large number of potentialloops in the system, and whether the results concur with the findings in Mass’sanalysis.

In the interest of brevity, we do not provide the details of the LEEA methoditself. The reader is referred to the literature cited above, in particular Kampmann(1996), Oliva (2004), and Kampmann and Oliva (2005). In the following, wepresent the analysis of each of the three models in turn. The Vensim® versionof the documented models, simulation output, as well as LEEA computationsand output are available at: http://iops.tamu.edu/faculty/roliva/research/sd/leea/. The paper concludes with our thoughts on the strengths and weaknessesof LEEA and suggestions for future work, including references to a set ofInternet-based tools that we have developed for others to apply the method.

Industrial structures model

The model

Figure 1 shows a flow diagram and behavior of the simple industrial structuresmodel (Mojtahedzadeh et al. (2004).1 Construction of new industry is assumedto be proportional to the stock of industry, and industry depreciates at aconstant fractional rate. Initially, when there is ample water available, thefractional investment exceeds the fractional depreciation, and industry growsat a constant exponential rate.

However, industry investment is affected by the availability of water, meas-ured by the “normal” water demand compared to actual consumption. Waterconsumption, in turn, is proportional to the industry stock, modified by aneffect of low levels of water resources. As industry stock rises and waterresources decline, the resource constraint affects water consumption, which inturn affects the growth of industry. Eventually, the effect is sufficiently large tostop the growth.

As water is continually depleted, the industry stock starts declining and thisprocess continues for the rest of the simulation as both stocks decay in parallel.The model is unusually simple in that it contains only six feedback loops that



Fig. 1. Flow diagram, feedback loops and behavior: Industrial Structures model.

Areas in behavior graph indicate the four phases identified by PPM method (Mojtahedzadeh et al., 2004)

Fig. 2. Feedback loop

gains: Industrial

Structures model

are all independent (see Figure 1). The evolution of loop gains gives a goodimpression of the shift in loop dominance (see Figure 2).

Inspection of the plot reveals that throughout the simulation the gain of thepositive loop 5 linking water consumption and industrial structures cancels



out exactly with the gain of loop 4, the balancing loop limiting industrialgrowth for a given level of water consumption (see the line “4 + 5” in Figure 2).It turns out that loop 5 is actually an artifact of the formulation for the effect ofwater adequacy on growth, which is a function of water consumption relativeto water demand. Since water consumption is in turn a function of the effect ofwater availability on water consumption, this means that the effect of watershortage on new industry construction is in reality a table function of a tablefunction, which is usually not good modeling practice. Given the formulation,those two loops cancel each other and could be eliminated from the model.

However, we have chosen to retain the original structure because it high-lights the issue of how formulation details in the model may create such“artificial” feedback loops.

PPM analysis

According to the PPM analysis in Mojtahedzadeh et al. (2004), the behavior ofthe industry stock can be divided into four phases, which are also indicated inFigure 1: a period of exponential growth (phase I), balancing growth (phase II),exponential decline (phase III) and balancing decline (phase IV). In phase I, themost influential structure identified by the method is the positive growth loop1. In phases II and III, the most influential structure is the negative loop 3 thatcontrols water depletion. Finally, in phase IV, the most influential structure isthe depreciation loop 2. It is of interest to see whether the same structures areidentified as important in the eigenvalue analysis, although the loop elasticitiesdo not allow direct comparisons with one-way pathways, as is the case inphases II and III.

LEEA

Figure 3 shows a plot of the two system eigenvalues over time; both are real-valued. Initially, one eigenvalue is positive and equal to the growth rate ofindustry, while the other is zero. The latter indicates a system behavior withno feedback, which is the case for the depletion of water resources, since theeffect of water availability is initially inactive. Hence, industry grows in-dependently of the water resources, and the water depletion is driven by thegrowing industry stock with no feedback from either stock.

As the effect of water availability is felt on both water consumption andindustry growth, both eigenvalues start to fall. Thus, from around t = 22 to t =25, two modes, one positive and one negative, are present. Subsequently, botheigenvalues are negative. The larger eigenvalue levels off slightly above thevalue of fractional depreciation rate of industry. (There is still some construc-tion even at the end of the run, though it is much smaller than depreciation).At t = 24, the time of the inflection point that defines the transition from phaseI to phase II in the PPM analysis, the absolute value of the larger eigenvalue



becomes smaller than the absolute value of the second eigenvalue. The posi-tive mode changes sign to become negative around t = 25, which is a few yearsbefore the industry stock peaks.

Figures 4 and 5 show the influence measure for the largest and smallesteigenvalue (numbers 1 and 2 in Figure 3), respectively.2 One can define aninitial phase in Figure 4 (until around t = 24), where the positive eigenvalue isaffected by loops 1 and 2. A stronger loop 1 (growth) increases the growth rate,while a stronger loop 2 (demolition) decreases the growth rate. The other loopsdo not play a major role in this mode. Note that the two “artificial” loops 4 and5 have a smaller equal and opposite influence during this phase—this is notthe case later in the simulation. The other eigenvalue (see Figure 5) is zero oralmost zero during this time, since the table functions are not in the activerange. All of this is as expected and in accordance with the PPM analysis,which also showed the growth loop being the most influential mechanism.The eigenvalue analysis, however, also reveals the roles of several parts of thesystem, such as the demolition loop in dampening the growth rate.

During the following few years (t = 24 to t = 25), the table functions resultingfrom water shortage become active and the loop gains, eigenvalues and influ-ence measures all change rather quickly (see Figures 2–5). Loops 1 and 2continue to play the dominant role in the growth mode, but a new mode(eigenvalue no. 2) now appears, and Figure 5 reveals that loop 3 (whichregulates water consumption as water resources fall) plays an importantrole in this mode. Hence, one may interpret the result as the activation of thefirst-order control on water resources. Note, however, that the growth mode

Fig. 3. Eigenvalues

evolution: Industrial

Structures model



Fig. 4. Loop influenceeigenvalue #1:

Industrial Structures

model.Loop influence of

a loop gain g on

eigenvalue λ is equalto (δλ/δg)g

Fig. 5. Loop influenceeigenvalue #2:

Industrial Structures

model.Loop influence

of a loop gain g on

eigenvalue λ is equalto (δλ/δg)g



(eigenvalue no. 1) is not much affected by loop 3 in the eigenvalue analysis. Inshort, the analysis reveals that loops 1 and 2 primarily regulate the growthmode, while loop 3 regulates the relatively rapid adjustment mode. The LEEAmethod, however, does not reveal how much of each mode is expressed in thetime behavior of a given variable.

Notice also that the two “artificial” loops 4 and 5 no longer cancel each otherout with respect to the influence on the eigenvalues. This may seem like aparadox, given that the loops are never effectively present in the model. How-ever, loop elasticities—and hence our defined measure of loop influence—measure the effect of changing the gain of one loop without affecting the othersand this, in principle, could be achieved by adding a new parameter or chang-ing the formulation.

The results show that one must take great care in interpreting loop elasticitiesand continually refer back to the model formulations to make sure that theloops are “real”.

After time t = 25, the largest eigenvalue becomes negative. At the same time,the significance of the growth loop 1 in determining this eigenvalue dropsquickly and loop 4, which conceptually limits growth via the effect of wateradequacy, shows increasing influence on this mode, indicating that were it notfor the fact that its gain is neutralized by loop 5, this loop could have arelatively large impact on how rapidly industry declines.

In subsequent years, loop 2 emerges to become the dominant structureregulating the gradual decay of industry (eigenvalue 1). This is in accordanceboth with intuition and with the PPM method. There does not seem to be,however, any evidence of changes in the first reference mode to match thetransition at t = 38 between phases III (“exponential decline”) and IV (“balanc-ing decline”) identified by the PPM method. Indeed, we question whether itmakes sense to speak of two distinct phases in this case—the time path of thevariables is simply a temporary transition not driven by structure but by therelative position of the state variables.

The other mode (eigenvalue 2) is mostly dominated by loop 3, except for afew years between t = 25 and t = 30, where loop 6 has a larger influence. Giventhat the influence measures shift so rapidly during this interval, and since thegains of loops 4 and 5 to some extent cancel each other, it is difficult to give aclear account of the dynamics during this period. After time t = 30, loop 3clearly emerges as the dominant loop for this mode. The eigenvalue is negativeand numerically quite large, indicating a rapid adjustment. An intuitive ac-count of what is going on could be that the level of water reserves is rapidlyadjusted to the path of the declining industry stock, via the first-order controlloop 3.

A few lessons can be derived from this first case study. First, loop influence(δλ/δg)g is a convenient, stable, and intuitive measure of a loop’s role in areference mode. Not only does it retain information of the relative importanceloops in the reference mode, but it also avoids problems when eigenvalues are



small or zero. Second, it is clear that eigenvalue analysis is capable of revealingmany subtleties in how structure affects behavior. The analysis, however, caneasily become rather complex, particularly in the presence of “artificial” loopsthat are not in fact active in the model because they cancel each other out.Third, when eigenvalues are stable, LEEA results match those of the PPMmethod. The current tool set, however, does not support an intuitive analysisof the transitions when eigenvalues are changing rapidly. Fourth, LEEA doesnot seem to identify the phase change identified by the PPM between exponen-tial decline (phase III) and balancing decay (Phase IV). Finally, since behaviormodes (eigenvalues) are not related to the time path of specific variables, it isnot easy to make intuitive interpretations of the results. The last three pointssuggest the need to consider the information available in the eigenvectors asthey determine how much of a given mode is expressed in the behavior of agiven variable (see, for example, Chen, 1970; Eberlein, 1984; Hines, 2005).

Lorenz model

The model

Lorenz was a meteorologist studying weather forecasting and its fundamentallimits. He constructed a simple model of fluid motion that was one of thefirst examples of deterministic chaos (Lorenz, 1963). The model has threestate variables governed by the following equations (see the flow diagram inFigure 6):

Fig. 6. Flow diagram, feedback loops, and behavior (phase plot X vs. Z): Lorenz model



X = −c(X − Y)Y = aX − Y − XZ,Z = b(XY − Z)

For values of a ≥ 24.5 the system exhibits deterministic chaos—a pattern ofmovement, typically oscillations that appear to have some random variation inthem but are in fact the result of the nonlinear equations of motion (see, forexample, Mosekilde et al., 1988; Ott, 1993). We use the parameter values thatLorenz used in his original model (a = 28, b = 8/3, c = 10).

Figure 6 also shows a phase plot of the model behavior in X–Z space. Themodel has three unstable equilibrium points, indicated in the figure: X = Y = Z= 0; X = Y = √(a − 1), Z = a − 1; and X = Y = −√(a − 1), Z = a − 1. The behaviortraces out a characteristic butterfly-shaped attractor, where the system initiallymoves in expanding oscillations around one equilibrium point. When theoscillations become sufficiently large, the system moves across the Z-axis tostart oscillating around the other symmetric equilibrium point. The furtheraway from one equilibrium point the system was when leaving it, the closer itmoves to the other equilibrium point as it starts its oscillations around thatpoint. The result is a “strange” attractor (Ott, 1993), which never returns to thesame point in state space, yet shows characteristic behavior patterns inter-cepted by seemingly random jumps between the two regions of state space.The model has six independent feedback loops (see Figure 6), three constant-gain first-order loops connected to each state variable and three loops withvariable gains: {X → X → Y → Y}, {Z → Z → Y → Y} and {X → X → Z → Z → Y →Y}. It was our original expectation that an eigenvalue analysis would revealthat the expanding oscillations were caused by a set of complex conjugateeigenvalues with positive real parts. However, this turns out not to be the case.

LEEA

Figure 7 shows the evolution of the three system eigenvalues during the timeinterval plotted in Figure 6. Although it is not obvious from the figure, ananimation of the movements of the eigenvalues reveals that during most of thetime when the system is oscillating around one equilibrium point, two of theeigenvalues are indeed complex conjugates, but they move around in syncwith the movement in state space. (The third eigenvalue is always real and thusmoving along the horizontal axis, making it difficult to discern in the figure).Thus, the eigenvalues change during each oscillation, obtaining both positiveand negative real parts—which would generally control the rate of expansionor contraction of the oscillations—and also showing great variation in theimaginary part—which would generally control the frequency of oscillation.

A plot of the evolution of the gains of the three loops that have variable gainswould show that the relative strength of the different loops shifts rapidlyaround during each revolution of the simulation. It seems clear, therefore, that



the nonlinearities in the system play an active role at all times. It is possiblethat one might focus on a single cycle of oscillation and look at the shifts inloop dominance over the course of that cycle. Similarly, one might look at thedynamics during a single shift from one region of state space to the other. This,however, is not much different from examining the behavior of the model instate space. It seems that not much additional insight is gained from the loopanalysis, and this approach fails to address the characteristic butterfly attractor.This finding confirms one of the lessons from the first case study that LEEA ismost beneficial when eigenvalues are sufficiently stable to explain behaviorover a period of time.

Mass’s economic model

The model

The third case in this study represents the area that we consider potentiallymost useful for applying the LEEA method: large-scale quasi-linear models,where behavior is for the most part close to an equilibrium point and where thesystem’s Jacobian,3 and hence the eigenvalues and eigenvectors, do not changerapidly over time. To explore the utility of the method, we chose a medium-sized model of economic cycles by Nathaniel Mass (1975).

The purpose of Mass’s study was to explore the mechanisms responsible forthe fluctuations in the macro-economy with a characteristic period of 3–7years, commonly known as the business cycle. Specifically, the work focusedon two alternative theories of the business cycle: one based on fluctuations ininventories and production, and one, more commonly accepted theory, based

Fig. 7. Eigenvalues

evolution: Lorenz

model



on fluctuations in capital investment, which is known to vary strongly with thebusiness cycle. The main conclusion of his study was that the investmentaccelerator theory of the business cycle with realistic parameters would pro-duce fluctuations at least twice as long as the typical business cycle whilefluctuations in inventories, employment and production would typically leadto the business cycle.

We decided to use the LEEA method to test whether it would lead to similarconclusions and also to explore whether this method would be more effi-cient—in terms of required effort on the part of the analyst—than the step-by-step partial model-testing approach used by Mass (1975). In the process, wefound it necessary to convert the table functions used in the original model toanalytical versions to allow for symbolic differentiation in computing theJacobian of the system. In the interest of focusing the analysis, we simplifiedthe model compared to the original version by removing the structures captur-ing variable labor efficiency, price adjustments, and endogenous consumption(§3.5–3.7 in Mass, 1975), but leaving in place the structure modeling fixedcapital as a factor of production (Chapter 4 in Mass, 1975). The resultingstructure is similar to the structure used by Mass in his analysis of the produc-tion sector including labor and capital as factor of production in Chapter 5 ofhis book. Our simplified version of the model contains only nine stocks and55 auxiliary variables (see Figure 8 for a partial flow diagram of the model).The independent loop set for the reduced version of the model has 38 loops(see Table 1 for a list of the loops in the SILS used in our analysis).

Figure 9 shows a simulation of the model starting slightly out of equilibrium.Since there are no additional perturbations, the model settles to equilibriumwithin 30 years. While the model has a relatively short transient, careful visualinspection reveals a pattern of fluctuation around the period of the businesscycle (4–5 years between peaks of production, labor, and inventory) and alonger-term cycle with a period of approximately 20 years (between peaks ofcapital), often referred as the Kutznets cycle.

LEEA

Using eigenvalue analysis, however, it is possible to characterize the behaviorwith more precision. Figure 10 shows the real and imaginary parts of thesystem eigenvalues over time. Both sets of measures are converted to years (astime constants and periods, respectively) to allow for easier interpretation. Allthe real parts of the eigenvalues are negative, as would be expected given thatthe model does not include an economic growth mode. It is, furthermore,evident from the figures that, apart from an initial relatively brief transient, theeigenvalues are largely constant over time, indicating that the model shouldlend itself to the LEEA analysis, being largely linear in nature.

The eigenvalue plots reveal that the model in fact exhibits three cyclicalmodes: a long-term and relatively lightly damped cycle with a period of



about 20 years (the Kutznets cycle), a fairly highly damped cycle with a periodof 4.3 years (the business cycle), and an even more damped cycle of an inter-mediate period of 8.5 years. This latter cycle is not discernible from the timeplots, indicating the limitations of the traditional system dynamics method ofvisual inspection of time plots.

What are the main feedback structures responsible for the observed cycles?Since eigenvalues and loop gains are fairly stable after year 4, it is possible to

Fig. 8. Flow diagram and feedback loops: Mass model (partial).Structure missing from the diagram relates to backlog and other inputs to the shipment rate



Table 1. Loops in Shortest Independent Loop Set: Mass model

Loop # Nodes Loop # Nodes

1 {KOO, KA}2 {K, KD}3 {L, TR}4 {APR, chAPR}5 {Vac, HR}6 {ANVC, chANVC}7 {AOK, chAOK}8 {ANVC, NVC, chANVC}9 {AOK, OK, chAOK}

10 {ANVC, RDOL, MLDOL, NVC, chANVC}11 {Vac, IVC, RDOL, MLDOL, NVC}12 {L, IVC, RDOL, MLDOL, NVC, Vac, HR}13 {ANVC, DVac, IVC, RDOL, MLDOL, NVC, chANVC}14 {AOK, RDKO, MKDOFC, OK, chAOK}15 {KOO, IKOC, RDKO, MKDOFC, OK}16 {KOO, KA, K, IKOC, RDKO, MKDOFC, OK}17 {AOK, DKOO, IKOC, RDKO, MKDOFC, OK, chAOK}18 {BL, BLR, BLMS, SR}19 {Inv, INVR, IMS, SR}20 {K, KR, MKDLC, ALK, KD}21 {L, LR, MLDDE, ADE, TR}22 {APR, PRR, KUF, EK, NormzEK, PR, chAPR}23 {APR, DProd, PRR, KUF, EK, NormzEK, PR, chAPR}24 {Inv, DProd, PRR, KUF, EK, NormzEK, PR}25 {APR, SR, Inv, DProd, PRR, KUF, EK, NormzEK, PR,

chAPR}

Key:

26 {APR, SR, BL, DProd, PRR, KUF, EK, NormzEK, PR,chAPR}

27 {APR, DInv, DProd, PRR, KUF, EK, NormzEK, PR,chAPR}

28 {APR, DInv, INVR, IMS, SR, BL, DProd, PRR, KUF,EK, NormzEK, PR, chAPR}

29 {APR, DBL, DProd, PRR, KUF, EK, NormzEK, PR,chAPR}

30 {APR, DBL, BLR, BLMS, SR, BL, DProd, PRR, KUF,EK, NormzEK, PR, chAPR}

31 {K, EK, NormzEK, PR, Inv, DProd, DK, KR, MKDLC,ALK, KD}

32 {KOO, KA, K, EK, NormzEK, PR, Inv, DProd, DK,IKOC, RDKO, MKDOFC, OK}

33 {K, EK, NormzEK, PR, EGRP, MGDK, DK, KR,MKDLC, ALK, KD}

34 {K, EK, NormzEK, PR, chAPR, APR, EGRP, MGDK,DK, KR, MKDLC, ALK, KD}

35 {L, EL, NormzEL, PR, Inv, DProd, DL, LR, MLDDE,ADE, TR}

36 {L, EL, NormzEL, PR, Inv, DProd, DL, IVC, RDOL,MLDOL, NVC, Vac, HR}

37 {L, EL, NormzEL, PR, EGRP, MGDL, DL, LR, MLDDE,ADE, TR}

38 {APR, PRR, MLDOT, MHY, RLWW, EL, NormzEL,PR, chAPR}

ADE average durationof employment

ALK average lifetime ofcapital

ANVC average newvacancy creation

AOK average orders forcapital

APR averageproduction rate

BL backlogBLMS backlog multiplier

for shipmentsBLR backlog ratiochANVC change in average

new vacancycreation

chAOK change in averagecapital order

chAPR change in averageproductionrate

DBL desired backlog

DInv desired inventoryDK desired capitalDKOO desired capital on

orderDL desired laborDProd desired

productionDVac desired vacanciesEGRP expected growth

rate in productionEK effective capitalEL effective laborHR hiring rateIKOC indicated capital

orders frominventory andbacklogcorrection

IMS inventorymultiplier forshipments

Inv inventoryINVR inventory ratio

IVC indicated vacancycreation forinventory andbacklog correction

K capitalKA capital arrivalsKD capital

depreciationKOO capital on orderKR capital rationKUF capital utilization

factorL laborLR labor ratioMGDK multiplier from

growth on desiredcapital

MGDL multiplier fromgrowth on desiredlabor

MKDLC multiplier fromcapital demand onlifetime of capital

MKDOFC multiplier fromcapital demandon orders forcapital

MLDOL multiplier fromlabor demand onorders for labor

NormzEK normalizedeffective capital

NormzEL normalizedeffective labor

NVC new vacancycreation

OK orders for capitalPR production ratePRR production ratioRDKO relative desired

orders for capitalRDOL relative desired

orders for laborSR shipment rateTR termination rateVac vacancies



Fig. 9. Behavior: Mass model

Fig. 10. Eigenvalues evolution: Mass model.1The time constant of the reference mode represented by an eigenvalue λ is equal to −1/Re[λ] (Ogata, 1990).2The period of the oscillatory reference mode represented by an eigenvalue λ is equal to 2π/Im[λ] (Ogata, 1990)



limit the analysis of loop influence to a representative point in time.4 Figure 11shows, for each of the model’s three complex eigenvalue pairs, a scatter plot ofthe absolute value and real part of influence measures for those 10 feedbackloops that have the largest absolute value of this measure with respect to theeigenvalue at time 9. Each point in the scatter plot represents a loop and theyare labeled with the same numbers as the loops in Figure 8 and Table 1. Thisgraphical arrangement focuses the analysis on the loops most influential onthe mode of interest (the loops with the largest absolute value, the x-axis)while also informing about the direction of loops’ influence in the y-axis(a negative influence measure always implies a stabilizing influence). Thescatter plot allows for visual groupings of loops, thus identifying pieces ofstructure that are “working together”, and for an assessment of distance, i.e.,relative power, between these groups or individual loops.

Fig. 11. Loop influences

on imaginary

eigenvalues: Massmodel.

Loop influence

of a loop gain g oneigenvalue λ is equal

to (δλ/δg)g



Figure 11(a) shows the most influential loops for the 4.3 years oscillation(eigenvalue 3), the “business cycle”. Significantly, we observe that capitalinvestment does not play a major role in the business cycle fluctuation in themodel, confirming Mass’s main result: the structures involved in the businesscycle center around the interaction of workforce, production, and inventory.Note that this result appears immediately from the full model simulationwithout the need for the series of partial model tests conducted by Mass. Morespecifically, the three loops on the lower-right corner of Figure 11(a) (12, 6 and24; labor adjustments and overtime production adjustment) are balancingloops and are stabilizing the business cycle, which accords with intuition,while the loops in the upper-middle (5, 8 and 23) have a destabilizing influ-ence. Both loops 8 and 23 are reinforcing loops related to the anchoring ofvacancy creation and production rates to past values, respectively. They repre-sent a planning inertia where labor and production continue to be adjustedeven after it may not be needed. Intuitively, it makes sense that this inertiadestabilizes the cycle. It is less intuitive that loop 5, which represents thehiring delay, should be destabilizing (a stronger gain equals a shorter averagehiring time, which ought to be stabilizing, one would think.) Our interpreta-tion is that it is an effect of the particular structure chosen for the hiringdecision, which anchors on past vacancy creation. It is interesting to note thatloop 36, the loop that links Labor and Inventory, does have the destabilizinginfluence on the business cycle one would expect from the classic inventory–workforce model, but is not as strong as the interplay of the “faster” loopswithin inventory and labor adjustment.

The intermediate cycle (eigenvalue 5), whose loop influences are depictedin Figure 11(b), is influenced by largely the same structures as the businesscycle, i.e., by inventory–workforce interactions. Loops 6, 12 and 24 (the mainstabilizing loops for the business cycle), however, play a destabilizing role,while loops 4 and 23, adjustments to the average production rate, are the mainstabilizing influence. The reversal of roles (from stabilizing to destabilizingand vice versa) for the loops across the two cycles suggests that, while the laboradjustments do dampen the business cycle, they “resonate” at the 8.5-yearperiod to create imbalances between labor and the desired production rate.

Similarly, Figure 11(c) shows that many of the loops involving capital in-vestment are responsible for the longer-term Kutznets cycle, with loops 7 and16 (capital adjustment) having the same stabilizing influence as their laboradjustment counterparts (loops 6 and 12) had for the business cycle and theanchoring of orders to past values (loop 9) having the main destabilizinginfluence.

A second significant result of the analysis is that the structures involv-ing backlogs do not play a major part in any of the cycles, suggesting thatthe model could be substantially simplified by eliminating all of thisstructure. This was not attempted in the present study but would be an obvi-ous next step.



In summary, one may say that the LEEA method appears to be very useful forboth an accurate identification of distinct behavior modes and for quicklyidentifying coherent pieces of structure (sets of loops) involved in a particularmode. Furthermore, the method yields an interpretation of the influence ofindividual loops within the dominant coherent structure.

Conclusions and discussion

The aim of the paper was to explore the application of the LEEA method to threedifferent models to assess its promise as a general analytical tool. The resultsshow that the utility of the method depends upon the nature of the dynamics.

In single-transient behavior, as seen in the Industrial-Structures model, thebehavior is typically caused by a relatively small number of one-time shifts inloop dominance. The analysis of the Industrial-Structures model showed thatthe method can reveal subtleties in the dynamics and shifts in loop domi-nance, but that the picture can easily become quite complex. The conclusionsregarding dominant structure were, to a large extent, in accordance with thefindings from the PPM analysis, even though the two methods are very different.

In linear and quasi-linear oscillations, demonstrated in the Mass economicmodel, behavior is dominated by a fairly constant set of loops, involving majornegative loops with delays. Here, the LEEA method clearly proved its utility inboth giving a precise description of the behavior modes and identifying theparts of the model structure most important to a given behavior mode. An issuethat we have not yet explored is that in nonlinear systems cycles of differentlengths may interact, as it is seen between the interaction of the business cycleand the long wave in the National model (Forrester, 1989; Forrester et al.,1976), where business cycles tend to grow larger and more severe duringdownturns in the long wave. Exploring this issue, along with phenomenalike mode-locking, would be an obvious next area to explore with LEEA5.

Finally, there are cases of chaotic behavior where the method is fundamen-tally unsuited to yield insights. The analysis of the Lorenz model showed theeigenvalues moving rapidly, in sync with the system variables. The fact thatthe eigenvalues in this model vary as much came as a surprise as we hadexpected that the gradually expanding oscillations occurring during much ofthe simulation were the result of a relatively constant pair of complexeigenvalues with positive real parts. In some ways, the LEEA did yield insightinto the model, because it clearly showed that the behavior is governed byhighly nonlinear dynamics at all times. However, using eigenvalues as expla-nations of system movement requires that they remain relatively constantwhile the system variables change.

The work done for this paper has addressed a number of technical issues.First, efficient algorithms have been implemented to identify the independentloop set and perform the eigenvalue elasticity computations. Second, a more



reliable and robust measure of loop influence has been identified. Finally, a setof tools has been developed to tackle LEEA in an integrated way—it is nowpossible to perform the analysis for any given model (once it has been linearized)in less than 20 minutes. The work with the three models, however, has re-vealed a number of challenges that need to be addressed before the LEEAmethod can be useful to a general audience.

At a technical level, the table, or lookup, functions used in system dynamicsare not very convenient for LEEA, partly because they are somewhat cumber-some to incorporate in the mathematical analysis of the linearized model(though this issue can be remedied), but more significantly because the kinksin the functions create sudden changes in the system Jacobian and hence theeigenvalues. As an aid to modelers, it would be a good idea to develop a libraryof typical functions based on analytical expressions, with a few parameters tofit specific data. Indeed, we believe that such a library may impose extradiscipline upon the model builder in formulating the nonlinear functions.

Another technical issue we encountered in working with the Mass modelwas that the matrix of eigenvectors would sometimes be close to singular evenwhen all eigenvalues were distinct (when the eigenvector matrix is singular,the matrix calculations in Kampmann, 1996, are no longer feasible). Thisshould not be possible in theory, indicating that numerical issues may arise inlarge-scale models. We found that the problem was greatly alleviated by rescalingthe extrinsic (scalable) model variables, but this is not a trivial matter to do in alarge complicated model.

A more fundamental issue arises when the eigenvalues are not distinct. Inthis case, one would have to resort to generalized eigenvectors (Chen 1970).We have not yet explored this issue, but it will have to be addressed at somepoint in time, since it is likely that large-scale models may have many non-distinct (or nearly non-distinct) eigenvalues. Perhaps the best time to take upthis problem is in connection with developing the eigenvector-based analysis,as we now discuss.

A major drawback of the LEEA approach is that it does not relate behaviormodes directly to the time paths of given system variables. In the case of quasi-linear oscillations this may be less of a problem than in the single-transient orlimit-cycle models. The next step in developing the LEEA method wouldtherefore be to use eigenvectors to determine how much a given mode isexpressed in the behavior of the variable. Some researchers are already begin-ning to explore this approach (see, for example, Güneralp, 2005; Hines, 2005;Saleh et al., 2005). Once the analysis can be shifted into the time domain, itremains to be seen how much it can inform the analyst in practice. For in-stance, it may be necessary to further develop approaches similar to the PPMmethod that identify pathways (e.g., to exogenous variables or links from otherparts of the system) in addition to loop structures. Indeed, the relationshipbetween PPM and LEEA needs to be further clarified, including the areaswhere the two methods yield different results.



A related issue is the very large number of loops to consider even inmodels of moderate size, and even if considering only an independent loopset. Creating useful summary measures or visualizations that create a betteroverview for the analyst will be a major challenge ahead. Our method inanalyzing the Mass model may be a useful avenue, where the loops are sorted(automatically) according to the absolute value of their influence measures,and where larger blocks of structure are considered, rather than single loops.But perhaps a form of color coding of the flow diagram of the model, wherehigh-elasticity links “glow” with a stronger color would be more useful, sinceit mobilizes our tremendous capacity to recognize visual patterns.

Eventually, the most revealing test of the LEEA method and its future exten-sions will come when it is applied in practice. To enable a larger group ofmodelers to experiment with the method, we have developed a set of web-based software tools, available to the public at http://iops.tamu.edu/faculty/roliva/research/sd/leea/. We hope that the tools we are providing to automateand standardize the LEEA method, and the insights we have presented con-cerning the power and limitations of the method, will enable researchers totackle the remaining technical challenges of the method.

Notes

1. In order to allow analytical differentiation of the equations, the original tablefunctions were replaced by analytical functions. The match in both structureand behavior is very close, giving us confidence that the two models are equivalent.

2. One problem with the elasticity measure normally used in LEEA,(∂λ/∂g)(g/λ), occurs when the eigenvalues are zero or very small. An alter-native measure, which we call the influence measure, normalizes only bythe gain; i.e., we use the measure (δλ/δg)g. This measure has essentially thesame properties as the elasticity measure, except that it is not invariantto the time scale (but is invariant to any other scales or choice of units inthe model). It shows the change in the eigenvalue relative to a small frac-tional change in the numerical gain. A negative influence measure alwaysimplies a “stabilizing” influence (positive eigenvalues are reduced innumerical values and negative ones are increased in numerical value), andvice versa. We shall henceforth use the influence measure in the analysis.

3. The Jacobian is the matrix {∂x·i/∂xj} of the flows and corresponding statevariables, x.

4. A continuous plot of loop influences indeed confirms that they are largelyconstant over the course of the simulation for all three cyclical modes. Forbrevity, these plots are not included here.

5. Mode-locking or entrainment refers to the tendency of oscillations of differ-ent periods to become synchronized – something that can only occur innonlinear systems (see, for example, Haxholdt et al., 1995).



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