Table of Contents
Using Digestto Implement the Pathway Participation Method for
Detecting Influential System Structure
Mohammad Mojtahedzadeh David Andersen George P. Richardson
Attune Group, Inc. Rockefeller College Rockefeller College
mohammad@ attunegroup.com University at Albany, University at Albany
State University of New Y ork State University of New Y ork
Albany, NY 12222 Albany, NY 12222
david.andersen@ albany.edu gpr@ albany.edu
Abstract
This paper briefly defines and describes the Pathway Participation Metric (PPM), a
mathematical calculation that can help to identify the linkages between the structure and
behavior of a dynamic system. PPM has been implemented in an experimental piece of
software called Digest, which we then present. Digest is not a simulation language, but
rather a companion to any commercial system dynamics package, which accepts a text
version of a simulation model and performs post formulation analysis of the model.
Digest detects and displays which feedback loops are most influential in explaining a
selected pattem of behavior in a model. Output from a sample Digest mn is presented
and described.
System Stories: Understanding How and Why Patterns of System Behavior
Arise from Most Influential System Structure
An important purpose of most system dynamics modeling efforts is to help managers
better understand the systems which they manage and in which they live. One key task in
this search for insightful, system level understanding is the telling of “system stories” --
coherent, dynamically correct explanations of how influential pieces of system structure
give rise to important pattems of system behavior!
A key task in creating system stories is accurately detecting exactly what part of the
system structure gives rise to (or contributes most importantly to) some pattem of
behavior identified in one or more simulation runs. Richardson (1996) has identified this
task as one of the key research problems presently facing the field of system dynamics.
Over the years, this problem has been examined using a range of approaches (Graham,
' Of course, other valid purposes for client-based systems work exist. For example Senge (1990) and many
others advocate the use of Systems Thinking approaches that do not rely on formal simulation. However,
these approaches may suffer from other conceptual limitations starting with something as simple as proper
interpretation of causal-loop diagrams (Richardson, 1986).
1977; Forrester, 1982; Eberlein, 1984; Kampmann, 1996; Davidsen, 1991; Ford, 1999;
and Salleh, 2002.
Despite the importance of understanding the linkages between the structure and dynamic
behavior in simulation models, tools to accomplish this task are lacking. Most skilled
practitioners approach the challenge of identifying the most influential structure with
some combination of intuition and analysis coupled to a program of repeated simulations,
testing hypotheses about what structure controls what behavior in a controlled way with
some of experimental logic? Years of experience with system dynamics models is
needed for launching artful hypotheses and testing them via repeated simulation, and no
satisfying accounts exist in the published literature prescribing a precise set of steps for
completing this key task. Even experienced modelers experience difficulty in testing
their hunches about the connection between structure and behavior.
For linear dynamic systems, some mathematical tools exist to make this trial-and-error
process more tractable. Indeed, modes of overall system behavior have a clearly cbfined
meaning for linear systems. System behavior is understood to arise from a linear
combination of the dynamics associated with the eigenvalues of the linear matrix of
system structure. Hence, the calculation of a system’s eigenvalues can go a long way
toward describing overall behavior modes of a linear system (Eberlein, 1989).
Closely coupled to eigenvalue analysis of dominant modes of system behavior is the
notion of “dominant loops.” Usually thought of intuitively as the feedback structure
“most important’ in determining some portion of the dynamics of a system, dominant
loops can be seen as a reduced set of closed feedback paths that contribute most to the
overall behavior mode of a model. Indeed, for linear systems, one can work out
mathematical relationships between a set of such dominant loops and the system's
eigenvalues (Forrester, 1983; Kampann, 1996). Nonlinear systems, however, have the
capability to shift loop dominance and therefore require more than what eignevalue
analysis can provide (Mosekilde, 1996; Forrester, 1987).
The work presented in this paper continues in the line of eigenvalue and dominant loop
analysis in that it continues the search for formal analytic approaches to support the
detection of which pieces of system structure contribute most to selected pattems of
system behavior. However, in contrast to previous attempts to solve this problem, our
approach does not focus on eigenvalues or on dominant loops as the key building blocks
of influential system structure’. Rather pathways, links of causal structure between two
? For example standard texts in the field such as Richardson and Pugh (1981)describe an
approach to model analysis that relies on repeated simulations, as does Goodman (1974)
and Sterman (2000).
3 Indeed, the PPM method can be related to eigenvalue analysis. Appendix B
demonstrates that for a second order linear system, the PPM method produces values that
are mathematically related to the two dgenvalues of the system. The same result can be
shown to hold for higher order linear systems.
system stocks, are envisioned as the primary building blocks of influential structure.t Of
course, one or more pathways can define closed feedback loops. Under this new
approach, some combinations of pathways (some of which form closed feedback loops)
define the most influential system structure. This most influential system structure,
explicitly linked to a pattem of behavior identified by the modeler, forms the basis for
creating insightful system stories.
This paper briefly reviews the conceptual underpinnings of this new pathway- determined
approach and then presents an experimental piece of software, Digest, that can be used to
implement this approach.
Pathway Participation Metrics (PPM): A Mathematical Algorithm for Detecting
Most Influential Structure
Mojtahedzadeh (1996, 1997) has proposed the Pathway Participation Metrics (PPM) as a
mathematical tool that could help support modeler intuition in dealing with the task of
unraveling relationships between system structure and system behavior. The basic
behavioral building block of the PPM is a single phase of behavior for a single variable.
A single phase of behavior for a selected variable is a time slice of the simulation where
the selected variable maintains the same slope and curvature (first and second time
derivatives). Hence, there are seven pattems of behavior that may exist within a single
phase: (1) reinforcing growth, (2) linear growth, (3) balancing growth, (4) reinforcing
decline, (5) linear decline, (6) balancing decline and finally, (7) equilibrium. Figure 1
depicts these seven pattems of behavior.
reinforcing
growth linear semis
growth oF
Tm Time Time
reinforcing
decline bess balancing
cline decline
Time Time Time
equilbrmm
‘Time
Figure 1: Seven pattems of over time behavior
* Actually, while most pathways are from one system state variable to another, some
pathways can connect a state variable to an ordinary auxiliary variable found in the
system “between” two or more state variables. See Mojtahedzadeh (1996) for a more
formal definition of a pathway.
The PPM approach begins when the modeler analyst selects a variable of interest. The
PPM approach will detect what structure of the model is most influential in determining
the behavior of this selected variable. Figure 2 below shows a typical Sshaped growth
for some variable X selected to be studied in a hypothetical system. The Digest software
slices the time path for X into discrete pattems representing the seven pattems of
overtime behavior. For the example shown in Figure 2, the trajectory of X consists of
only two time slices—an initial time slice of reinforcing growth followed by a second
time slice of balancing growth. Once the time trajectory for the selected variable has
been decomposed into separate patterns, the PPM approaches answers the question,
“What structure is most influential in explaining one pattem of over time behavior for the
selected variable?" For the example shown in figure 2, these questions reduce to "What
structure in the model most influences the initial phase of reinforcing growth in this
system?” and then sequentially, “What structure in the model most influences the
balancing growth phase of the simulation?”
Fast Time tice
Re mforcing Growth
Fart Yiaue tlice
Balacme Gx wih
Tame
Figure 2: Digest "slices" the hypothetical time trajectory for X into separate pattems of
over time behavior according to its slope (X ) and curvature (X )
The mathematics of the PPM sets out to determine which pathway from a system state to
the variable of interest contributes most to the current behavior pattem of that variable.
This apparently simple question requires some mathematics to be answered.
The PPM calculates how much the net-flow (X ordX/dt) could change given a small
change in the state variable under consideration, (dX /dX ); this is called Total Pathway
Participation Metrics. SincedX /dX can be transformed into dX /dt divided bydX /dt, the
Total Participation Metric contains information about both slope and curvature of the
variable of interest and is thus an appropriate tool for analyzing behavior’. This measure
of the Total Participation Metric for state variable X is then partitioned among pathways
coming into the net-flow’. The most influential pathway is defined as the one whose
patticipation is the largest in magnitude and has the same sign as the total changes in the
net-flow x-dot when it is disturbed by a infinitesimal change in the state variable at the
5 Mohamed Saleh, 2002, also use slope (X ) and curvature (X ) and calls it BPI
(Behavioral Pattem Index) to characterize behavioral pattems.
® Richardson (1995) proposed that the net time derivative of a state variable with respect
to the state variable itself (dX /dX ) can be an important measure of when a loop shifts
dominance. The PPM approach calls dX /dX the Total Pathway Participation Measure.
tail of the pathway. For a more complete description of pathway participation metrics see
Appendix A).
A simple example might help to clarify what is going on here. Figure 3 shows a
hypothetical fourth order system showing only 4 of what might be a much larger number
of pathways. Two pathways, P; and P2, lead from state X2 to change X;. Only one
pathway, P3, connects the state variable X3 with X;. And finally pathway P, represents
the linkage between X; and X;. If X; is the selected variable and is showing reinforcing
growth as indicated in Figure 2, the PPM approach asks the specific question, "Which of
these four pathways dominates the initial reinforcing growth of X; (contributes the most
todX/dX )?".. The PPM approach answers this question by calculating the partial
contribution of each of the four pathways to the total pathway measure, dX /ax , and then
selecting that pathway that has the same sign and greatest magnitude as dX/dX . Let us
assume that all these calculations identify P» as the most influential of the four pathways
shown in Figure 3. We now know that X2 has the strongest influence on X2 and
furthermore that influence is exerted through the pathway P2. But what structure now
influences X2 the most? The process of analysis continues.
Pl P4
Figure 3: The PPM approach selects pathway P2 as most influential in the behavior of X1
Figure 4 gives a more complete look at the structure of our hypothetical system indicating
10 pathways and numerous closed loops. The second iteration of the PPM approaches
now seeks to identify which pathway (Ps or Ps?) contributes most to the behavior of X.
The calculations are the same as in the first iteration. The PPM approach computes
dX 2dot/dX2 and the relative contributions of pathways P; and Pe to that total. Let us
assume that pathway P6 is selected as the most influential at this stage. We now know
that Xi is most strongly influenced by X2 through the pathway P2 and that X2 is most
strongly influenced by X3 through the pathway Ps. The next iteration of the PPM
approach would ask "Which pathway, Pg or Pio, most strongly influences X3. If the
answer in our hypothetical example were to be RB, then we have identified a closed loop
that begins with and ends with X1.
P6
Figure 4: Schema of major and minor loops in hypothetical fourth order system
That closed loop is isolated and displayed in Figure 5. The interpretation of this figure,
similar to figures generated by the Digest software, is that the reinforcing feedback loop
involving X1, Pg, X3, Pe, X2, and finally B, is the loop most influential in determining the
initial reinforcing growth of X;. Figure 4 contains a large number of major as well as
minor loops that could have contributed to the initial reinforcing growth in the selected
variable X;. What the PPM has done is to select three pathways that are connected into a
single loop and has identified that loop as the most influential of all other possible loops
in determining the initial growth in the system. The PPM approach does not always
identify a single major loop. Sometimes the most influential structure may be a minor
loop or in some cases the system's dynamics may be mainly influenced by an exogenous
time series. Frequently a pathway will lead from the selected variable of interest to
another minor or major loop located far from it in the overall causal structure of the
model. However, it can be shown that repeated application of the PPM mathematics at
each step does converge on a unique piece of structure identified as most influential for a
given behavior pattem of the variable of interest.
The PPM approach concludes by moving on to the next time slice that differs from the
previous one in slope or curvature. The analysis for each time slice is similar. Note that
the most influential structure identified for each slice of time may vary.
Third pathorsy
P2 closes the loop
First pathoray Pa
[= | =
x P6 2
identified
Figure 5: Schema depicting dominant major loop from hypothetical fourth order system
shown in Figure 4.
This very brief explanation of how the PPM approach works skips over all of the
interesting mathematical details of how the contributions of each pathway are actually
computed. An overview of the mathematics underpinning the PPM calculations is
provided in Appendix A and full details are provided in Mojtahedzadeh 1996. An
example using a model characterized by system overshoot is presented below.
One problem with PPM as an algorithm is that it is cumbersome and difficult to compute
and no existing commercial simulation software packages support these calculations.
Digest is a piece of experimental software that automatically computes the PPM and then
uses information from the PPM to automatically detect and display influential pathways
and feedback loops.
Digest is not a simulation package such as iThink, Stella, Vensim, or Powersim. Digest
cannot support most of the simulation functions that these languages can. Digest is
designed to be used after the model has been constructed to detect and display influential
structure. Of course, at some point in the future, the relevant and most useful features of
Digest could be integrated into any of the commercial simulation packages.
Digest accepts model equations from any commercial simulation package in text form.
In its present version, some hand editing of the text equations may be necessary if the
model uses macros or functions that are not yet parsed by the Digest equation translator.
Once a text version of the model equations has been edited and accepted by Digest, the
software leads the modeler through a series of step-by-step procedures that uses the PPM
calculation to first detect and then display model structure’.
Using Digest to Analyze a Simple Overshoot Model:
This section analyzes the behavior of one variable in a simple overshoot model using
Digest. In doing that we need the @yuations of a “simulatable” model saved in a text file
format. The model used as an example is a classic structure that illustrates how Industrial
Structures in a particular region grow over time until all the resources needed to support
the growth of Industrial Structures are depleted. Figure 6 depicts the structure of the
model. (A list of the equations of the model is provided in A ppendix C).
” Digest is currently available in a Beta test version. Readers interested in experimenting
their own with this Beta version are encouraged to contact the authors fora copy of
Digest.
Industrial
new
industies -SUCTES = demolision
=z or aus
4
water
effect of water demand
shortage wee
Water Reserves
water consumption
"
availabilit
Figure 6: A simple model for the growth of Industrial Structures
The model captures three real- world processes:
1; Industrial Structures grow with new industries through a reinforcing loop
and demolish by a balancing loop;
2. Industrial Structures consume water, which decreases water reserves;
3. Water shortage (defined as the ratio of water consumption to water
demand) affects new industries indirectly;
4. Water availability (defined as the ratio of water reserves to water demand)
controls water consumption.
For an appropriate set of parameters and initial values, the model generates an overshoot
in the behavior of Industrial Structures, while Water Reserves follows an Sshape decline.
In explaining the behavior of the model the question is what feedback loops are more
influential in generating the behavior of any variable of interest. For example, what is
making Water Reserves to decline rapidly and what controls it? What is driving
Industrial Structures to grow rapidly in the first few years? What part of the structure is
responsible for the decline of Industrial Structures followed by its growth? For modelers
who have worked with this sort of model, it is not difficult to explain the growth phase
and the declining phase of the behavior of this simple structure. However, it may not be
as easy to distinguish what part of the structure contributes most to the behavior of
Industrial Structures in the transition from reinforcing growth to a balancing decline.
Using Digest one can identify the most influential structure as the behavior of the model
unfolds.
The outputs of Digest:
Once a model is loaded in Digest environment, using the information embedded in the
equations for the model, it could produce four different outputs. These outputs are:
1. A list of variables of the model by which the user could select the variable of interest.
2. Digest automatically identifies pathways associated with the user-selected variable of
interest
Once the variable of interest is chosen, the causal route associated with the behavior of
interest will appear in the second window. This diagram reveals how the variable of
interest is determined by other variables in the model. For the Industrial Structure as the
variable of interest, Figure 7 shows the causal route diagram that is associated with
Industrial Structures. Arrows with a plus sign indicate a direct (positive) impact of the
variable at the tail of the arrow at the dependent variable and an arrow with a negative
sign refers to an indirect impact of the cause on the effect (a negative or indirect
relationship).
‘The causal route for industrial_Structures
demolition
va
oe Me _ F aie Structures
—
Industries avarentt
:
Structures
Figure 7: Causal route diagram for Industrial Structures
3. Digest identifies distinct phases in the behavior pattern of user-selected variable of
interest
Digest produces the overtime behavior of the variable of interest and identifies the shifts
in the pattem of behavior. Figure 8 shows the behavior phases of the variable of interest,
Industrial Structures.
industrial
Structures
Behavior snd phases of Industril_Structures
Set
‘el
2et
tet
0.00 12.50. 25.00 37.50. 50.00
Figure 8: The behavior of Industrial Structures and its four phases
The first phase of Industrial Structures is a reinforcing growth. The reinforcing growth
lasts for 24 years. During the first 24 years of simulation time, both slope (first time
derivative) and curvature® (second time derivative) of the variable of interest, Industral
Structures, remain positive. The next distinct phase in the behavior of Industrial
Structures, identified by Digest, is balancing growth. In this phase the slope and
curvature of the variable of interest have opposite signs. The third distinct behavior
phase in Industrial Structure is reinforcing decline. And finally, in its fourth phase, the
variable of interest experiences a balancing decline in its over time behavior.
4. Digest detects and displays most influential structure contributing to behavior pattern
ineach phase
Corresponding to the first phase of the behavior of Industrial structures, there is a
reinforcing feedback loop that, according to Digest, is the most influential feedback loop
in generating the reinforcing growth in Industrial Structures. The reinforcing feedback
loop is shown in Figure 9. Based on this feedback loop a higher level of Industral
Structures attracts more new industries, which in tum increases Industrial Structures. By
inspecting the structure of the models in Figure 6, one could identify about 6 feedback
loops. Using pathway participation metrics, Digest automatically selects the reinforcing
feedback among all the other loops in the model without intervention by the model
builder or analyst.
Wai] Froninentstotures for halstil_Sruchires arise [=]
From 0.00 to 24.00
—_*
(R) nein
+
rh
Figure 9: The most influential structure in creating the first phase of the
behavior of Industrial Structures
The most influential structure in creating the second phase of the behavior of the variable
of interest shifts from the reinforcing feedback loop to a balancing feedback loop
associated with Water Reserves. Figure 10 depicts the balancing loop that controls water
consumption as Water Reserve continues to fall, along with a pathway that caries the
effect of the balancing feedback loop to the variable of interest, Industrial Structures.
5 Actually Digest calculates neither the first nor second time derivative of the variable of
interest; it merely determines dX /dX at any time. This derivative is related to first and
second time derivatives of the variable of interest. A positive sign of the derivative
indicates that both slope and the curvature of the variable of interest have the same signs.
(See Mojtahedzadeh 1996 for details).
10
This structure remains most influential in the third phase of the behavior of the variable
of interest.
<[ 202 [>] Prominent structures for Industril_Structures LAlktoop [2
From 24.26 to 38.00,
+ + + + :
[costae —newindust. a aftect of amg —water_shor
~\ ;
ian sce i woes
ed
‘effect_of_
Figure 10: The most influential structure in the second and third phases of
the behavior of Industrial Structures
It may not be difficult for the modelers to see the role of the balancing feedback loop that
controls water consumption, when striving to explain why Industrial Structures is
generating a balancing growth in its second phase. Water availability is dropping and,
therefore, new industries are restricted. The subtlety in explaining the behavior of the
variable of interest is the subsequent reinforcing decline in the behavior of Industrial
Structures in phase four. Some novices may even look for a reinforcing feedback loop to
explain the reinforcing decline. Digest reveals that what forces Industrial Structures to
fall faster and faster is exactly the same process that controls it. The balancing loop that
controls water consumption continuously lowers new industries and once new industries
falls behind industrial demolition, the Industrial Structures generates a reinforcing
decline.
The last phase of the behavior of the variable of interest, Industrial Structures, is
influenced the most by the balancing feedback loop associated with demolition, as shown
in Figure 11.
W2e2 DI Prominent structures for Indusirl_ Structures [Antes
From 36.2 to 60.00 |
a,
(B) demotion
Figure 11: The most influential structure in the fourth phases of the
behavior of Industrial Structures
Digest could redraw distinct phases in the behavior pattem of user-selected variable of
interest based on shifts in the most influential structure, as shown in Figure 12.
Set
et
fet 4 Time
0.00 12.50 25.00 37.50 50.00
Figure 12: The essential structure for explaining Industrial
Structures growth model
Conclusion
Model analysis, the process of understanding and then describing how the structure of a
complex dynamic system gives rise to over time behavior, is still in its relative infancy.
Well developed mathematical techniques exist for linear systems as well as for some
regimes of complex non-linear dynamics such as deterministic chaos (Andersen, 1982;
Mosekilde, 1996). However, in common practice with client- based modeling, skilled
modelers create dynamic insights using carefully crafted simulation experiments to
formulate explanations about what pieces of model structure drive the overall system
behavior. But this intuition is difficult to codify and develops slowly over a career of
practice. Modem software promises to help. For example, the latest release of Vensim
allows for real time visualizations of structural sensitivities using brute force computing
power and speed to create these visualizations.
This paper explores a promising additional approach. The Pathway Participation Metric,
described in overview form in this paper, relies on the analysis of individual linkages or
pathways between nodes of a model as the basic building blocks of structure. The
approach leads to dominant feedback structure, if that’s appropriate, but does not begin
with the feedback loop as the basic building block. Using a recursive heuristic systematic
analysis, the PPM calculations always yield a reduced structure of a key feedback loop
plus one or more pathways that contribute most to a given mode of behavior for a
selected model variable.
Important questions remain about this approach. Do the automatically identified “most
influential structures” yield important insights for clients working on real world
problems? Do clients and modelers alike have a strong enough intuition about the PPM
12
to “trust” the structure that it identifies?? How can a set of loops, each of which is
connected to a single phase of behavior, be combined into a fuller explanation of the
complete dynamic trajectory of a single variable? How can analyses for two or more
variables be merged into a coherent story of the system taken as a whole?
Digest is an experimental piece of software that can help us begin to answer these
questions. Because Digest automatically and quickly analyzes and displays the results
from the PPM calculations, we now have a tool that will allow us to experiment with yet
another approach to the critical question of how to quickly and reliable relate system
structure to system behavior.
In the near future, all system dynamics simulation software packages will contain new
functions that support automatic model analysis'°. We view Digest as an early
experimental tool to move the field toward this future. We hope to encourage a vigorous
experimental program to move questions and results in this critical area of inquiry
forward.
° Mojtahedzadeh (1996) has begun an investigation of these last two questions by working with a number
of models, such as the simplified Urban Dynamics model presented by Alfeld and Graham (1976).
However, this work needs to be extended and deepened.
19 Automation of model analysis functions within standard software packages is essential for their uptake in
practice. For example, the “Reality Check” feature advocated by Peterson and Eberlein (1994) was made
possible as a practical tool by being integrated into Vensim.
13
Appendix A: A Mathematical Definition of Pathway Participation Metrics
This appendix introduces the mathematics of pathway participation metrics (PPM).
Consider the following n- order non: linear system:
=f(x;p) (1]
Where x is the vector of state variable x is the vector of derivative of x with respect to
time, and p is the vector of the parameters of the system. The equation of the k® state
variable as the variable of interest may look like:
Re = ACK, Xp yeer Kyi D) (2]
Taking the derivative of the net changes in the state variable of interest, x, , with respect
to the state variable of interest, x,, yields:
di, _ af, dx, af, dx, af, dx, af, ax,
= beseech Paseh [3]
dx, ax, dx, "Ox, ax, ax, dx, ax, ox,
Or simply,
= = Y = (for % #0) 14]
Each term in equation [4] represents all minor feedback loops and pathways leaving i
state variable and coming into the variable of interest, x, We can decompose the effect of
each minor feedback and pathway coming into the state variable xx.
n mi) =
= “22 ax aK [5]
Where mii) is number of minor loops and pathways that leave a i state variable and
come into the k® state variable, and af,! /ax, is the polarity of the pathway or minor
feedback loop. The ratio x, /, represents the net changes in the i” state variable and the
net changes in k" state variable. The total effect infinitesimal change in xj, of the net rate
of Xx is the not only driven by the polarity of the feedback loops and pathways but also
the ratio of net changes in the two state variables.
The effect of each pathway can be normalized in such a way that it varies between - 1 and
1. Thus for each pathway coming into variable of interest we could have a metric that
14
measure the impact of that pathway (or minor feedback loop) in creating the behavior of
the variable of interest. This metric is called pathway participation metrics (PPM).
of, X&
icane ox, &,
PPM(i,j) = ice (6]
ax &,
The most influential pathway (or minor feedback loop) is defined as the one whose
participation metrics (PPM) is the largest and has the same sign as dx,/x, . For i#k
the same calculation is done until there is a feedback loop.
If the variable of interest is a non-state variable, we need to determine the net changes the
variable of interest and the follow the same procedure. Suppose a presents the vector of
non-state variables and it is related to state variables through g and a vector of parameters
q. Thus we have,
a =g(x;q)
If the axis the variable of interest, that is a non state variable, the net changes in a, over
the period of dt will be,
, — OG, .
HV Os: 7
ay 2 ag [7]
Taking the derivative of the net changes in the variable of interest, a,,, with respect to the
state variable of interest, a. yields:
ag, 495, a 8
ae axaa, * ax ead Bl
Which can be rearranged as:
= Dery ag, dk 9
ao Re OXAx; a, rs EB
The pathway participation metrics can be determined after decomposing the impact of
each pathway leaving a particular state variable and coming into the variable of interest.
15
Appendix B: Pathway Participation Metrics and Eigenvalues
In linear systems there is a close relationship between the pathway participation metrics
and eigenvalue of the system. In fact we show that
¢ Inthe steady state condition, the total participation metric is equal to the largest
eigenvalue of the system
In doing so, we use a second order linear system and derive pathway participation metrics
for a state variable. Then, we show that the sum of the pathway participation metrics, or
total participations metrics, for any state variable equal the largest eigenvalue of the
system.
Consider the following second order system:
x =axthy (1)
y =cx +dy [2]
The pathway participation metrics for state variable x is:
oe aap [3]
dx x
There are two pathways coming to the state variable x whose participation metrics are:
Participation metrics for Pathway 1: a
Participation metrics for Pathway 2: bp»
x
The pattem of behavior of x is determined by the total participation metrics, which is the
sum of participation metrics for these two pathways. If total participation metrics is
positive the state variable x experiences a reinforcing growth and if it is negative, x shows
a balancing behavior. The most influential pathway then is the one whose participation
metrics is the largest in magnitude and has the same sign as the total participation
metrics.
Now we calculate y/xthrough the response of state variables x and y. We can rewrite
the second order linear system presented in [1] and [2] as:
Oe “f one (41
d
The above system has two eigenvalues, A, and A,. For each eigenvalue we have:
16
ar, +br, =An; [5]
cr, +dr, =A, [6]
Where r, and nr, are the elements ¢ the right eigenvector associated with A. The time
response of the state variables is:
aor [7]
Where x, and y, are the initial values of the state variables and ¢(t) with the dimension
of 2*2 is the transition matrix of the system which can be calculated as:
o=Se Ras fy] [8]
Where f,, and f,,are the elements of the left eigenvector associated with A.
Substituting [8] in [7] and expanding it yields:
Hee” if, erate bf anh (9]
t 12. fi i, fy 0 22 f, Ty fn 0
The value of x and y at any time is:
x =(f, £.% +H fy e™ +5 £.% +h frye” [10]
Y=(hf.% He frye +5, 6% +n, frye’ (11]
We can calculate x and y by taking derivatives of [10] and [11] with respect to time.
K=A(H £% +H Lyle" +05: £% +h Lyle” [12]
YHA(n, £.% +h, Lr yle™ +A, G, £.% +t) frye” [13]
Using [12] and [13], we calculate the ratio of y/ x.
fa fii) His fe WE" +A, Gy f% +t frye” 14]
m1 £.% tH faye” +A, £:.% ee
as
Or
ACE
AC
¥ AG fi% He frye +A fr% + fn Yo) [15]
x AG
1f.x, th fy lel +A, (n, fy +0, fy)
17
Assuming A, is the largest eigenvalue, when time approach infinity tems
Atm fim Hp frye”? and A(q, £.% +m, fy, Je"? in [15] approaches zero.
Thus, for y/ x we have
(h) £.%) Hy fo¥0)
us
[16]
A (G1 f% +h fev)
ve
x
The above equation can be rewritten as:
Ya 17
x Th {17]
Now we can substitute [17] in [3],
dx G
— =a th 18
a " [18]
Equation [18] according to [5] is equal to A,.
an [19]
It can be easily shown that the above proposition is true for an n-order system.
18
Appendix C: A list of the equations of the Industrial Structures Growth
Model (iThink version)
Industrial_ Structures(t) = Industrial_Structures(t - dt) + (new_industries - demolition) *
dt
INIT Industrial_ Structures = 10
new_industries = Industrial_Structures*effect_of_water_shortage*normal_growth
demolition = Industrial_Structures*dem_fre
Water_Reserves(t) = Water Reserves(t- dt) +(- water_consumption) * dt
INIT Water_Reserves = 10000
water consumption = effect_of_water_availability*water_ demand
dem_fre = .05
normal_growth =.12
water demand = Industrial_Structures*water demand_per industry
water demand_per industry = 10
effect_of_water_availability = GRA PH(0.1*Water_Reserves/water_demand)
(0.00, 0.00), (0.1, 0.06), (0.2, 0.14), (0.3, 0.255), (0.4, 0.395), (0.5, 0.535), (0.6, 0.685),
(0.7, 0.825), (0.8, 0.92), (0.9, 0.975), (1, 1.00)
effect_of_water_ shortage = GRA PH(water_consumption/water_ demand)
(0.00, 0.00), (0.1, 0.06), (0.2, 0.14), (0.3, 0.255), (0.4, 0.395), (0.5, 0.535), (0.6, 0.685),
(0.7, 0.825), (0.8, 0.92), (0.9, 0.975), (1, 1.00)
19
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