Dominance Analysis Using Pathway Force Decomposition
Jeremy B. Sato
July 16, 2017
Abstract
This paper proposes a formal and mathematically rigorous framework for defining dom-
inance based on nec
ry and sufficient conditions. A procedure is outlined for identifying
dominant structure which is tested against several forms of the logistic growth model. The pro-
cedure is compared with other dominance methods such as PPM, LEEA, Ford’s method, and
the Loop Impact method. Relationships are identified between each method. The framework
also indicates the robustness or fragility of a system. Shifts in dominance were found to occur
when systems exist in fragile states in the dominance framework. The framework captures
both structural and behavioral aspects of dominance found in other dominance methods. It
also establishes a mathematical basis and definition for observed phenomena such as shadow
loop domi ce and Itiple loop domi: e.
1 Introduction
Loop dominance methods have at times produced conflicting or inconsistent results. One of the
challenges in model analysis is the lack of a formal and rigorous definition of loop dominance.
This paper proposes a formal definition and criteria for dominance. Using this definition, a
procedure is constructed for identifying dominant structure which is then tested against several
forms of the logistic growth model. The procedure is then compared with other dominance methods,
and relationships are identified between each method. The paper concludes with a discussion on
how the proposed dominance framework can be used to characterize the robustness or fragility of a
system.
2 Definitions and Implications
Consider the dynamic system described by the following n“’-order ordinary differential equation
(ODE):
X(t) = f(x(#), u())
where,
Ty fi Uy
12 fa u2 (1)
x= : f=|°. u= :
En fn Un
where each f; is differentiable in x and in u;
dL.
Towards the goal of identifying elements of system structure which dominate behavior, the
following definitions are proposed for structure, behavior, and dominanc
Behavior. Behavior is defined as the sign of the second time derivative of a state variable of
interest, a(t), evaluated at a specific point to along the state’s trajectory.
behavior of x;(t) at time ty = sgni,(to)
Structure. The explanatory elements of system structure are the immediate causal pathways
from state variables (21, v2,...,2,) to the first time derivative of the state variable of interest
a, (t) = fj(x, uj), as shown in Figure 1. A causal pathway pjj,(xi) is a scalar function representing
a causal process which maps a cause, state x;, to an effect, state derivative 7;. There may exist
more than one causal pathway (representing different causal mechanisms) from 2; to #;. In such
cases, pijp is the k” pathway from 2; to tj.
pathway Py ja ()
pathway Pw) x,
pathway Pj(-)
—_—— a ae),
x, = f/(%4,) :
pathway Pra),
pathway Pao)
n
pathway P,jp,(-)
Figure 1: Immediate causal pathways from state variables (x1, v2, ... ,%,) to the derivative of the
state variable of interest (;).
Dominant. Given a state variable x;(t) whose behavior is of interest, and point tp along its
trajectory, causal pathway pijx is dominant if and only if it is both necessary and sufficient for
determining sgn #;(to).
2.1 Decomposing Behavior into Pathway Contributions
From Equation 1, the dynamics of #; can be succinctly written as a function of the state variables:
&; = fj(@1, £2, ..., Lay Uj) (2)
However, in order to distinguish between multiple causal pathways from a common state variable,
the dynamics can alternatively be expressed as a function of the pathways depicted in Figure 1, as
shown in Equation 3.
sg
9g = FilDigas Digby «++ y Digks <9 Pagar Drgbs «+ +4 Prijs tty) (3)
From Equation 3, is derived using the chain rule of differentiation, and expressed as a sum of
contributions from each immediate causal pathway pjjx:
af. , fj
. Of; .
ae) Bie + = ipt aa $ Bigg to +
Oprja | Opin”? Opign
In Equation 4, the term
quantifies the contribution of causal pathway pj,j, to the second derivative (%;). By analogy to
Newton's second law of motion, this term can also be conceptualized as the force exerted by pijx
on the variable x;, causing an acceleration. The first factor of this term (the partial derivative)
represents the gain of p;;,, which is the change in f; with respect to changes in p;;,. The second
factor is the rate of change of p;;x. The second derivative, or acceleration, of the variable of interest
(%;) is expressed as the sum of the force contributions from each immediate pathway, and is visually
depicted using a free body diagram as shown in Figure 2.
acceleration " ah By tent o; Put Wh ay
a = jg tees ye Paes ain
decomposition 1 Oy, Dx Dim
force analogy net _ pathway athway p, athway Py», input u
= Pathway Pye, Pathway py, pathway Py, input uy
Yo force yer force fore *"* fore *"* force force
% '
free body | | :
i 0
iagram | 7
Figure 2: Free body diagram illustrating the contribution of each pathway to the acceleration of
the variable of interest.
2.2 Illustration: Pathway Decomposition for the Logistic Growth Model
The logistic growth (or population growth) model, also known as the Verhulst equation as discovered
by Pierre-Francois Verhulst in 1838 (see [11]) is a common model for examining shifting influence
3
population
birth fraction ratio of population to death fraction
ee | capacity
+ ae +
; ‘ \
normal birth normal death
fraction carrying fraction
capacity
Figure 3: Stock and flow diagram of population growth.
between reinforcing and balancing feedback loops. Figure 3 shows a stock and flow diagram of the
logistic growth model.
The center box variable, population, is the single stock (state variable). The double-lined arrows
with valves flowing into and out of population are the flow variables (components of the state
derivative), associated with births and deaths. The clouds represent unconstrained population
sources and sinks. R1 is the reinforcing feedback loop associated with the birth process, in which
births add to the population, further increasing the birth rate. B1 is the balancing feedback loop
associated with deaths, in which deaths decrease the population, which subsequently slows the death
rate. B2 and B3 are the balancing loops associated with the constraints of a fixed environment
with a carrying capacity (maximum population size the environment can sustain based on finite
resources). The ratio of the current population size to its carrying capacity affects the birth fraction
and death fraction (also known as the birth and death fractional rates), which represents the
fraction of births and deaths per unit of time. Normal birth fraction and normal death fraction
are the fractional birth and death rates in an unconstrained environment. The dynamics of logistic
growth are governed by the following equations:
d
ap Population = births — deaths
A P P
P=b- (1-2) “P- a-(Z) -P
where (5)
P = population
b = normal birth fraction
d= normal death fraction
C = carrying capacity
4
Through a simple change of variables, system 5 can be re-written as a function of the four
distinct pathways from P to P, representing feedback loops R1, B1, B2, and B3:
P = pua(P)- pir(P) + pus(P) - puo(P)
where
R1: pii(P) = P
B1: py2(P) = —P
Pp:
B2: pyi3(P) =d- (Z)
B3: pua(P) =b- (1 - )
However, logistic equation 5 is also sometimes expressed in the following equivalent form [11, 12]:
P=aP-P?
where
a= (7)
8 (b+ d)
Cc
This alternate form illustrates how for a >> 6, when P is relatively small, the reinforcing growth of
births drives the dynamics P, but as P gets larger and reaches its carrying capacity C’, the squared
term increases and slows the rate of growth P. System 7, while mathematically equivalent to 5, can
be written as a function of two pathways representing one reinforcing loop and one balancing loop.
P = piu(P) + Pur(P)
where
R1: pu(P) = bP (8)
b+d
Bl: pualP) = — ("4 ) PP
This first-order example illustrates how even for simple first-order models, there can be multiple
ways to define causal pathways. Systems 6 and 8, while mathematically equivalent, represent two
different sets of causal mechanisms. The choice of pathways depends on the causal mechanisms
and the correspondence between model variables and real-world processes. The logistic equation
has been applied to the fields of biology, chemistry, demography, ecology, economics, sociology, and
political science. In each instance, variable and pathway decomposition depends on the phenomena
being described. While pathway selection does not change the behavior of the model, as will be
shown, it does change how explanations are developed in terms of structure dominating behavior.
Since different pathway decompositions represent different variable transformations of the system,
the chain rule of differentiation allows the modeler to choose the most appropriate decomposition
of the second derivative into contributing forces or causal mechanisms.
2.3 Identifying Necessary and Sufficient Pathways
With the second derivative of the variable of interest expressed as a function of causal pathways
(Equation 4), dominance is evaluated by examining each pathway and its effect on the sign of the
5
second derivative. Testing for necessary and sufficient conditions requires pathways to be isolated
and their effects somehow removed or deactivated, independent from one another. This raises the
question, What does independent removal or deactivation look like mathematically?
2.3.1 Pathway Removal Versus Deactivation
The practice of isolating and deactivating partial model structures (while leaving the rest of the
model intact) and examining the impact on behavior through simulation, has a long history in
em dynamics (SD) [10]. For the purpose of identifying dominant structure, there are two
primary ways of isolating and deactivating structure. The first is complete removal, and the second
is deactivation by holding the partial structure constant [2, 9, 5}.
To illustrate the difference between these approaches, consider a second-order system with state
variables x and y, in which y is the variable whose behavior is of interest and pathway x to y is to
be removed or deactivated, as shown in the upper left corner of Figure 4.
Pathway Deactivation Pathway Removal
Figure 4: Illustration of pathway deactivation versus pathway removal in a second-order system.
This system has two feedback loops: L1 which is a minor feedback loop from y to itself, and L2,
a major feedback loop going through both y and x. The intent behind deactivating pathway x to
y is to test the influence of feedback loop L2 on ij. The bottom left diagram in Figure 4 shows an
equivalent representation of the causal structure using the decomposition of i, highlighting the role
of both the states and their derivatives. The middle column of the figure represents the deactivation
approach. The pathway is deactivated by holding the value of the pathway constant, which requires
holding x constant (accomplished by setting & to zero). The result (lower middle figure) is that the
impact of the dynamics of the pathway are eliminated (hence L2 is eliminated), while the pathway
itself remains constant and still appears in the equation for 7, as a gain applied to feedback loop
L1. The right column represents the approach of fully removing the pathway from x to y. Under
this approach, the pathway is completely removed from the equation of y, and hence neither x nor
& affect 7.
For this paper, the deactivation approach (middle column) is used to test for necessary and
sufficient pathways. The primary reason is because the full removal method appears to test two
effects simultaneously, which is undesirable. It tests the effect of both the existence of a pathway as
well as its dynamics. This is evident in the lower right diagram of Figure 4 in which both the effects
of a and & are eliminated, whereas in the deactivation method (lower middle diagram), only the
effect of & is eliminated. In other words, the deactivation approach tests the effect of the dynamics
of the pathway (generated by feedback loop £2), but not its existence. This is especially preferable
if pathway « to y is an integral aspect of the theory for y (for example, is required to make the
equation y logically and dimensionally consistent). Through deactivation, one may desire to know
if x can be replaced by a parameter or exogenous variable, or if it being a state variable within a
feedback loop is required in order for y to exhibit certain behaviors.
Furthermore, to test the full removal of a pathway goes beyond the question of dominant struc-
ture, and to one of model simplification. To test the full removal of pathway x to y requires the
formulation of an alternate, dimensionally consistent theory about the causal mechanisms affecting
y. There is no general approach for removing a variable from an equation - the method of removal
depends on the nature of the equation and requires thoughtful consideration from the modeler. In
some cases, when pathways are nonlinearly coupled, it may not be possible to remove one path-
way without affecting other pathways, which violates the principle that the influence of different
elements of structure should be isolated and tested independently [10]. In some cases, it may not
be possible to reformulate a coherent rate equation without the causal pathway of interest! In
contrast, the theory (equation) of y remains unaltered and dimensionally consistent if the pathway
is deactivated and held constant, but not removed. This also results in a stronger test in which
only the dynamics of the pathway are being isolated and tested, as opposed to both dynamics and
existence.
There are two important additional observations about the deactivation of pathways, made
evident in Figure 4. First, with respect to model behavior, is that deactivation only impacts y,
maintaining the smoothness of trajectories, whereas full removal impacts both y and y at the same
time, resulting in non-smooth trajectories at the time of removal. This also illustrates how, when
conducting pathway deactivation, the second derivative of y alone is sufficient for determining dom-
inance and shifts in dominance, providing further support for the proposed definition of behavior.
Second, with respect to model structure, suppose there exists an additional feedback loop, L3,
through state variable z, between y and #, as shown in Figure 5. Also, suppose that after deactivat-
ing and testing the pathway from 2 to y, it is shown to be necessary for determining the sign of j.
A natural question might be which feedback loop, £2 or L3, is the most influential loop influencing
y at that time, since both contain the necessary pathway from x to y. Figure 5 shows that, with
respect to jj, this question is unanswerable. At the time of testing, the influence of loops L2 and
£3 cannot be distinguished from each other. Their effect is mediated through #, and thus one
additional simulation time step is required to observe their impact on 7 ?. A similar observation
was also made by Hayward and Boswell in their discussion of the limitations of the Loop Impact
method [4]. It would be more appropriate to ask about the relative influence between loops L2 and
L3 with respect to #, instead of 7. Because the systems are state-determined, and states accumulate
1For example, in the Lotka-Volterra predator prey model, the causal pathways of the major balancing feedback
loop between predators and prey cannot be removed without also removing other minor feedback loops as well (due
to their nonlinear coupling), in order to maintain a dimensionally consistent and coherent set of equati
2Deactivating a pathway n state variables removed from the variable whose behavior is of in
simulation time steps before an effect is observed. This fact makes some loop de
the sed d of d
requires n
vation methods inadequate for
(9)
Figure 5: Inability to distinguish between loops containing a common pathway.
(integrate) the effects of feedback loops from history to the present, the influence of loops sharing
the same immediate pathway to the variable of interest cannot be distinguished from each other.
Their influence, rather, unfolds as the simulation progresses over time and their impacts observed
on the pathway they have in common. This lends further support to considering pathways, rather
than feedback loops, as the explanatory element of system structure.
2.3.2 Tests for Necessary and Sufficient Pathways
The following table summarizes the tests for necessary, sufficient, and contributory pathways.
Type | Description Necessary? | Sufficient? | Test
sgn az — pathway
1 necessary pathway yes no ° B v x
. no pathway — —sgn%
me athway + sgn
2 sufficient pathway no yes P me 8
a i —sgnz%— no pathway
; necessary and pathway > sgnz
3 orn yes yes “
sufficient pathway no pathway + —sgn¢
4 contributory pathway no no sgn pathway = sgn#
5 none of the above no no sgn pathway = — sgn
Table 1: Five types of causal pathways.
Equation 4 expresses behavior #; as a sum of force contributions (F);;,) from each pathway
=Fija + Figg +. + Page +... + Bajm + Py
where: of
Of;
Fiju = 5 Bije
ij OPijx ij
Pathway pj;, is necessary if its deactivation changes the sign of %;. Deactivating pathway pj;;, is
accomplished by setting pj; = 0 in Equation 4 at the time of deactivation, which zeros out the
force contribution F);,. Thus, p;;, is a necessary pathway if the following inequality holds:
sgn i; # sen (4; — Fy) (9)
Pathway pj;, is sufficient if it alone guarantees the sign of the %;. The first condition for this to
hold is that the sign of F;;, must be the same as the sign of
sgn; = sgn Fi;
Secondly, sufficiency of p;;, requires that sgn; be unaffected by any combination of other pathways.
Equivalently, disproving the sufficiency of p,j, requires finding some combination of other active
pathways which results in a sign change of #;. Because pathway contributions to #; are purely
additive, this is equivalent to the absolute value of Fj; being greater than the absolute value of the
sum of all opposing pathway contributions (i.c., pathways whose contributions are opposite in sign
of Fj;,). Therefore, pj, is a sufficient pathway if the following inequality holds:
|Fige| > > Pipe (10)
{sen Fijm#sen Fije }
Lastly, if pathway p;;,, is neither sufficient nor necessary, but contributes in the same direction as
the observed behavior,
sgn a; = sgn Fijx
then it is called a contributory pathway.
2.4 Illustration: Dominance Framework Applied to a Linear Model
The above criteria for necessary, sufficient, and contributory pathways creates a dominance frame-
work which is applied to a first-order linear model containing multiple pathways, in order to il-
lustrate the different possible combinations of pathway types. Consider the following first-order
system (Figure 6 and Equation 11) with state variable x, governed by three linear inflows and a
single outflow.
= AC + A9x +032 — Bx
where, (11)
a1, a2, a3, 8 20
In this model, the four pathways could be aggregated into a single pathway with fractional rate
(ay +a +a — 3), however, assume each pathway represents a unique causal process and the desire
is to understand the relative influence of each pathway on the behavior of x. The pathways are
identified as:
path1:
path2:
path3:
path4 :
The second derivative of x is expressed as the sum of each pathway contribution:
= at + at + a3t — Be (12)
in flow3
Figure 6: Stock and flow diagram of first-order linear model with four causal pathways.
The relative contributions of each pathway are solely determined by the parameter values which are
constant. Therefore, there can be no shifts in dominance over time. However, postulating different
sets of values for the parameters illustrates the different possible outcomes with respect to necessary,
sufficient, and contributory pathways. Without loss of generality, assume at the time of interest to,
(to) = 1. Equation (12) reduces to:
H(to) = a, + a2 +03 — 8 (13)
Six sets of parameter values in Table 2 illustrate the possible outcomes of necessary, sufficient, and
contributory pathways, corresponding to different points on the dominance framework in Figure 7.
on a2 ag B
case 1 3 3 3 5
case 2 6 3 3 5
case 3 6 6 a 5
case 4 4 2 2 5
case 5 6 1 1 5
case 6 3 3 1 5
Table 2: Six sets of parameter values in the first-order linear model.
Using a visual representation similar to a free body diagram, the results of each case are displayed
in Figure 8, in which #(to) is represented by the first vertical bar, labeled sum, and the subsequent
vertical bars are the individual contributions of each pathway.
Figure 9 illustrates how the six cases map onto the dominance framework.
Case 1: No necessary or sufficient pathways. Multiple pathways (1, 2, and 3), each
exerting a force smaller than opposing pathway 4 (and thus insufficient), collectively exert a force
10
Dominance Framework
multiple necessary
g (0,n) pathways
&
3
<=
P=]
wo
a
2 single necessary and
Hf sufficient pathway
Oo
8 (0,1)@ @(1.1)
z multiple sufficient
* pathways
e e eo -
(0,0) (1,0) (m,0)
no necessary or _.
sufficient pathways # Sufficient Pathways
Figure 7: Dominance framework depicting the number of necessary and sufficient pathways that
determine the behavior of a variable of interest.
greater than opposing pathway 4 with margin such that no pathway is necessary. Pathways 1, 2,
and 3 are therefore considered contributory pathways. An analogy is a game of tug of war in which
on one side of the rope is a team of many weaker individuals who collectively greatly over-power a
single opposing strong person. claim: A minimum of four pathways are required for a system to
be in this condition. proof: Suppose there are only three pathways, one opposing the other two.
The one opposing pathway cannot exert greater strength than the other two together, otherwise it
would be necessary and sufficient. Of the remaining two pathways, if one is not necessary, then the
other must be sufficient for determining the sign, which contradicts the claim that no pathways are
necessary or sufficient.
Case 2: One sufficient and no necessary pathways. Pathway 1 is sufficient (has magnitude
greater than that of opposing pathway 4), but is not necessary, in that pathways 2 and 3, together,
also exert a greater force than opposing pathway 4. As in case 1, pathways 2 and 3 are neither
necessary nor sufficient. Using the same tug of war analogy in case 1, this case occurs when one of
the weaker individuals on the winning team is replaced by a strong person. claim: A minimum of
four pathways are required for a system to be in this condition. proof: Suppose there are only three
pathways, one opposing the other two. The one opposing pathway cannot exert greater strength
than the other two together, otherwise it would be necessary and sufficient. Of the remaining two
pathways, only one of them is not sufficient. The other then is not only sufficient, but must also be
ry, other ts removal would cause the sign to change, which contradicts the claim that
11
Case 1: Case 2:
No necessary or sufficient pathways One sufficient pathway
sum pathl ~—path2--—path3-—pathd sum pathl — path2—path3-—_pathd
‘pooo (Pies
2 cococ |]: @c«c [|
6 ag 6 5
Case 3: Case 4:
Two sufficient pathways One necessary pathway
sum pathl ~—path2—path3 path sum pathl —path2—ppath3—— path
10 10
2 © ® 2
2 c 2 W) c oc |
€ 5 6 5
Case 5: Case 6:
One necessary and sufficient pathway | Two necessary pathways
sum pathl = path2—path3_—— path sum path! = path2— path3_——path4
10
oT oe le OO
GS c 2 ® ® « ||
N= Necessary $= Sufficient C= Contributory
Figure 8: Six c combinations of necessary, sufficient, and contributory pathways.
none are necessary.
Case 3: Two sufficient and no necessary pathways. Pathways 1 and 2 are both sufficient
(their force magnitudes are each greater than the opposing force from pathway 4), and therefore
neither alone are necessary since their individual deactivation does not change the sign. Pathway
3, as in the previous cases, is neither sufficient nor necessary, but contributes in the same direction
as the observed behavior. claim: A minimum of two pathways are required for a system to be in
this condition. proof: By definition, a minimum of two pathways are required for there to exist
two sufficient pathways. To illustrate how only two pathways are required, consider the current
example of Case 3 but with pathways 3 and 4 completely removed. Pathways 1 and 2 would
remain individually sufficient for producing the sign, but each alone unnecessary since they are
both contributing in the same direction with no opposing force.
Case 4: One necessary and no sufficient pathway. Pathway 1 is necessary (deactivating it
12
# Necessary Pathways
(0,0) (4,0) (2,0)
Case 7 Case 2 Case 3
# Sufficient Pathways
Figure 9: Six cases mapped onto the dominance framework.
results in a change in sign), but it alone is not sufficient (its force magnitude is less than that of the
opposing pathway 4). Pathways 2 and 3, as in Cases 1 and 2, are contributory, but neither necessary
nor sufficient. In this case, consider the analogy of a sports team with a single all-star player whose
talent is required for the team to win, but who alone does not make up a team and therefore is
insufficient. The remaining team members contribute to the team’s success but are neither sufficient
nor necessary because they are easily replaced. claim: A minimum of four pathways are required
for a system to be in this condition. proof: Suppose there are only three pathways. At least one
pathway must exert a force in an opposite direction from the other two, otherwise all pathways
would be sufficient. Consider one pathway exerting an opposite force from the two others. Of the
remaining two, if one is necessary but not sufficient, then the other must also be necessary, which
contradicts the assumption that only one is necessary.
Case 5: One necessary and sufficient pathway. Pathway 1 is both ne
If pathway 1 is deactivated, #(to) changes signs, so it is necessary. The absolute value of pathway
1’s contribution is greater than that of all opposing paths, so pathway 1 is also sufficient. This is
the only case in which a pathway meets the established criteria for dominance. There can be at
most one necessary and sufficient pathway at a point in time, and that if a system has a single
necessary pathway and a single sufficient pathway, they must be one and the same. claim: A
minimum of one pathway is required for a system to be in this condition. proof. Consider this
example but with pathways 2, 3, and 4 removed. Pathway 1 would remain necessary and sufficient
for determining the behavior since its removal would change the sign to zero, and since there are
no opposing pathways.
Case 6: Two necessary and no sufficient pathways. Pathways 1 and 2 are both necessary
in that their individual deactivations result in a change in sign, but they are also individually
insufficient in that their force magnitudes are less than the opposing force magnitude from pathway
4. To use another sports analogy, consider a team with no back-up players and in which every
sary and sufficient.
13
member of the team is necessary to play their position. claim: A minimum of three pathways are
required for a system to be in this condition. proof. Consider a system with only two pathways. As
demonstrated earlier, if both contribute in the same direction then both are sufficient and neither
necessary. If they contribute in opposite directions, then the one with larger magnitude will be
ry and sufficient. Therefore, this case is not possible with only two pathways. For a three
pathway system, consider the current example but with pathway 3 removed. Pathways 1 and 2
would still both be necessary but not individually sufficient.
nec
3 Pathway Force Decomposition Procedure
The above process for identifying necessary, sufficient, and contributory pathways, derived from
the proposed definitions of structure, behavior, and dominance, is now summarized in the following
steps, and is referred to as the pathway force decomposition (PFD) procedure.
a
. Select state variable whose behavior is of interest.
No
. Specify initial conditions and time horizon of interest, over which to perform dominance
analysis.
wo
. Specify immediate pathways to the state variable of interest, as functions of other state vari-
ables. Express the derivative of the state variable of interest as a function of the pathways.
i
. Using the chain rule, express the second derivative of the behavior of interest as a sum of
contributions from each pathway, by computing each partial derivative.
. For each time step within the time interval of interest, calculate the force contributions of
each pathway and test each pathway for necessary, sufficient and contributory conditions.
on
4 Application to Logistic Growth Model
The PFD procedure is applied to the logistic growth model, which exhibits transient S-shape be-
havior. This model is chosen due to its association with general growth processes from which the
term feedback loop dominance originally emerged. The logistic growth model is also the canonical
example of shifts in loop dominance and is therefore an important test for a formal definition of
dominance. Three forms of the logistic model are analyzed and results compared with current
methods.
4.1 Logistic Model Form 1: Reinforcing Loop and Constraining Loop
Richardson analyzed the two-loop logistic growth model (Figure 10) in which R1 is reinforcing
growth, and B1 is the constraining effect from carrying capacity C' on the fractional growth rate a
[11] (also, see [12, p. 296]).
Step 1: Behavior of interest. Since population P is the only state variable, it is the variable
of interest.
Step 2: Initial conditions and time horizon. The population begins with a single member,
P(0) = 1. The model is analyzed from t = 0 to t = 100.
alpha
unconstrained
growth
Figure 10: Stock and flow diagram of the logistic growth model (form 1).
Step 3: Express P as a function of its immediate pathways.
P= Put * Piz
Pui = aP (14)
Pathway pi; is the pathway from P through auxiliary variable unconstrained growth, and rep-
resents the reinforcing growth feedback loop R1. Pathway pj. is the pathway from P through
auxiliary variable constraining factor, and represents the balancing feedback loop B1 from carry-
ing capacity C which constrains growth.
Step 4. Express P as a function of pathway force contributions.
P=Fut+Fir
oP. P ,
Fin = y Pin = (: = z) -aP
Opin Cc
ap -P
Fy = ——Piy = aP -
112 Opa? a. ( C }
Step 5. For each time step, calculate force contributions and identify necessary,
sufficient, and contributory pathways. Figure 11 shows the results of the simulation and
analysis. The top graph is the behavior over time of state variable P. The middle graph plots P
(Net Force) along with the force decomposition of Path 1 (p11) and Path 2 (pii2). The bottom
graph shows which paths are necessary and which are sufficient for determining the sign of P.
4.1.1 Results
P exhibits the expected S-shape growth: divergent exponential growth followed by convergent goal-
secking growth. The inflection point (P = 0) occurs around time t = 46, when P reaches half its
carrying capacity (50). Path 1 (associated with reinforcing feedback) always exerts a positive force,
while Path 2 (associated with the balancing feedback loop) always exerts a negative force. From
120
100
population
time
cee Path1 Force
Path2 Force
Net Force
-0.05 20
acceleration
time
os Sufficient
Necessary
paths
ie) 20 40 ws 60 80 100
time
Figure 11: Simulation results for logistic growth model (form 1).
t = 0 to the inflection point t = 46, the force exerted by Path 1 is greater in magnitude than the
force exerted by Path 2, and thus is both necessary and sufficient for creating positive acceleration
(and by definition is dominant). After t = 46, Path 2 exerts a greater force than Path 1 and is
necessary and sufficient for creating negative acceleration (deceleration), and therefore dominance
shifts to Path 2. As time continues, both forces decrease as P approaches equilibrium.
4.1.2 Excursion
Consider the case in which the population begins above its carrying capacity (Figure 12). P exhibits
exponential decay as it approaches its carrying capacity, C. The net force is always positive,
slowing the decline. Path 2 (representing the balancing loop associated with the carrying capacity)
exhibits the greatest force, however both Paths 1 and 2 exhibit positive forces and so both are
sufficient (neither are necessary) for generating the observed behavior. Therefore, neither pathway
is dominant. The result that Path 2 is sufficient is intuitive since it is associated with the balancing
feedback loop and has a negative closed-loop gain. To understand, however, how Path 1 (associated
with the reinforcing growth loop) is sufficient for generating this behavior, observe that if P > C,
Path 2 is negative and therefore, when deactivated, switches the polarity of the feedback loop
associated with Path 1. This illustrates how nonlinear coupling of pathways can change the polarity
of feedback loops and lead to shifts, not just in loop dominance, but the polarity of feedback loops
16
200
c 150 Mo
2
= = 100
3
Q 50 J
a
0
40 50
5 sooneee Path1 Force
3 Path2 Force
2 —Net Force
oO
°
°
o
40 50
3
wn a Sufficient
£
5 Necessary
a 1
Q
0 10 20 30 40 50
time
Figure 12: Simulation results for logistic growth model (form 1) (P > C).
in different regions of the state space.
4.1.3 Discussion and Comparison with Other Methods
The results from applying the PFD procedure agree with the conclusions from Richardson [11] and
Sterman [12] who use the concept of dominant polarity to determine loop dominance for first-order
two-loop systems, where dominant polarity is positive if population is less than half the carrying
capacity (and thus the reinforcing loop dominates), and dominant polarity is negative if population
is greater than one half (and thus the balancing loop dominates). The results also agree with
numerous examples from SD instructional literature which informally introduce the concept of shifts
in dominance based on the logistics model [1, 7]. However, while applying the concept of dominant
polarity to identify loop dominance is straightforward when P < C, for P > C, Richardson and
Sterman are silent on the specific roles of the two feedback loops. Other literature on logistic growth
is also silent on loop dominance when P > C. The PFD procedure provides a formal and rigorous
answer to this question.
Mojtahedzadeh performs dominance analysis on the logistic growth model, in the form of the
Susceptible-Infectious (SI) epidemic model, using the Pathway Participation Method (PPM) [8].
The SI model uses the following interpretation of variables which make-up the two pathways in
system 14:
population (P) = infectious (I)
a@ = contact rate (c) x infectivity (i)
carrying capacity (C) = population size (N)
Contagion Reinforcing Loop : py, = cil
I
Depletion Balancing Loop : pii2 = (: - x)
PPM, which uses the total pathway participation metric (TPPM) to determine dominance, con-
cludes that the contagion loop (pi11) is dominant in the first phase of S-shape growth, followed by
the dominance of the depletion loop (pj12) in the second phase. For this two-loop model, because
PPM identifies the loop which contributes most to (I/J) as the dominant loop, and each loop al-
ways has opposite contributions, the one with largest magnitude measure is also guaranteed to be
both necessary and sufficient, and so the results agree with the PFD procedure. The case in which
population starts above the carrying capacity does not apply to the SI model, since the infectious
population can never be greater than the total population, so this case was not analyzed by PPM.
Similarly, Kampmann and Oliva [6] apply PPM to a simple diffusion model of adopters (A) and
potential adopters (N — A) having the same mathematical and causal structure as the SI model, in
which:
population (P) = adopters (A)
a = contact rate (c) x adoption fraction (i)
carrying capacity (C) = population size (N)
Word of mouth Reinforcing Loop : pi. = ci A
Market Saturation Balancing Loop : py12 = (: - =)
The PPM results agree with the PFD results and identify the wordof mouth reinforcing loop as
dominant in the first phase, and the market saturation balancing loop as dominant in the second
phase. Kampmann and Oliva also apply loop eigenvalue elasticity analysis (LEEA) to the same
diffusion model, using influence metrics based on the derivative of eigenvalues of the linearized
system with respect to loop parameters. LEEA also identifies the word of mouth loop as dominant
in the first phase, followed by the dominance of the market saturation loop, agreeing with both
PPM and the PFD procedure.
Taylor and colleagues [13] test their experimental statistical screening procedure on the same
diffusion model, which identifies highly influential elements of structure by measuring the correlation
between structural and behavioral changes. Initial adopters is identified as most influential in the
beginning because it defines the initial state. Shortly after, adoption fraction and contact rate
both have the highest correlation coefficients, which agrees with the PFD procedure in that both
contribute to the gain of the causal pathway associated with the reinforcing loop. In the last phase,
initial potential adopters has the highest correlation coefficient which also agrees with the PFD
procedure, in that it restricts the system through the balancing market saturation loop.
PFD results are now compared to Ford’s behavioral method of loop deactivation. First, consider
the case in which population P starts below the carrying capacity C. Loops B1 and R1 (Figure
10) are each individually deactivated once during the divergent growth phase (t = 10) and once
again during the convergent growth phase (t = 50). The resulting simulated responses are shown
in Figure 13.
18
180 | 81 deactivated /
120 B1 deactivated
(t=10) Baseline
R1 deactivated
population
3
a (t=50)
“ R1 deactivated
/ (t=10)
20 eo
he
‘ ° .° °° 0 100
time
Figure 13: Ford’s behavioral test for dominance applied to the logistic growth model (form 1).
During the divergent growth phase, deactivating B1 does not change the atomic behavior pattern
(exponential), and thus, according to Ford’s method, is not dominant. Deactivating R1 does change
the atomic behavior pattern from exponential to logarithmic, and thus R1 is dominant. In the
second phase, deactivating B1 changes the atomic behavior pattern from logarithmic to exponential,
and thus B1 is dominant. Deactivating R1 does not change the atomic behavior pattern, and is
not dominant. Because Ford’s criteria for dominance employs a counterfactual test, it identifies
necessary pathways. In this model, since the dominant pathways are both necessary and sufficient,
and there are no pathways which are necessary but not sufficient, Ford’s test identifies the same
dominant structure as the PFD procedure.
Now consider the case in which P > C. The model exhibits only one behavior pattern (loga-
rithmic) and loops B1 and R1 are individually deactivated at time (t = 10), as shown in Figure
14. Deactivating R1 results in no noticeable change in behavior, and thus R1 is not dominant.
Deactivating B1 causes a noticeable change but does not change the atomic behavior pattern, and
thus B1 is also not dominant. However, deactivating Rl and B1 simultaneously changes the atomic
behavior pattern from logarithmic to linear, indicating what Ford calls a shadow dominance condi-
tion. However, because their are only two loops, deactivating both loops always results in a change
in atomic behavior pattern unless the model is in equilibrium, therefore, Ford’s test for shadow
dominance may be degenerate in this case. None-the-less, Ford’s test agrees with the PFD pro-
cedure in not identifying any dominant loops, since Ford’s test identifies necessary pathways, and
in this case, both loops are found by PFD to be sufficient and not necessary. This also indicates
a potential relationship between shadow dominance and cases in which there exists one or more
sufficient pathways and no necessary pathways. In examining Ford’s method, it appears that a
pair of shadow loops are identified when cither loop is sufficient for changing the behavior, in the
absence of the other, which indeed indicates a sufficient condition. He describes this as a different
type of dominance. He also admits that his algorithm is more difficult as the number of shadow
feedback structures increase (i.e. the number of sufficient pathways increase).
One observation is that if the two sufficient loops are considered as a set, then together they
are both sufficient and necessary for creating the observed behavior, and thus can be considered
as a dominant set, meeting the established criteria of dominance. This expands the definition of
19
R1 deactivated
150
(10) —
400 ~ EY Baseline
5 B1 deactivated
% 50 (#10)
= ~
0
2 80 100
-50 iiss
~ -_
~100 B1 and R1 deactivated cao
(t=10)
-150
time
Figure 14: Ford’s behavioral test for dominance applied to the logistic growth model (form 1)
(P>C).
structure to include not just single pathways, but sets of pathways. Under this expanded definition,
Bland B2 would be identified by the PFD procedure as a dominant pair of pathways. This directly
corresponds to Ford’s results which identifies B1 and B2 as a pair of shadow feedback structures
which are together dominant.
4.2 Logistic Model Form 2: A Second Balancing Loop (Deaths)
A three-loop version of logistic growth is analyzed and compared against the Loop Impact method
[4]. This version (Figure 15), adds a second balancing feedback loop B2 representing population
decline through deaths based on a constant fractional death rate b. Accordingly, a third pathway,
Pug, Tepresenting balancing loop B2, is added to the decomposition of P:
P = Pu’ Pur + Pus
Pi =aP
16
Puz=1-—a ( )
Cc
Pug = —bP
P=Fint+ Fiat Fis
Fin = (1-2) -aP
: (17)
+P
Fin =aP- ( CO }
Fis = —bP
The simulation begins from an initial population of 1. Figure 16 shows the results of the simulation
and force decomposition analysis.
20
constraining factor
decline
unconstrained
growth
alpha
Figure 15: Stock and flow diagram of the logistic growth model (form 2).
25
— Pathl Force
Path2 Force
75 Path3 Force
— Net Force
3 —
seseee Sufficient
v
i 2 —— Necessary
1
0)
0 5 10 15 20 25
Figure 16: Simulation results for logistic growth model (form 2).
4.2.1 Results
As before, the inflection point occurs when P = C/2. Path 1 (R1) always exerts positive force,
while Path 2 (B1) and Path 3 (B2) always exert negative force. There are three distinct phases
of dominance. Before the inflection point (phase 1), Path 1 is necessary and sufficient (dominant).
21
After the inflection point, there is a brief period in which Paths 2 and 3 are both necessary, and
neither sufficient, to produce deceleration. Then, Path 2 becomes necessary and sufficient and
dominates for the remainder of the trajectory.
Phases 1 and 3 agree with the Loop Impact method, identifying Rl and B1 as dominant,
respectively. In phase 2, the Loop Impact method identifies B1 and B2 as dominant together, or
as a set, and concludes that in this phase there is no single dominant pathway or loop. The PFD
method also concludes that in phase 2, there is no single dominant pathway (i.e. no pathway is
both necessary and sufficient). However, if Path 1 and Path 2 are considered as a set, then as a
set they are necessary and sufficient in phase 2, and mect the criteria for dominance. Thus, if the
definition of dominance is expanded to include as elements of structure sets of pathways, then the
results are the same as in the Loop Impact method.
The Loop Impact method, by construction, finds the minimum combination of loops of like
polarity (i.e forces contributing in same direction) whose combined impact is greater than the sum
of all loops of opposite polarity [4]. This criteria, by construction, identifies cither a single sufficient
loop’, or a minimum set of loops which are sufficient. Thus, it is expected that the PFD procedure,
which identifies dominant structures as those which are both necessary and sufficient, would agree
with the Loop Impact method, however the Loop Impact method also classifies loops as dominant
which are sufficient and not necessary, and thus identifies dominance more frequently than the PFD
procedure.
Next, Ford’s behavioral method is applied to this model. In Ford’s procedure, locations of
deactivation are based on changes in behavior patterns. Therefore, in this form of the model,
deactivating in two locations is insufficient for identifying the three phases of dominance discussed
earlier, which is a shortcoming of Ford’s method 4. The dominance transition from phase 2 to phase
3 is not associated with a behavior mode change. Regardless, Ford’s deactivation method is applied
to each loop in each of the three phases (¢ = 5, t = 10, t = 15) to see what insights are discovered.
Figure 17 shows the results of Ford's procedure.
baseline
TET B1 deactivated
-——— R1 deactivated
—— B2 deactivated
population
time
Figure 17: Ford’s behavioral test for dominance applied to the logistic growth model (form 2).
5Tt is not
4Extens’
behavior mode transition points, and thus address t
lear how the Loop Impact method works when there are multiple single suffici
to Ford’s method include automated testing at every point along the tr
s shortfall in Ford’s method [9].
t loops.
ry and not just at
22
In phase 1, deactivation of R1 is the only case which causes an immediate behavior mode change
(exponential to logarithmic), and thus R1 is dominant. In phase 2, deactivating B1 results in a
behavior mode change, as does deactivating B2, so both are found to be individually dominant.
Ford refers to this as simultaneous multiple loop dominance. In phase 3, deactivation of B1 is the
only case resulting in a behavior mode change, and thus B1 is dominant.
Ford’s behavioral method agrees with PFD and the Loop Impact method in phases 1 and 3. In
phase 2, it differs from both PFD and Loop Impact in that it identifies Bl and B2 as individually
dominant. This is because, as observed earlier, Ford’s criteria for dominance is a necessary condition
(not sufficient), thus the PFD procedure explains why Ford’s method produces a different result.
Any time loops are necessary (regardless of whether or not they are also sufficient), Ford’s method
will identify them as dominant, which leads to the potential identification of multiple dominant
loops.
4.3 Logistic Model Form 3: Alternate Two Loop Version
Figure 18 shows an alternative two-pathway representation of the logistic model (compare with
form 1, Figure 10). Whereas in form 1, pathways represent the reinforcing growth loop and the
Change in P
Figure 18: Stock and flow diagram of logistic growth (form 3).
loop constraining the fractional growth rate, in this form, pathways represent the reinforcing growth
loop and the balancing death loop:
Pa=pu + Piz
Pir =aP
(18)
—a
pi = (=) op?
P=Fiunt+Fin
Fy =aP (19)
—2aPP
fas =
2 fal
Note that Equations 18 and 19 are equivalent to Equations 14 and 15, respectively, but have different
pathway decompositions. Both forms are often found in literature on sigmoid growth, as observed
earlier. For P < C, the simulation and analysis results are shown in Figure 19.
4.3.1 Results
The results are the same as in form 1 (P < C), in which Path 1 (R1) is necessary and sufficient
(dominant) in the first phase of exponential growth, and Path 2 (B1) is necessary and sufficient
(dominant) for logarithmic growth in the second phase.
120
100
population
0 20 40 60 80 100
senna Path1 Force
Path2 Force
——Net Force
acceleration
time
----- Sufficient
Necessary
paths
ie) 20 40 7 60 80 100
time
Figure 19: Simulation results of logistic growth (form 3).
Next, the results for condition (P > C) are shown in Figure 20.
These results differ from form 1 (compare with Figure 12). Whereas in form 1, both paths
were sufficient for producing the exponential decay behavior, in form 3, the balancing loop (path
2) is necessary and sufficient for producing the behavior while the reinforcing loop (path 1) has an
opposing force. In form 1, it was noted that the balancing loop changed the polarity of the reinforcing
loop to behave like a balancing loop, so both were sufficient for generating the behavior. In form
the balancing and reinforcing loops are added together instead of multiplied, and therefore alwa:
represent positive and negative forces strictly associated with births and deaths. The necessary and
sufficient pathways for each case are summarized in Tables 3 and 4.
In summary, while the pathway decomposition does not change the dynamic behavior of the
24
population
0 40
ned Path1 Force
Path2 Force
acceleration
20 30 40
——Net Force
paths
time
os Sufficient
Necessary
10
20 . 30 40
time
50
Figure 20: Simulation results of logistic growth (form 3) (P > C).
Loop P<o 5 <P<c P>C
R1 N&S S
Bl N&s S
Table 3: Dominance of loops in logistic equation form 1.
Loop P<5 5 <P<e P>C
R1 N&S
Bl N&S N&S
Table 4: Dominance of loops in logistic equation form 3.
system, this example illustrates how pathway definition and decomposition affects the explanation
for how structure determines behavior, and the nature of dominant structure.
5 Conclusions
5.1 Procedure for Identifying Dominant Structure
Using the proposed definitions for behavior, structure, and dominance, behavior is formally expressed
as a sum of contributions from individual elements of structure (pathways). There may be more
than one way to decompose behavior, depending on how causal mechanisms are defined. The
dominance criteria requires deactivation of pathways to identify necessary and sufficient conditions
and it was shown that examining the second derivative is sufficient for detecting dominance and
shifts in dominance.
Furthermore, it was shown that immediate pathways (as opposed to feedback loops) are adequate
for identifying necessary and sufficient structure. Concise mathematical tests were proposed for
detecting necessary and sufficient pathways. A procedure, pathway force decomposition (PFD),
was developed for identifying necessary and sufficient pathways and was applied to several forms
of the logistic model. The PFD procedure has qualities similar to exploratory /behavioral methods
of dominance analysis in that the criteria for dominance is anchored in a behavioral criteria (sign
change of the second derivative).
The PFD procedure also has qualities similar to formal/structural methods in which the criteria
for dominance is developed from the structure (equations) of the model. Thus, the PFD procedure
may be considered both a behavioral and a formal/structural dominance method. One reason
for this is that the criteria for dominance has behavioral/objective qualities in that dominance
requires an objective behavior change, as well as structure-relative qualities in that the metric of
force contribution can be directly compared for different pathways. Therefore, the PFD procedure
captures both the behavioral-objective dimension of dominance as well as the structural-relative
dimension.
When tested against the logistic growth model, the PFD procedure showed that the loops com-
monly associated as dominant satisfied both necessary and sufficient conditions, providing further
support for the proposed definition of dominance. The procedure also illustrated how pathway
choice affects structure-behavior explanations.
5.2 Relationship Between Dominance Methods
All methods of dominance analysis have been applied to the logistic growth model and produce
consistent results, but for different reasons.
Ford’s behavioral test employs a counterfactual criteria and identifies necessary structure as
dominant. Conversely, the Loop Impact method, by construction, identifies sufficient elements
of structure as dominant. Where the structure is both sufficient and necessary, Ford’s behavioral
method, the Loop Impact method, and PFD identify the same dominant structure. Where structure
is necessary but not sufficient, or sufficient but not necessary, the methods will not identify the same
dominant structure. PFD applies Ford’s behavioral criterion analytically when it identifies necessary
structure, and thus can also be viewed as an automated, analytical version of Ford’s method. PFD
also improves upon Ford’s method in that it directly identifies multiple shadow feedback structures
analytically. PFD is also easily applied at each time step along a trajectory and does not require
manual deactivation of feedback structure. This is important because, as observed, not all shifts
in dominance are associated with changes in behavior modes. Additionally, manual deactivation
and testing of models can be a tedious exercise and clutter up the model with switch variables.
In Ford’s method, deactivation points must be manually identified, and multiple combinations of
26
loops deactivated simultaneously in the case of shadow structure. By using a completely analytical
procedure, these challenges are avoided.
Likewise, the Loop Impact method has many similarities to PFD, but uses the Impact metric
which is the acceleration contribution divided by the first derivative of the variable of inter:
Thus, it uses the same metric as the PPM method.
PPM results agree with PFD and the behavioral methods when the dominant loop also happens
to have the largest TPPM contribution, as in the case of the logistics model. In fact, when there
only exists a single loop contributing in the direction of the observed behavior, PPM will always
agree with the behavioral method and the PFD. TPPM is simply the second derivative divided by
the first derivative, and thus is very similar to the proposed definition of behavior. However, because
PPM relies on a relative metric for determining dominance, it will always identify a single dominant
loop. Whenever there is a change of loop with the largest TPPM, a shift in loop dominance will
be identified, thus it is expected that PPM identifies dominance and shifts in dominance more
frequently than behavioral-based methods such as Ford’s procedure and the PFD procedure. The
PFD procedure improves upon the PPM method in that it precisely determines each pathway’s
contribution to the observed behavior, as opposed to a normalized proxy measure of behavior. The
PFD procedure also allows for the possibility that multiple loops or no loops dominate, and it is
well-defined when the derivative of the variable of interest is equal to zero.
For similar reasons as for PPM, LEEA (also based on a normalized influence metric), agrees
with the results of PFD and the behavioral methods in the case of the logistic model since there
is only one reinforcing and one balancing loop. Like PPM, LEEA will also always identify a single
dominant loop and detects shifts in dominance more frequently than in behavioral methods. One
way to interpret the results of PPM and LEEA is that they identify highly influential structures
which are contributory and potentially dominant, and are good candidates for subsequent testing
for dominance [3].
Ford’s behavioral method identifies cases of shadow loop dominance, which others have also
described as shared dominance [9]. It also describes situations of multiple loops dominating simul-
taneously. The phenomenon of shadow loop dominance is associated with multiple sufficient and
unnecessary pathways, and the phenomenon of multiple loop dominance is associated with multiple
necessary and insufficient pathways.
5.3 Implications of Dominance Framework for Policy Design
5.3.1 Expanded Definition of Dominance
Ford’s method considers pairs of shadow feedback structures (i.e. sets of sufficient pathways/loops)
as dominant together. Similarly, the Loop Impact method considers sets of pathways/loops which
are collectively sufficient to be a dominant set. One observation is that multiple sufficient pathways
may form a set that is both sufficient and necessary. Likewise, multiple necessary pathways may
form a set that is both necessary and sufficient.
Expanding the definition of dominance to consider not just single pathways which are necessary
and sufficient, but sets of pathways which together are necessary and sufficient, permits a broader
use of the term.
Therefore, the following modified definition is proposed:
Given a state variable x;(t) whose behavior is of interest, and point to along its trajectory,
a set of causal pathways are dominant if and only if the set is both necessary and
sufficient for determining sgn 2 ;(to).
27
5.3.2 System Robustness and Fragility
Ford’s method and the Loop Impact method above illustrate three possible compositions of domi-
nant sets:
1. A dominant sect contains a single necessary and sufficient pathway.
2. A dominant set contains some combination of necessary and contributory pathways, but no
sufficient pathways.
3. A dominant set contains some combination of sufficient and contributory pathways, but no
necessary pathways.
The robustness or fragility of a variable’s behavior at a given time depends on the number of
necessary and sufficient pathways to that variable at that time. Consider Figure 21, which expands
upon the dominance framework introduced earlier.
fragile systems
with multiple
yop leverage points
8
S g
g § 5
: 3
S 8
z 8 | 3 ;
s 3
Pos 5
a EI system with
o Ela e single dominant
3 pathway
= robust systems
1 exhibiting
policy resistance
robustness / resilience
e ra
e
system with 1 2 3 oo m
multiple weak q i
contributing y ;
pathways “shadow feedback” or “shared dominance”
# Sufficient Pathways
Figure 21: System robustness depending on the number of necessary and sufficient pathways.
Figure 21 shows the typical case of dominance associated with a single necessary and sufficie’
pathway. It also shows that shadow feedback or shared dominance occurs when there exists one
or more sufficient and no necessary pathways. Similarly, multiple loop dominance or simultaneous
dominance occurs when there exists two or more necessary and no sufficient pathways. A system
with no sufficient or necessary pathways contains multiple contributing pathways. The robustness
or resilience of a system increases as the number of necessary pathways decrease and as the number
of sufficient pathways increase. Here, the terms robustness or resilience are used to describe the
likelihood of system behavior change when there are changes in causal pathways.
5.3.3 Design Implications for Robust Systems
General principles for making a system more or less robust can be inferred from the dominance
framework. First, robustness can be increased by decreasing the necessary pathways. This is
28
accomplished by adding new supporting causal pathways, increasing the force of the weaker con-
tributing pathways, or by reducing the force of the opposing pathways. Second, robustness can
be increased by increasing the sufficient pathways. This is accomplished by increasing the number
of individual pathways which are stronger than the opposing pathways, either by increasing the
force of supporting pathways, adding new strong supporting pathways, or reducing the strength of
opposing pathways.
This also raises new questions of how to identify the best way to intervene in a system that is
operating in one of these six states. Also, can these insights be used to develop design heuristics
for sustainable policies and interventions? Can they be used to better understand policy resistance
observed in real systems?
6 Summary
A formal definition of dominance is proposed and tested against a simple model alongside other
methods. In the process, some discrepancies between current methods were addressed, and am-
biguities associated with shadow loop dominance, multiple loop dominance, and pathway/loop
representation were also addressed. Additionally, based on the tests, the definition of dominance
was expanded to include not just single pathways, but sets of pathways. Based on the proposed
definition of dominance and the PFD procedure, new insights were offered regarding the relationship
between dominance and system robustness, policy resistance, and leverage points.
Additional analyses and comparison between PFD and other loop dominance methods on larger
and more complex models is warranted.
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