DRAFT
Objective Analysis of Subjective Feedback Structures:
The Problem of Consistency in Explaining Model Behavior
Mohammad Mojtahedzadeh
Attune Group, Inc.
January 2009
Abstract
Real-world concepts can be operationalized into variety of feedback structures which may be
mathematically identical but diverse in the number of feedback loops. Factors including model purpose,
the modelers’ perspective and the intended audience all influence the final layout of a feedback rich
model. One challenge in the analysis of model behavior is to account for the variations in the appearance
of its structure and the feedback loops. This paper focuses on consistency in explaining model behavior
and illustrates some of the issues related to the cancellation problem and figure-8 loops. Both conditions
can potentially lead to poor and even contradictory explanations of model behavior based on its
idiosyncratic feedback structure. The paper concludes by illustrating how the pathway participation
approach addresses these two issues and calls for comparative studies to using alternative approaches to
model analysis to better understand the general principles and subtleties in connecting the structure to
the behavior and explaining observed dynamics. Different methods in formal analysis can learn from one
another and expedite the development of user-friendly tools to aid model analysis that serve a wider
audience.
Introduction
At the heart of system dynamics are consistent, coherent and dynamically correct causal explanations
about how the system’s structure influences its behavior over time. It has been persuasively argued that
intuitive understanding of dynamic systems is prone to error (Forrester, 1994; Peterson et al, 1994).
Although simulation reveals the dynamics of complex systems and facilitates performing “what if”
analysis, it is insufficient in providing consistent explanations about why the system does what it does.
Significant progress has been made in the last several decade in developing tools and methods that can
enhance modelers’ intuition on dynamics consequences of feedback structures. Yet more challenges are in
the way. (Richardson, 1996; Sterman, 2000)
One of the difficulties facing formal model analysis is that real-world concepts can be operationalized
into a variety of feedback structures, which may be mathematically identical', but diverse in the number
of feedback loops. Such diversity can potentially lead to inconsistent and incorrect stories about how the
structure contributes to the behavioral dynamics. “Non-dynamic” feedback loops (Lyneis and Lyneis,
2006), “phantom” loop (Kampmann and Oliva, 2006, 2008) and “figure-8” loops (Mojtahedzadeh, 1997;
* Mathematically identical models are defined, here, as models that may differ the number of feedback loops they
contain, but have the same reduced form. Reduced form of dynamic models can be obtained by substitution of
auxiliary variables into the corresponding net rate equations (Sterman 2000, p. 203).
DRAFT
Giineralp, 2006) present a subset of a bigger challenge related to feedback representation of model
structure that explains its dynamic behavior.
This paper aims to explore the issue of consistency in explanations for systems’ behavior based on its
feedback structure. It discusses how the pathway participation metric (PPM) approach maintains
consistency in mathematically identical models despite variations in the number of feedback loops they
contain. In doing so, the paper focuses on two outstanding issues, cancellation and figure-8 loops that
defy intuitive descriptions of structure-behavior relationships. These two problems are merely a subset of
the larger problem arising from different feedback-loop structures in mathematically identical models;
they, nevertheless, help to understand the consistency issue in model analysis. The paper includes four
case studies that present the cancelation and figure-8 loop problems. The first case study draws upon the
recent work by Lyneis and Lyneis (2006) on simple epidemic models with mathematically identical
equations but different feedback structure. The second and third case studies focuses on figure-8 loops
and illustrate how adding or omitting auxiliaries can hide important feedback loops in the visual
diagrams which are vital in explaining the observed dynamic behavior. The forth case study illustrates
more subtle examples of cancellation problem. Both problems of cancelation and figure-8 loops come
from including additional auxiliaries (or flows) into the structure and can potentially lead to incorrect
and inconsistent explanation of model behavior. The case studies show how the PPM approach detects
the dominant structure and avoids inconsistencies in explaining model behavior, regardless of the choice
in auxiliaries, algebraic expressions and the layout of the model.
Consistency in Explaining Model Behavior
Mathematical descriptions of dynamic systems require state variables and net-flows. It is only these two
classes of variables, as well as their relationships, that determine the dynamics of the systems. However,
for the purpose of better communication and clarity (Sterman 2000), it is often helpful to include
auxiliaries, and to break down the net-flow into meaningful inflows and outflows in the model.
Auxiliaries help to operationalize the model based on real-world concepts and variables. They,
nevertheless, do not impact the dynamics of the closed-loop structure, although, they greatly influence
the visualization of the structure and the number of feedback loops that the structure contains.
Auxiliaries are intermediate variables that “are algebraically substitutable into the subsequent rate
equations and are structurally part of the rate equations” (Forrester, 1968). According to Road Maps?, an
auxiliary is “a subdivision of rate equation that allows a model to be disaggregated into easier to
understand equation statements.” (Road Maps 9, D-4509-2). Sterman (2000, p. 203) encourages modelers
to avoid “economizing on the number of equations” and to include auxiliaries that help to clearly express
the main idea and relevant real world concepts. Similarly, Lyneis and Lyneis (2006, p. 4) state “using
multiple algebraic expressions within variables violates a standard system dynamics modeling practice”.
Despite their contribution in enhancing clarity and ease of communication, the inclusion of auxiliaries
may increase the number of feedback loops within the structure?. Kampmann (1996) derived the number
2 Road Maps is a self-study guide to learning system dynamics developed at MIT under Jay Forrester’s direction. For
more information visit: http://sysdyn.clexchange.org/road-maps/home.html
3 Additional flows (other than net flow) can also increase the number of feedback. Furthermore, any inference about
the net rates of state variable may change the number of feedback loops visible in the structure. (For examples and
further discussions see cases studies reported in next sections of this article).
DRAFT
of feedback loops in a maximally connected system—where each net rate is determined by all state
variables-- given the number of state variables and auxiliary variables. The result of his study is
summarized in Table 1. While most models are not maximally connected, Kampmann’s calculations
show the potential impact of auxiliaries in the expansion of feedback loops in a dynamic model.
Table 1 makes the point that in large-scale models, adding one auxiliary to a model may have a larger
impact on the number of feedback loops than adding one state variable (stock). For instance, adding the
first auxiliary to a third order model may increase the number of feedback loops from 8 to 34 while an
additional state variable will increases the number
of feedback loops to 24. The difference is greater
for higher order models. In a fourth-order system, State | Auxiliary variables
while an additional auxiliary may increase the variable |p " 3
number of feedbacks by a factor of seven, with rqasidal 3 mi
virtually no impact on the dynamics of the system; 5 3 3 1.088
an additional state variable, which may “
dramatically change the dynamics, will only . Bs] 38 [168/708
increase the number of feedbacks by a factor of i ie | oe 10"
three. 5 89 | 1,458 108
Table 1: Number of Feedback Loops in a Fully
Connected System
From: CE Kampmann, 1996, Feedback loop gains
and system behavior
Notwithstanding their contribution in
understanding the equations, enhancing clarity
and ease of communication, auxiliaries present a
challenge to model analysis, that is, how to maintain
consistency and correctness in the explanations for observed dynamics? The generous use of auxiliaries
exponentially increases the number of feedback loops in a model, greatly complicating the task of
consistently explaining mathematically identical models. Since auxiliaries do not create any dynamics of
their own, the 192 feedback loops in a fully connected fourth-order system with one auxiliary are
essentially a “breakdown” of the 24 feedbacks found in the reduced form. Unless carefully analyzed, the
additional feedbacks produced by the auxiliaries introduced in a model can mislead and cause errors and
inconsistencies in detecting the dominant structures.
Although the issue of consistency in explaining model behavior is not adequately addressed in formal
approaches to model analysis, a number of scholars have pointed out to the potential problems and
challenges in working diagrammatic tools to understand the structure (Richardson 1986; Lane 2008)
connecting the its feedbacks loops to the observed behavior. Lyneis and Lyneis (2006) show that due to
alternative formulations some of the feedback loops in a model may be “non-dynamic” that can “obscure
the focus on the essential dynamic loops” (page 19) and distort “an understanding of the direct
relationship between feedback structure” (page 12). In reviewing eigenvalue elasticity approach to formal
model analysis, Kampmann and Oliva (2008) recognize the problem of what they call “phantom” or
“artificial” feedback loops and define as loops that “cancel each other by logical necessity and are
essentially artifacts of equation formulation used in model” (page 513). The authors suggest that the
phantom loops “could nonetheless have large elasticities and thus seriously distort the interpretation of
the results”. Giineralp (2006) also notes that “elasticity of a feedback loop can be negated by elasticity of
another if the gains of these loops contribute to exactly the same compact loop gains” (page 286) and calls
for caution in “interpreting the weighted loop influence plots” when “opposing loops” are present. The
problem of “artificial” feedback loops, according to Kampmann and Oliva (2008), “may not be intractable
but their resolution will require careful mathematical analysis” (page 513).
DRAFT
Consistency in Pathway Participation Approach
The story told about the observed behavior based on the underlying feedback structure using pathway
participation approach remains consistent regardless of the number of auxiliaries and the number of
feedback loops in the model. The main reasons that the PPM approach to model analysis avoids the
problems of figure-8 loops and non-dynamic loops are two folds:
¢ Both identification of feedback loops and detection of dominant structure in this approach are
based on pathway that are recognized with model equations.
¢ The search algorithm identifies the dominant structure in multiple stages.
In pathway participation approach, pathways, links of causal structure between two system stocks, are
envisioned as the primary building blocks of feedback loops. Pathways as construct of the feedback loops
are identified by the model equations and not the schematic display of the model structure. As discussed,
the second and third case studies, the method of indentifying feedback loop by visual inspection of the
structure can be misleading. The second case study demonstrates how PPM reveals a hidden second-
order feedback loop that is needed to explain an oscillatory behavior in the system.
Detecting the dominant structure in pathway participation approach is also based on pathways and not
loops. According to pathway participation approach, the dominant structure is a set of most influential
pathways that connects one state variable to another (and form a feedback loop). In most cases, in the
absence of any auxiliaries there is only one pathway connecting two state variables, however, when
auxiliaries are added to the model, the number pathways connecting two state variables increases, but
one of the pathways will always be dominant. As a result, the order of dominant loop will remain the
same when auxiliaries are inserted or eliminated. The third case study shows that a second order
feedback loop, figure-8 loop, is dominant in the growth phase of urban dynamics regardless of the
presence auxiliaries in the model. Indeed, to assure the consistency in explaining model behavior, it is
essential that the order of the detected dominant feedback loop to remain identical regardless of the
number of auxiliaries.
The pathway participation approach does not compare feedback loops around two different stocks; it
only compares pathway (and first-order loop) reaching the state variable of interest. Any cancellation that
occurs around a state variable is reflected in the total and partial derivatives in the participation metrics.
Consequently, the “non-dynamic” or “phantom loop” will not be selected as dominant. The first case
study demonstrates how non-dynamics feedback loops remain dormant even with alternative
formulation and operationalization of the model. Furthermore, to avoid choosing a feedback loop or a
pathway with large participation metrics that may be canceled out by another, the search algorithm for
selecting dominant structures groups (aggregates) pathways according to the state variables at the head
and the tail of the pathways and select the most influential aggregate pathways. The pathway with
largest the participation metric at the aggregate and individual level will be considered as dominant. The
fourth case study provides two examples containing opposing feedback loops and shows how pathway
participation approach can avoid the potential errors and inconsistencies related to the cancelation
problem.
DRAFT
Case 1: Alternative Feedback Structure for Epidemic Model:
In their article Lyneis and Lyneis (2006) present different versions of the epidemic model which are
“exactly the same equation structure, all producing identical behavior” but they differ in the number of
feedback loop they contain. Figure 1 shows the feedback structures for the four versions of the epidemic
model. All these four structures in Figure 1 are all correct and pass alternative tests developed for
examining the system dynamics models (Sterman 2000), including dimensional consistency, integration
error and extreme condition tests. These models are mathematically identical and produce identical
behavior. Through careful comparison of the four epidemic model versions, Lyneis and Lyneis (2006)
raise an important challenging question related to the analysis of model behavior: “How can the same
?” While only the reinforcing Contagion and
balancing Depletion (or Saturation) feedback loops are sufficient to explain the S-shaped growth in the
behavior of the model, the challenge is to make sure that the additional feedback loops do not become
behavior be explained with such different feedback structures
part of the dominant structure in the three and four-loop models.
Figure 1.a: Two-Loop Epidemic model.
Adopted by Lyneis et al from Business
Dynamics (Sterman, 2000)
Healthy —— Infected Healthy Infected
{ getting sick \ / getting sick =,
{ 3 RA i) { ( AA (+)
\ ae } \ a \ Lo) we
\_ Depletion , Contagion \\ Depletion Contagion
/ Nose at / \ \
Ni fraction /—S Somiaels BY Na traction /— \ santaets by |
hpalthy healthy infected 4 /
\ 4
\ a
Figure 1.b: Three-Loop Epidemic model.
Adopted by Lyneis et al from Road Maps
(Glass-Husain, 1991)
Healthy +—_-S<#_ >| Infected
CaN
if etting sick
Fey Bee t+)
\ Kod 7
\\ Depletion Contagion, 4}
Sa + / \. contacts by
fraction // WA }
‘ healthy « infected /
total ,.
~> population
Figure 1.c: Four-Loop Epidemic model.
Adopted by Lyneis et al from the WPI introductory
system dynamics course (Hines and Lyneis, 2005)
Infected
Pm
t+)
)
getting sick
fi}
an
\ Contagion |
{ ‘
| \\ contacts by
™ infected <~
\
) Saturation
Fraction
~——~ Healthy
y
Figure 1.c: Two-Loop, one Stock
Epidemic Model.
Adopted from Lyneis and Lyneis
DRAFT
The concepts of “relevance”* and “elegance” help to compare and contrast the four alternative structures
of the epidemic model. Lyneis and Lyneis (2006) argue that the two-loop epidemic models are more
elegant because they contain just the two loops needed to explain the S-shaped pattern in the behavior of
Healthy and Infected population. The four-loop model, Figure 1.c, is more relevant to the real world
situation as total population is operationalized in the model; however, they are not as elegant as the two-
loop model because the two extra loops can be misleading in the analysis of simulation outcomes. The
relevance criterion seems to influence some tests outlined for model validation process such as the one
developed for structural assessment (Sterman, 2000). Practitioners working on developing real-world
business models are often concerned about relevance to build on their clients’ confidence that the model
adequately and accurately represent the system. The elegance principle, however, merely help modelers
to assure correct and consistent explanations of the system’s behavior based on its feedback structure.
The balance between elegance and relevance that Lyneis and Lyneis (2006) arguably call for is a nice
solution that may be hard to achieve, if not impossible, in complex large scale models. In fact, the need for
elegance principle mainly comes from lack of tools and techniques for model analysis regardless of how
the model is operationalized and formulated.
The three-loop model possibility leans toward the relevance criterion because the total population is
implicitly operationalized. In the other hand, it only comes with one additional loop which diminishes
the chance of incorrectly picking the dominant structure. However, it is at odd with another standard
which suggests avoiding multiple algebraic expressions within a variable. Compliance with this standard
can lead to even more additional feedback loops. Adding total population as an auxiliary to model
equations helps to avoid multiple equations in fraction healthy, but as Lyneis and Lyneis (2006) have
pointed out, it increases the number of feedback loops as shown in Figure 1.b and 1.c.
The biggest challenge is in commercial models. It is likely that in larger models, a number of feedback
loops will often remain dormant and do not contribute much or even at all in creating the observed
dynamics of the system. Much of these feedback loops are needed to include different perspectives on
how the system work in order to build confidence. However, identifying and highlighting the part of
feedback structure that remain dormant require extensive experience in working with large scale models.
As a result, formal model analysis is perhaps the only practical solution to this outstanding issue of
connecting system’s behavior to its structure.
Using PPM Approach to detect the Dominant Structure in Epidemic model:
The application of pathway participation metrics in the four different feedback structures shown in
Figure 1 suggests that the reinforcing Contagion loop is dominant in the early phase of the behavior of
the Healthy and Infected population which later shifts to Depletion loop. Figure 2 depicts the two phases
of the behavior of Healthy and the dominant structure in each phase, according to pathway participation
metrics. In the first phase that lasts until day 9.5, Healthy population follows a reinforcing pattern.
During this period, Healthy population is highly influenced by the pathway that connects Infected to
Healthy population through contacts by infected and getting sick. At the same time, the behavior of
Infected is driven by the reinforcing Contagion loop. As a result, the Infected-Healthy pathway (that goes
through contacts by infected, getting sick and riches Infected) together with the Contagion loop explain
the reinforcing decline in the Healthy population. Around day 9.5, the Health population experiences a
shift in its behavior pattern from reinforcing to balancing and the Depletion loop dominates for the rest of
+Lyneis and Lyneis (2006) use the word “operational” to describe the concept of “relevance” to the real
world situation.
DRAFT
the simulation. Different algebraic expressions that led to different feedback structures shown in Figure 1
does not seem to change the dominant structure detected by pathway participation metrics.
Reinforcing Phase Balancing Phase
Pi<
Healthy A Infected
0 5 10 15 20
Time (Day)
Healthy 4 ie por
: z <<
Infected f ie
; Healthy | 4 \ \
a |
x Depletion
\. contacts by,“ Me ge
infected getting sick
Figure 2: For All Four Structures in Figure 1, the Contagion Loop is Dominant in the
First Phase of Healthy (and Infected) Followed by Depletion (or Saturation) Loop
The difference between the three and four-loop model is the use of multiple algebraic expressions in the
variable total population. This leads to additional feedback loop. Figure 3 and Figure 4 depict the
pathways that are involved in the Healthy population in three-loop (Figure 1.b) and four-loop (Figure 1.c)
models, respectively. Healthy population in the three-loop model contains one first-order feedback loop
and two pathways the starts with Infected, whereas in the four-loop model, it involves two first-order
loops and two pathways starting with Infected. The difference in the number of loops and pathways in
the two models comes from additional auxiliary, total population, explicitly formulated in the four-loop
model.
+ | Healthy | depletion loop
fraction «~~
~ healthy « Ce
ee ES Infected |.iNfected-healthy
Healthy }¢— Setting « pathway
"sick
= contacts by fanfoclod’heati
: YMinfected |. ifected-healthy
aes uw Contact pathway
Figure 3: Causal Pathways involved in Healthy in Three-loop Epidemic Model
DRAFT
Healthy |-- depletion loop
+
fraction « + _| Healthy [----. "ew reinforcing
4% healthy « wa 2 i healthy loop
a , + 4
- getting « population ~. reckon Wendt
Healthy «—° _. Infected {-----ifected-healthy
—— sick * iia pathway
+
contacts Infected | infected-healthy
by infected + Contact pathway
Figure 4: Causal Pathways involved in Healthy in Four-loop Epidemic Model
Table 1.a, 1.b, 2.aand 2.b show the participation metrics for pathways involved in Healthy and Infected
population for the three-loop (Figure 1.b) and four-loop model (figure 1.c), respectively. According to the
tables, for both models, the total participation metrics for Healthy and Infected population and the
aggregate pathways (and first-order loops) are equal. In both models Infected-Healthy Contact pathway
is dominant in early phase of the Healthy population. During this phase, in both models Contagion loop
is mainly responsible for the reinforcing growth in Infected population. In both models, Depletion loop
remains dominant as long as Healthy population follows a balancing pattern -- total pathway
participation metrics is negative. Therefore, avoiding multiple algebraic expressions and including
additional auxiliary, total population, the new feedback loop that it creates does not change the dominant
structure identified by pathway participation approach.
‘Time (Days) 0 2 4 6 8 95 10 12 14 16 18 20
‘Total PPM for Healthy 1 i 0.99 | 0.94 | 0.61 |-0.02] -0.27 | -0.86 | -0.98 1 al 1
Aggregate first-order loops of] o | o [ -0 | -0.04 |-026| -o4 | -087|-098| 1 | -1 | -1
Depletion loop o {| o | o | © | 0.04 |-026] -04 |-087|-098] 1 | a | 2
Aggregate Infected-Healthy pathways 1 1 0.99 | 0.94 | 0.65 | 0.24] 0.13 0 0 0 0 0
Infected-Healthy pathway 0 | 0 | 0 | -0.03 | -0.16 |-0.25] -0.23 | -0.06 |-o01] -0 | 0 0
Infected-Healthy Contact pathway 1 | 1 | 099 | 097 | 081 |049| 037 | 007 [001 | 0 | 0 0
Table 2.a: Pathway Participation Metrics for Healthy in Three-loop Epidemic Model (Figure 1.b)
‘Time (Days) o [2 4 6 8s [95] 10 | 12 | 14 | 16 | 18 | 20
‘Total PPM for Infected 1_| 1 | 099 | 0.94 | 0.61 |-0.02| -0.27 | -0.86 | -0.98 | -1 | -1 | 2
Aggregate first-order loops 1 [| 1 | 099 | 0.94 | 0.65 | 0.24] 013 | 0 0 0 0 0
Contagion 1 | 1 | 0.99 | 097 | 081 | 0.49] 037 | 007 | 001 | 0 0 0
New balancing infected loop o [| -o | -0 | -0.03 | -0.16 |-0.25] -0.23 | -0.06 | -0.01 | -0 0 0
Aggregate Infected-Healthy pathways | 0 | 0 o | -0 | -0.04 |-0.26] -04 |-087]-098/ -1 | 1 | 2
Healthy-Infected Contact pathway 0 [0 o [ -0 | -0.04 [-0.26] -04 |-087|-098[ 2 | 1 | 4
Table 2.b: Pathway Participation Metrics for Infected in Three-loop Epidemic Model (Figure 1.b)
DRAFT
[Time (Days) 0 2 4 6 s [95] 10 | 12 | 14 | 16 | 18 | 20
Total PPM for Healthy 1 1_| 099 | 0.94 | 0.61 |-0.02| -0.27 | -0.86 | -0.98 | -1 | -1 | -2
Aggregate first-order loops 0 0 0 0 | -0.04 |-0.26| -0.40 | -0.87| -0.98 | -1 | -1 | -1
Depletion loop 0 0 0 | -0.03 | -0.19 |-0.51| -0.63 | -0.93 | -0.99 | -1 | -1 | -1
New reinforcing healthy loop 0 0 0 | 0.03 | 0.16 | 0.25 | 0.23 | 0.06 | 0.01 | 0 0 0
Aggregate Infected-Healthy pathways |_1 1 _| 0.99 | 0.94 | 0.65 | 0.24| 0.13 | 0.01| 0 0 0 0
Infected-Healthy pathway 0 0 0 | -0.03 | -0.16 | -0.25| -0.23 | -0.06 | -0.01 | _ 0 0 0
Infected-Healthy Contact pathway 1 1_| 099 | 097 | 0.87 | 0.49 | 0.37 | 0.07 | 001 | 0 0 0
Table 3.a: Pathway Participation Metrics for Healthy in Four-loop Epidemic Model (Figure 1.c)
‘Time (Days) 0 2 4 6 8s [95] 10 | 12] 14 | 16 | 18 | 20
Total PPM for Infected a 1_| 0.99 | 0.94 | 0.61 |-0.02| -0.27 |-0.86| -0.98 | -1 | -1 | -1
Aggregate first-order loops 1 1_| 0.99 | 0.94 | 0.65 | 0.24 | 0.13 |0.01| 0 0 0 0
Contagion Loop 1 1_| 099 | 097 | 0.87 | 0.49 | 0.37 | 0.07| 0.01 | 0 0 0
New balancing infected loop 0 0 0 | -0.03 | -0.16 | -0.25| -0.23 |-0.06| -0.01 | 0 0 0
Aggregate Infected-Healthy pathways | 0 0 0 0 | -0.04 |-0.26| -0.40 |-0.87| -0.98 | -1 a a
Healthy-Infected pathway 0 0 0 | 0.03 | 0.16 | 0.25 | 0.23 | 0.06 | 0.01 |_ 0 0 0
Healthy-Infected Contact pathway 0 0 0 | -0.03 | -0.19 |-0.51 | -0.63 |-0.93| -0.99 | -1 | -1 1
Table 3.b: Pathway Participation Metrics for Infected in Four-loop Epidemic Model (Figure 1.c)
In both models the new feedback loops, new reinforcing healthy loop and the new balancing infected
loop that emerge because of explicitly formulating total population are dormant. Consequently, the new
feedback loops do not become part of the explanation of the observed behavior of Infected and Healthy
population, and the story about the connection between behavior and structure remains consistent
Inspecting Table 3.a and 3.b indicates the new balancing loop and reinforcing loop have the same
participation metrics in magnitude and different in sign. Because of the similarities between the two
feedback loops, one might conclude that the two first-order loops cancel each other out. Pathway
participation approach does not compare the feedback loops around different state variables. It is true the
two new feedback loops in Figure 1.c disappear by replacing the equation for total population with its
constant value. In practice, it is hard to make such generalization particularly when additional structure
can change total population while maintaining the similarities between the two first-order loops. Using
algebraic equation for total population, in three and four-loop models, introduces additional nonlinearity
that can potentially have adverse impacts on the role of different feedback in the model. In the current
model, such nonlinearity has no dynamic impact since total population is constant. Further, applying the
concept of “cancelation” in the three-loop model does not help much to explain the role of the third loop
around Infected in Figure 1.b.
Case 2: Figure-8 Loop: When Epidemic Model Oscillate
Interestingly, adding one reinforcing loops around Healthy population and one balancing loops around
Infected population in the epidemic model shown in Figure 1.a causes the model to oscillate, while the
addition do not have such impact on the three-loop and four-loop models®. The new feedback loops
® The reason that the three and four-loop models does not oscillate in the presence of the two new loops is that the
total population does not remain constant. In fact, the structures shown in the Figure 1 are mathematically identical
given their existing structure but they may not be, if new structures are added.
DRAFT
brings the number of first-order loops in epidemic model (in Figure 1.a) up to four: Two first-order loops
around Healthy and two first-order loops around Infected. The structure with the additional feedback
loops may not necessarily be meaningful but it helps to analyze the system under different conditions.
Figure 2 shows the feedback structure and the resulting behavior of the new “epidemic model”.
Infected
Za _
new a . Infected
healthy | getting sick y \ \ death
y aA +)
\ | \ V4 |
\ Contagion /
/ Bs
“& fraction pf \\ contacts by
healthy infected
a“
Healthy
i \ Infected
30
Time (Day)
Figure 5: The Feedback Structure and Oscillatory Behavior of the New Epidemic Model
The challenge in the analysis of the ‘epidemic model’ shown in Figure 5 is to explain its oscillatory
behavior based on the feedback structure involving four first-order loops around Healthy and Infected.
However, at least one second-order loop (or higher) is necessary to make sense of cycles in oscillatory
systems (Graham, 1974, Sterman, 2000). Another classical example is the Prey-Predator model. In fact, the
modified ‘epidemic model’ is a special case of Prey-Predator model, when the outflow of Prey, prey
death, happens to be exactly equal to the inflow of Predator, predator birth. Figure 6.a shows the
similarity between the new epidemic and Prey-Predator models. The figure clearly depicts the four
feedback loops around Prey and Predator, still no second-order loop visible in the system to explain the
cycles in the behavior of variables of interest. Figure 6.b presents another layout of the Prey-Predator
model that is exactly the same as the structure in Figure 6.a, and produces exactly the same behavior. The
latter resulted from the omission of auxiliary variables from the former structure and including the
corresponding equations in the inflow of Predator and the outflow of Prey. Consequently, the two
structures in Figure 6.a and 6.b are mathematically identical. The second-order feedback loop around
Prey and Predator is now visible as a result of removing the auxiliaries. With the evident second-order
loops, it is possible explain the cycles in the oscillatory behavior in Prey and Predator in terms of their
feedback structure.
10
DRAFT
Predator
{areca}
=
predator /
sation. TREN CeanRENY
N 5 i ; wa Predator
prey ~—___—“encouters ~~» ,
SN predator. 4%
death
Figure 6a: Prey-Predator Model as Epidemic model Figure 6b: Prey-Predator model:
balancing second-order loop visible
The emergence of the fifth feedback in Figure 6.b comes with the possibility of, at least, two diverse
explanations for the behavior of two algebraically identical structures. The feedback structure shown in
Figure 6.a is known as figure-8 loops and present a special structure where, at least, two loops passing
through two stocks have at least one auxiliary variable in common. In the presence of the auxiliary, the
graphical representation of the structure tends to hide the second-order loop that may play an important
role in explaining the observed behavior. The issue of figure-8 loops is discussed in Mojtahedzadeh (1997)
while analyzing the dynamics of URBAN1 model (Alfeld et al, 1976) using pathway participation metrics.
Giineralp (2006) applies the concept of pathways to identify the hidden loop in Prey-Predator model and
Kampmann and Oliva (2008) describe figure-8 loops as “specific puzzles relating to pathological cases”
(page 518). Figure-8 loops are indeed puzzling as they can lead to inconsistent and incorrect explanations
of observed dynamics.
Using PPM Approach to detect the Dominant Structure in Observed Cycles:
Characterizing the Structure:
Indentifying pathways and feedback loops based on the model equations eliminate the error that can
occur in visual inspections of the diagrams. The pathway participation approach indentifies feedback
loop in the model based on the pathways that connect on state variable recognized by the equation of the
model, not the visual diagrams. As a result, the presence of auxiliaries and alternative algebraic
expression does not prevent identifying dominant loops that remain hidden in schematic diagrams.
depletion healthy
47 [Healthy f=" Toop | Healthy pathway
getting “
- sick *¥__ infected * 7S Ccontagion
ee + Infected } pathway ie Infected | Joop
Healthy Infected —=
jr ie
+ new growth - infected death
healthy —| Healthy | " loop death ~— Infected | “ioop
Figure 7a: Causal Pathways (and feedback loops) Figure 7b: Causal Pathways (and feedback loops)
involved in Healthy involved in Infected
DRAFT
It is, in fact, the presence of the auxiliary variable, encounters, in Figure 6.a that makes the second-order
loop in the Prey-Predator model invisible. To identify the hidden second-order loop in the new epidemic
model in Figure 5, the pathways involved in Healthy and Infected population are identified based on the
model equations. Figure 7.a and 7.b depicts pathways involved in Healthy and Infected population of the
new epidemic model, respectively. According to Figure 7.a, there are three pathways involved in Healthy
population: The two pathways that begin and end with Healthy are in fact first-order loops. The Infected
pathway starts with Infected, goes through getting sick and ends with Healthy population. On the other
hand, Infection population, as shown in Figure 7.b, contains two first-order loops and one pathway that
start with Healthy population, passes through getting sick and riches Infected.
The pathway the starts with Infected, goes through getting sick and riches Healthy population in Figure
7.a along with the pathway involve in Infected in Figure 7.b that starts with Healthy form a second-order
loop. This second-order loop, Interaction, as well as the four first-order loops in the new epidemic model
is depicted in Figure 8.
Dominant Structure in Observed Cycles
The application of pathway participation approach for identifying the dominant structure for observed
cycles in Healthy and Infected population suggests the second-order loop, Interaction, is responsible for
the periodicity of the cycles. Both Healthy and Infected experience longer and shorter half-cycles. For
Healthy population, the longer half-cycles are unstable and their instability is mainly influenced by the
Growth loop. The shorter half-cycles in Healthy population are stable and their stability is driven by the
Depletion loop. In Infected population, the longer half-cycles are unstable and are mainly influenced by
the balancing Death loop while the shorter half-cycles are unstable and are influenced by Contagion loop.
fraction ———-» getting
_~* healthy sick ~~ _
a \ / ‘
nee 4.) a = #8 | ~ Lo
\ eal rd )
healthy y y | a |X
Growth 7 Depletion |; ). 4: | Contagion —7—
LS J \ / Interaction | J~,
—
contacts by 4~
i a
~ ettins
oe keg infected
sick +—
Figure 8: Feedback Structure of the Oscillatory Epidemic Model: The Second-order Balancing Loop
(Interaction) Responsible for the Cyclical Behavior becomes Visible
Table 1 and 2 summarize the pathway stability and pathway frequency factors® for half-cycles’ in Healthy
and Infected. According to Table 1, the first half-cycle in Healthy population begin at time 6.46 and last
° Pathway frequency (pff) and stability factors (psf) are derived from the participation metrics (ppm) in the
beginning of the middle of a half-cycle.
= ppm,
reff, =———
Dp Jno and psf, = ppm |,o/> OF psfy =— pif, /tanlz a; /c)
where ppm is the pathway participation metrics in the beginning of a half is-cycle, @ is the duration of a
half-cycle and ppm’ is the pathway participation metrics in the middle of a half cycle. Notice that these
12
DRAFT
about 5.33 days followed by another longer half-cycle that is 17.9 days long. These two half cycles form a
complete 23.23 days cycle. Table 2 shows that Infected population also experiences 23.23 days cycles,
however, the first half-cycle is 14.88 long and the second half-cycle is shorter and it is only 8.23 days.
Half- |_.. ‘ Infected | Healthy first-order loops 2
Time | duration | factors == = Total Dominant
cycles pathway |Total| Depletion |Growth
1 leas} 533 [Ftea_| 059 | 0 0 0 | 059 Infected pathway
Stab. -0.62 |-0.93| -1.13 0.2 _|-1.55| Depletion loop
> lize] izo [Erea-| 028 | 0 0 0 | 0.18 [Infected pathway
Stab. 0 0.2 0 0.2 0.20 Growth loop
3 |r060| 333 |Frea| 059 | 0 0 0 | 059 Infected pathway
Stab. -0.62 |-0.93| -1.13 0.2. |-1.55| Depletion loop
a laso2] 179 [ftea-| 028 | 0 0 0 | 0.18 [Infected pathway
Stab. 0 0.2 0 0.2 0.20 Growth loop
Table 4.a: Pathway Frequency and Stability Factors for the Half-Cycles in Healthy
Half- |_. , healthy [Infected first-order loops F
Time| duration | factors Total dominant
cycles pathway | Total | Contagion | Death
1 |ao3| agg Frea| o21 | 0 0 0 | 0.21 [Healthy pathway
Stab. -0.06 | -0.61 0.19 -0.8 | -0.67 Death loop
Freq. 0.. 0 0 0 0.38 |Health) thy
2 |23.11) 8.35 x ae = wipe way
Stab. 0.34 1.03 1.83 -0.8 | 1.37 | Contagion loop
Freq. | 0.21 | 0 0 0 | 0.21 [Healthy pathway
3 |31.46) 14.88
Stab. -0.06 | -0.61 0.19 -0.8 | -0.67 Death loop
Freq. . 0 0 0 0.38 |Health thy
4 |46.34) 8.35 a fae = ype wey.
Stab. 0.34 1.03 1,83 -0.8 | 1.37 | Contagion loop
Table 4.b: Pathway Frequency and Stability Factors for the Half-Cycles in Infected
As indicated in Table 4.a, the dominant pathway in creating the frequency of Healthy population is the
Infected pathway that starts with Healthy, passes through getting sick and ends with Infected. Table 2
indicates that the dominant pathway responsible for the frequencies in Infected population is the Healthy
pathway that starts with Healthy and ends with Infected. Therefore, based on pathway frequency factors
the balancing second-order feedback loop (named Interaction in Figure 8) containing both Healthy and
Infected is dominant for the cyclical behavior of the systems.
Table 4.b shows that the short half-cycle in Healthy has a negative total pathway stability factor (-1.55)
which indicates stability. The Depletion loop is mainly responsible for the stability of the short half-
cycles. The longer half-cycles in Healthy population is unstable because of its positive total pathway
stability factor (0.2). The dominant structure for the instability of this half-cycle is the Growth reinforcing
loop around Healthy population. The longer half-cycles in Infected population, based on the information
properties holds for linear systems in steady-states, therefore applying them to nonlinear systems is just
rough approximations. (Mojtahedzadeh, 2009)
” 4 half-cycle defined as a period in which the total pathway participation metric changes its sign from
positive to negative.
13
DRAFT
in Table 4.b, is stable (total stability factor is -0.67) and balancing death loop appears be dominant. For the
shorter half-cycle in Infected population is unstable, the total stability factor is positive and the
reinforcing Contagion loop in mainly responsible for the instability of the half-cycle.
Case 3: Figure-8 Loop in Urban Dynamics
Another structure involving Figure-8 loop is the URBAN1 model (Alfeld et al, 1976). Overshoot behavior
of the URBAN1 model is created by the in interactions among Business Structures, Population and
Housing. Figure 9 shows the part of the model structure the entails the Figure-8 loop and its dynamic
behavior. According to this structure, business constructions is determined by the size Business
Structures, a fraction, business construction normal, and business labor force multiplier. In the other
hand, in-migration is propositional to the Population size and is influenced by attractiveness of job
multiplier. Both Attractiveness of job multiplier and business labor force multiplier are functions of labor
to job ratio.
foo
vw Business
ZN Structures >
~~ business \
Va constructions é Population
/ jobs
business labor
force multiplier
wn «
Sue labor to ag
job ratio ks Business
attractiveness of Structures
jobs multiplier
| 0 4 8 12 16 20
Population
Time (year)
in-migration
mm
Figure 9: Partial Structure of URBAN1 containing a figure-8 Loop and it behavior
For the acceptable range of the model parameters, the structure shown in the above produces a
reinforcing growth observed in the early phase of overshoot patterns. Visual inspection of the diagram
indicates four first-order feedback loops; Business Structures and Population each contain one reinforcing
and one balancing first-order loops. Given the four feedback loops identified from the structure, there are
number of possible ways to explain the reinforcing growth in Business Structures and Population. One
may explain the reinforcing growth in terms of one or both reinforcing first order loops around the state
variables. Conversely, the balancing loops around one state variable may play the role of “catching up” if
the reinforcing loop around another state variable is perceived as dominant. While this explanation for
the reinforcing pattern may sounds plausible, the question is whether it holds if the very same model is
presented with different auxiliaries and algebra?
Depending on the operationalization and the use of auxiliaries, the model can be represented in different
feedback structures. Figure 10.a and 10.b depict two alternative feedback structures for the figure-8 loop
in URBAN1 model. The two new structures are mathematically identical to the original model in Figure
9. The structure in Figure 10.a appears as a result of substituting for labor to job ratio in attractiveness of
job multiplier and business labor force multiplier. It contains five feedback loops including the balancing
14
DRAFT
loops. The structure also reveals a second-order loop involving both Business Structures and Population
with reinforcing polarity. Figure 10.b depicts the feedback structure of the reduced form where the
equations are written in terms of net rates and state variable. The structure contains three reinforcing
feedback loops; two first-order loops around Population and Business Structures and one second-order
loop containing both stocks.
EN a Se
Busmess 7. Business
Structur
rasivioss: fractures He a \ Structures |~
constructions ~Sobs / business
e \ (constructions
business labor
foe
force multiplier
\ we
_
in-migration
a
_ | Population}¢——
in-migration
A
Fi
gure 10b: Three-Loop Representation of the
Figure-8 Loop in URBAN1 Model
Figure 10a: Five-Loop Representation of the
Figure-8 Loop in URBAN1 Model
The “catching up” story told for the four-loop model (Figure 9) is supported by the five-loop model
depicted in Figure 10.a as the structure preserves both balancing feedback loops. However, these
balancing loops disappear in the three-loop model and thus the “catching up” story is no longer valid. On
the other hand, the appearance of the reinforcing second-order feedback in Figure 10.a and 10.b can easily
become part of the explanation of the exactly the same dynamics observed by the figure-8 loop, but it is
essentially inconsistent with the original story. The challenge in model analysis is to provide consistent
explanation for the observed behavior of mathematically identical models despite their differences in the
number of feedback loop they contain. What happens when some feedback loops appear and disappear
as a result of different representation of the same equations? Which one is correct? Which one is more
elegant? Which feedback loops are dominant under what conditions?
Using PPM Approach to Detect the Dominant Structure in URBAN1:
Pathway participation approach indentifies the same dominant feedback loop, and therefore tells
consistent stories, regardless of variations in the number of feedback loop resulting from the choice of
auxiliaries and algebra. Since pathway participation approach detects feedback loops based on pathways
that connect one state variable to another, it recognizes the reinforcing second-order loop in the structure
in Figure 9. In fact, according to pathway participation metrics, the second-order loop is dominant in
creating the reinforcing growth in the model.
Figure 11.a and 11.b depicts pathways for Business Structures and Population in original figure-8
structure. The pathways involved Businesses Structures and Population form a second-order feedback
loop which is difficult to detect by visual inspection of the structure in Figure 9. When both pathways are
identified as the most influential in creating the reinforcing growth in Business Structures and
Population, the second-order loop will be considered as dominant.
15
DRAFT
Business | Reinforcing
su s| loop .
z i ibd P Business] Balancing
jusiness | _ business: “ jobs <——| Toop
Structures fs Ed Structures
constructions. z& +
Sy business labor labor to Bustos
force multiplier + job ratio « labor . SNE
+ force Population Structures
+ Pathway
Figure 11a: Causal Pathways for Business Structures in Figure-8 Loop presented in Figure 9
Reinforcing
Population] —- loop
; 7” . Business | —_ Population
—, ‘an obs «| Business ]__ opulatio
Population Bea - aaa Structures Pathway
2! +—_ attractiveness of labor to « .
+ “jobs multiplier © job ratio + labor Balancing
“4 [Population | ____
+ force ~ P| loop
Figure 11b: Causal Pathways for Population in Figure-8 Loop presented in Figure 9
Table 5.a and 5.b display the participation metrics for pathway involved in Business Structures and
Population shown in Figure 11.a and 11.b. The total participation metrics for Business Structures and
Population is positive indicating a reinforcing growth in the behavior of both state variables. According
to Table 6.a, the dominant pathway for Business Structures is the pathway that begins with Population
and ends with Business Structures. The dominant pathway mainly responsible for the behavior of
Population based on Table 6.b is the Business Structures-Population pathway. The two dominant
pathways are in fact the second-order feedback loop that is invisible in Figure 9 but easily detectable in
Figure 10.a and 10.b. Note that the participation metrics for pathways involved in Population and
Business Structures in Figure 10.a and 10.b are the same those of Figure 9 indicating the appearance of the
structure and the number feedback does not change the outcome of the analysis.
Time (Years) 0 2 4 6 8 10 | 12 14 16 18 20
Total PPM for Business Structures 0.12 | 0.11 | 0.09 | 0.09 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08
Agg. Business Structures first-order loops| -0.04 | -0.05 | -0.05 | -0.05 | -0.05 | -0.05 | -0.05 | -0.05 | -0.05 | -0.05 | -0.05
Business Structures reinforcing loop 0.07 | 0.07 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08
Business Structures Balancing loop -0.10 | -0.13 | -0.13 | -0.13 | -0.13 | -0.13 | -0.13 | -0.13 | -0.13 | -0.13 | -0.13
Agg. Business Structures-Population path| 0.16 | 0.16 | 0.14 | 0.14 | 0.13 | 0.13 | 0.13 | 0.13 | 0.13 | 0.13 | 0.13
Business Structures-Population path 0.16 | 0.16 | 0.14 | 0.14 | 0.13 | 0.13 | 0.13 | 013 | 0.13 | 0.13 | 0.13
Table 5.a: Pathway Participation Metrics for Business Structure in Figure 9
Time (Years) 0 2 4 6 8 10 12 14 16 18 20
Total PPM for Business Structures 0.04 | 0.04 | 0.06 | 0.07 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08
Agg. Population first-order loops -0.07 | -0.16 | -0.18 | -0.19 | -0.19 | -0.19 | -0.19 | -0.19 | -0.19 | -0.19 | -0.19
Population reinforcing loop 0.10 | 0.09 | 0.09 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08
Population Balancing loop 0.17 | -0.26 | -0.26 | -0.27 | -0.27 | -0.27 | -0.27 | -0.27 | -0.27 | -0.27 | -0.27
Agg. Population-Business Structures path] 0.11 | 0.20 | 0.24 | 0.25 | 0.26 | 0.27 | 0.27 | 0.27 | 0.27 | 0.27 | 0.27
Population-Business Structures path 0.11 | 0.20 | 0.24 | 0.25 | 0.26 | 0.27 | 0.27 | 0.27 | 0.27 | 0.27 | 0.27
Table 5.b: Pathway Participation Metrics for Population in Figure 9
16
DRAFT
In pathway participation approach the balancing loops around Population and Business Structures will
not be identified as dominant as long as both stocks experience a reinforcing growth. As a result, the
“catching up” story that is based on a significant role for the balancing loops is not supported by the
pathway participation approach.
Case 4: Back to Loop Cancellation:
Adding an inflow and outflow to the partial URBAN] structure in Figures 9, 10a or 10b would not change
the behavior of the model, if both flows are sufficiently close to one another. Figure 12 displays the partial
URBANI structure shown in Figure 10b. In the new structure, h, and h, are assumed to be large, but the
difference between the two is very small, so
the impact of the new reinforcing loop on the
behavior is almost cancelled out by that of the
balancing loop. The question is whether the
formal methods would mistakenly pick the
new first-order loops for large values of h, ,
instead of the second-order feedback loop
containing Business Structures and
Business
Structures
A...
/ business
Population. Of course, this is a very simple s
/constructions
example and it is easy to recognize the two
new first-order feedback loops cancel each
other out. In more complex models,
identifying what Giineralp (2006) calls
“opposing” loops, especially higher order
ones, may be more difficult. , x
| Population <——~—
in-migration
A
As discussed in the previous section, according —
to pathway participation approach, under
Figure 12: Additional Feedback Loops in
certain conditions, the second-order URBAN] Model
reinforcing feedback loop is dominant. The
presence of the two new loops, regardless of
the value of hi and hz, does not influence the outcome. The reason lies in the search algorithm for
choosing the dominant structure based on pathway participation metrics. In choosing the dominant
structure, the PPM approach first groups all pathways with the same state variables in the head and tail
of pathways and then it picks the most influential aggregate pathway. In the second round, the most
influential pathways is selected within the grouped pathways leaving and reaching the same state
variables. In the partial URBAN] structure in Figure 12, the two new feedback loops are almost cancelled
out, therefore the aggregate pathway participation metrics for pathway that starts and end with Business
Structures remains unchanged.
Table 6 provides the pathway participation metrics for Business Structures in Figure 12. The aggregate
participation metrics for the first-order loops in Business Structures do not change in the present of the
new loops as they cancel each other out although not exactly. As a result the second-order loop involving
Business Structures and Population remains dominant.
7;
DRAFT
Time (Years) 0 2 4 6 8 10 | 12 | 14 | 16 | 18 | 20
0.13 | 0.12 | 0.11 | 0.11 | 0.11 | 0.10 | 0.10 | 0.10 | 0.10 | 0.10 | 0.10
‘Agg. Business Structures first-order loops| -0.03
‘Total PPM for Business Structures
-0.04 | -0.04 | -0.04 | -0.04 | -0.04 | -0.04 | -0.04 | -0.04 | -0.04 | -0.04
Business Structure reinforcing loop 0.07 | 0.07 | 0.07 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08
Business Structure balancing loop -0.10 | -0.12 | -0.13 | -0.13 | -0.13 | -0.13 | -0.13 | -0.13 | -0.13 | -0.13 | -0.13
New Business Structures balancing loop | -100
-100 | -100 | -100 | -100 | -100 | -100 | -100 | -100 | -100 | -100
New Business Structures reinforcing loop |100.01|100.01| 100.01 | 100.01
[Agg. Business Structures-Population path | 0.16
100.01 | 100.01 | 100.01 | 100.01 | 100.01 | 100.01 | 100.01
0.17 | 0.16 | 0.15 | 0.15 | 0.15 | 0.15 | 0.15 | 0.15 | 0.15 | 0.15
Business Structures-Population path 0.16 | 0.16 | 0.14 | 0.14 | 0.13 | 0.13 | 0.13 | 0.13 | 0.13 | 0.13 | 0.13
Table 6: Pathway Participation Metrics for Business Structures in Figure 12
Another and even more subtle problem of cancellation can be observed in the epidemic model. To
illustrate the problem, two additional inflow and outflow are included in the one-stock-two-loop model
shown in Figure 1.d. The new structure is shown in Figure 13. Again, k, and k, , are assumed to be large
numbers but the difference between the two is very small. As a result the impact of the new reinforcing
loop on the behavior is almost cancelled out by that of the balancing loop. The challenge is to select the
dominant feedback for the balancing phase of the behavior of Infected for large values of k, .
getting sick. \ \
pa ) \ |
\ Contagion / —\ |
X /
J /
‘contacts by | k
— infected
/ Saturation = /
/
fraction
~ healthy <~
Figure 13: Adding a reinforcing and balancing loop around Infected
For large values of k, and therefore k, , the two new feedback loops may sound reasonable candidates to
replace Contagion and Saturation feedback. Drawing upon the elasticities of the feedback loops around
Infected may also lead to the similar conclusion as the eigenvalue elasticities of the new loops appear to
be larger than those of the old ones. However, the new balancing first-order loop is obviously a wrong
choice for explaining the balancing growth in Infected as it lacks the nonlinearity to facilitate the shift in
from reinforcing to balancing growth.
DRAFT
According to the pathway participation approach, the new reinforcing feedback loop is dominant in the
reinforcing growth of Infected population, however, the dominant loop shifts to the Saturation balancing
loop and drives a balancing growth in Infected. Table 7 shows the pathway participation metrics for
Infected broken down by linear and nonlinear feedback loops. According to the table, the total pathway
participation metrics for Infected remains positive in the first 9.5 day indicating a reinforcing growth in
Infected. During this period, aggregate first-order linear loops, including Contagion, new reinforcing and
balancing loops, and therefore, the new reinforcing loop is dominant. After day 9.5, the total participation
metrics are negative and the dominance shifts to the Saturation loop.
[Time (Days) 0 2 4 6 8 | 950 | 10 | 12 | 14 [| 16 | 18 | 20
‘Total PPM for Infected 1 1 | 0.99 | 0.93 | 0.62 | -0.02 | -0.26 | -0.86 | -0.98 | -1 - -l
Agg. linear first-order loops 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99
Contagion loop 1 1 1 1 1 1 1 1 1 1 1 1
New reinforcing first-order loop| 100 | 100 100 100 | 100 100 100 | 100 100 100 100 100
New balancing first-order loop |-100.01}-100.01) -100.01 |-100.01}-100.01| -100.01 |-100.01}-100.01| -100.01 |-100.01] -100.01 }-100.01)
Agg. nonlinear first-order loop | 0.00 | 0.00 | -0.01 | -0.06 | -0.38 | -1.02 | -1.26 | -1.86 | -1.98 | -2.00 | -2.00 | -2.00
Saturation Loop 0.00 | 0.00 | -0.01 | -0.06 | -0.38 | -1.02 | -1.26 | -1.86 | -1.98 | -2.00 | -2.00 | -2.00
Table 7: Pathway Participation Metrics for Infected Population in Figure 13
As a heuristic, the search algorithm in pathway participation approach also groups pathway, according to
their degree of nonlinearity and then selects the dominant structure based on the magnitude of the
participation metrics. In the above example, the Contagion and two new loops are aggregated in one
group because they have the same degree of nonlinearity. In the second phase of the behavior of Infected,
the Saturation alone outweighs the significance of the other three loops together and dominant.
Conclusions and Discussions:
Real-world concepts can be operationalized into variety of feedback structures which may be
mathematically identical but diverse in the number of feedback loops. Auxiliaries that help to better
operationalize system dynamics models, achieve clarity and avoid confusion in algebraic equations can
increase the number of the feedback loops without contributing to the dynamics of the system under
study. As a result, consistency in explaining model behavior in terms of its feedback structure can present
a challenge for formal approaches to model analysis.
The case studies reported in this paper focus on two important issues of loop cancellation and figure-8
loops. These two problems are merely a subset of the larger problem arising from the role of auxiliaries
and the variations in the feedback-loop structures of the same model; they, nevertheless, help to
understand the consistency issue in model analysis. Although the issue of consistency in explaining
model behavior is not adequately addressed in formal approaches to model analysis, a number of
scholars have pointed out to the potential problems and challenges in connecting the structure and
behavior. Kampmann and Oliva (2008) describe figure-8 loops as “specific puzzles relating to
pathological cases” that can “seriously distort the interpretation of the results” of eigenvalue analysis.
Giineralp (2006) calls for caution in “interpreting the weighted loop influence plots” when “opposing
loops” are present. Lyneis and Lyneis (2006) recognize the problem of “non-dynamic” loops that can
“obscure the focus on the essential dynamic loops”.
19
DRAFT
Both cancellation and figure-8 problem arise from adding auxiliaries --as well as additional flows and any
inference about the flows-- to the model to achieve various purposes such as clarity, ease of
communications and relevance to real-world. The first case study shows that it is, in fact, focusing on
relevance to real-world concepts and introducing new auxiliaries leads to additional feedback loops,
some of which may be non-dynamics or dormant. The second case study makes the point that merging
the outflow in Healthy population that happens to be equal to the inflow of Infected population—to
make a different point not related to the concept of feedback loops-- hide a second-order loop (Figure 5).
This feedback loop would have been visible if the two flows were not merged (Figure 6.b).
The third case study focuses on the partial structure in URBAN1 model and demonstrates that
operationalizing labor to job ratio and formulating it as a separate auxiliary masks the second-order
feedback loop. The second-order loop would be visible in the diagram when labor force to job ratio is
formulated in the multipliers (Figure 10.a) or the rates (Figure 10.b). The point of examples in the fourth
case studies is to reiterate the additional flows (or auxiliaries) can lead to new pathways and feedback
loops that may cancel each other out.
Modelers developing real-world business models are often required to focus on “relevance” to assure
that the model adequately and accurately represent the actual system. As a result, the final model may
contain more feedback loops needed to explain observed behavior. User friendly tools and easy to
interpret techniques for model analysis may be a viable solution to achieve consistency in explaining
observed dynamics regardless of how the model is operationalized and formulated.
The story told about the observed dynamics based on the underlying feedback structure using pathway
participation approach remains consistent regardless of the number of auxiliaries and the number of
feedback loops in the model. The application of pathway participation metrics in case studies reported in
this paper demonstrate that identifying feedback loops in the model through pathways and model’s
equation, and not the schematic display of the structure, help to avoid the problem of cancellation and
figure-8 loops.
The paper calls for comparative studies using alternative formal methods in model analysis. Comparing
the outcome of different formal as well as intuitive approaches to model analysis can help to better
understand the general principles and subtleties in explaining observed behavior in terms of its feedback
structure. Different methods in formal model analysis can learn from one another and expedite the
development of user-friendly tools to aid model analysis and serve a wider audience.
20
DRAFT
Appendix A:
This Appendix present the equations for the models used in the case studies.
Case Study 1: Epidemic Models
Two-Loop Epidemic Model
Healthy = Healthy, + J-getting sick *dt
Infected = Infected, + J getting sick “dt
getting sick = fraction healthy * contact by infected * infectivity
fraction healthy = Healthy/total population
contacts by infected = Infected * contact rate
contact rate = 2, infectivity = 0.5, total population, = 10000 . Healthy, = total population, - Infected, .
Infected, =1 , , ,
Three-Loop Epidemic Model
Healthy = Healthy, + J-getting sick “dt
Infected = Infected, + J getting sick “dt
getting sick = fraction healthy * contact by infected * infectivity
fraction healthy = Healthy /(Infected + Healthy)
contacts by infected = Infected * contact rate
contact rate = 2, infectivity = 0.5, total population, = 10000 , Healthy, = total population, - Infected, .
Infected, =1 , ‘ ‘
Four-Loop Epidemic Model
Healthy = Healthy, + J-getting sick *dt
Infected = Infected, + J getting sick “dt
getting sick = fraction healthy * contact by infected * infectivity
fraction healthy = Healthy /total population
contacts by infected = Infected * contact rate
total population = Infected + Healthy
contact rate = 2. infectivity = 05, total population, = 10000 ; Healthy, = total population, - Infected, ;
Infected, =1
Two-Loop, One Stock Model
Infected = Infected, + J getting sick “dt
getting sick = fraction healthy * contact by infected * infectivity
fraction healthy =(total population, - Infected) /total population,
contact rate = 2, infectivity = 0.5, total population, = 10000 . Infected, =1
21
DRAFT
Case Study 2: Epidemic Model that Oscillates
Healthy = Healthy, + j (new healthy - getting sick) “dt
Infected = Infected, + J (getting sick - Infected death) *dt
new healthy = Healthy * healthy fraction
Infected death = Infected * infected fraction
getting sick = fraction healthy * contact by infected * infectivity
fraction healthy = Healthy/total population,
contacts by infected = Infected * contact rate
healthy fraction = 0.2 , infected fraction = 08. contact rate = 2. infectivity = 0.5,
total population, = 10000 . Healthy, = total population, - Infected, . Infected, =1
Case Study 3: URBAN1 Partial Structure, figure-8 Loop
Business Structures = Business Structures, + [business constructions *dt
Population = Population, + Jinmigration “dt
business constructions = Business Structures * business construction normal * business labor force multiplier
inmigration = Population * inmigration normal * attractiveness from job multiplier
business labor force multiplier = business labor force multiplier function (labor fouce to job ratio)
business labor force multiplier function = (0,0.2),(0.2,0.25),(0.4,0.35),(0.6,0.5),(0.8,0.7),(1,1),(1.2,1.35),(1.4,1.6),
(1.6,1.8),(1.8,1.95), (2,2)
attractiveness from job multiplier = attractiveness from job multiplier function (labor force to job ratio)
attractiveness from job multiplier function = (0,2),(0.2,1.95),(0.4,1.8),(0.6,1.6),(0-8,1.35),(1,1),(1.2,0.5),(1.4,0.3),
(1.6,0.2),(1.8,0.15),(2,0.1)
jobs = Business Structures * jobs per business structures
labor force = Population * labor participation fraction
Business Structures, = 1000 ; Population, = 50000 ; business construction normal= 0.07 ;
inmigration normal = 0.1; labor participation fraction = 0.35; jobs per business structures = 0.18
Case Study 4: Back to the Cancellation Problem
Example 1:
Infected = Infected, + J (getting sick + k, * Infected - k, * Infected) *dt
getting sick = fraction healthy * contact by infected * infectivity
fraction healthy = (total population, - Infected) /total population,
k, = 100 ; k, = 100.01 ; contact rate = 2, infectivity = 05, total population, = 10000 ; Infected) =1
Example 2:
7 7 business constructions + h, *Business Structures
Business Structures = Business Structures, +] “dt
- h,*Business Structures
Population = Population, + Jinmigration “dt
22
DRAFT
business constructions = Business Structures * business construction normal * business labor force multiplier
inmigration = Population * inmigration normal * attractiveness from job multiplier
business labor force multiplier = business labor force multiplier function (labor fouce to job ratio)
business labor force multiplier function = (0,0.2),(0.2,0.25),(0.4,0.35),(0.6,0.5),(0.8,0.7),(1,1),(1.2,1.35),(1.4,1.6),
(1.6,1.8),(1.8,1.95),(2,2)
attractiveness from job multiplier = attractiveness from job multiplier function (labor force to job ratio)
attractiveness from job multiplier function = (0,2),(0.2,1.95),(0.4,1.8),(0.6,1.6),(0-8,1.35),(1,1),(1.2,0.5),(1.4,0.3),
(1.6,0.2),(1.8,0.15),(2,0.1)
jobs = Business Structures * jobs per business structures
labor force = Population * labor participation fraction
Business Structures, = 1000 ; Population, = 50000; business construction normal= 0.07 ;
inmigration normal = 0.1; labor participation fraction = 0.35; jobs per business structures = 0.18 ;
h, = 100.01; h, = 100
23.
DRAFT
References
Alfeld LE, Graham A. 1976. Introduction to Urban Dynamics. Cambridge, Mass.: MIT Press. Reprinted by
Productivity Press, Portland, OR. (Now available from Pegasus Communications, Waltham, MA.)
Forrester J. 1994. System dynamics, systems thinking, and soft OR. System Dynamics Review 10(2-3):
245 - 256.
Forrester, J. W. (1968). Market growth as influenced by capital investment. Industrial Management
Review (MIT), 9(2). 8, 16
Graham AK. 1977. Principles of the relationship between structure and behavior of dynamic systems. PhD
dissertation, Sloan School o Management, Massachusetts Institute of Technology: Cambridge, MA.
Glass-Husain W. 1991. Teaching System Dynamics: Looking at Epidemics, MIT SDEP Road Maps, D-
4243-3, http://sysdyn.clexchange.org/sdep/Roadmaps/RM5/D-4243-3.pdf
Giineralp B. 2006. Exploring structure-behavior relations in nonlinear dynamic feedback models. PhD
dissertation, University of Illinois: Urbana-Champaign, IL.
Giineralp B. 2005. Towards coherent loop dominance analysis: progress in eigenvalue elasticity analysis.
System Dynamics Review 22(3): 263-289.
Hines JH, Lyneis JM. 2005. Lectures Slides for SD550, System Dynamics Foundations: Managing
Complexity, WPI, Session 7: 56-64.
Kampmann CE, Oliva R. 2008. Structural Dominance Analysis and Theory Building System Dynamics.
Systems Research and Behavioral Science 25, 505-519.
Kampmann CE, Oliva R. 2006. Loop eigenvalue elasticity analysis: Three case studies. System Dynamics
Review 22(2): 141-162.
Kampmann C. 1996. Feedback loop gains and system behavior. Proceedings of the 1996 International
System Dynamics Conference, Boston. System Dynamics Society, Albany, NY.
Lane David C. 2008. The Emergence and Use of Diagramming in System Dynamics: A Critical Account.
Systems Research and Behavioral Science 25, 3-23.
Lyneis JM, Lyneis, DA. 2006. Two Loops, Three Loops, or Four Loops: Pedagogic Issues in Explaining
Basic Epidemic Dynamics. In Proceedings of the 2006 International System Dynamics Conference, New
York, NY. System Dynamics Society: Albany, NY.
Mojtahedzadeh M. 2009. Do the Parallel Lines Meet? How Can Pathway Participation Metrics and
Eigenvalue Analysis Produce Similar Results? System Dynamics Review 24(4): ?-??.
Mojtahedzadeh M, Andersen D, Richardson GP. 2004. Using Digest to implement the pathway
participation method for detecting influential system structure. System Dynamics Review 20(1): 1-20.
Mojtahedzadeh M. 1997. A path taken: Computer assisted heuristics for understanding dynamic systems. PhD
dissertation, University at Albany, SUNY: Albany, NY.
Peterson D, Eberlein R. 1994. Reality check: a bridge between systems thinking and system dynamics.
System Dynamics Review 10(2-3): 159-174.
Richardson, GP. 1996. Problems for the future of system dynamics. System Dynamics Review 12(2): 141 -
157.
Richardson, GP. Dominant structure. System Dynamics Review, 1986. 2(1): 68-75.
Sterman JD. 2000. Business Dynamics: Systems Thinking and Modeling for a Complex World.
McGraw-Hill: New York: 301.
24
