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CONFUSION IN THE POLARITY OF MAJOR LOOPS
Mohammad T. Mojtahedzadeh
George P. Richardson
The Rockefeller College of Public Affairs and Policy
State University of New York at Albany, Albany, NY, 12222
Abstract:
There is an a priori in the system dynamics paradigm which implies that the behavior of a
system is tightly related to its structure. A structure specifies the relationships between the
elements of the system in terms of stocks and flows and feedback loops. The attempt is to
explain the behavior of a system based upon the accumulations in a system and its positive
and negative feedback loops. This note introduces the concept of implicit loops or
pathways, as components of a major loop, to explain how changes in the behavior of a
variable of the system occur over time. The note demonstrates that the presence of a
negative loop (explicit or implicit) around a positive loop (explicit or implicit) is necessary
in order for a system to oscillate. In particular, it explains how a pure positive major loop
can oscillate.
There is an a priori in the system dynamics paradigm which implies that the
behavior of a system is tightly related to its structure. A structure specifies the relationships
between the elements of the system in terms of stocks and flows and feedback loops. The
basic building block of dynamic models is the feedback loop. The feedback loop is a path
coupling decision, action, level of the system, and information, with the path returning to
the decision point (Forrester, 1968). The behavior of the system is then the result of the
interaction between different feedback loops embodied within the system boundary. In
control theory, the behavior of a system is usually characterized by eigenvalues,
eigenvectors, poles, zeros, gain, phase, and so on. These concepts, however, help us to
understand the system, but they are not relevant to the real process. Since the feedback
loops are based on the real process in a phenomenon, the attempt is to explain the behavior
of the system in terms of feedback loops. It is well known that a negative minor loop
corresponds to a goal seeking behavior, and a positive minor loop generates exponential
growth. There are no general rules about the behavior of negative and positive major
feedback loops.
The more complicated behaviors, such as overshoot, S-shape, and oscillation, are
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created by feedback loops with different polarity and most likely the possibility of shifts in
their polarity. Literally, the growth phases of these behavior are the result of positive
loops, and the goal seeking phases are generated by negative loops. Therefore, a necessary
condition for oscillation is the presence of a negative loop around a positive loop. The
growth phase and goal seeking phase of an oscillatory variable are shown in Figure 1.
a \
Goal-Seeking | Exponential
Growth
Exponential Goal-Seeking
Growth
Time
Figure 1: Growth Phase and Goal-seeking Phase of an Oscillation.
A simple example which contradicts this theory is a single major loop with negative polarity
that create oscillation. Another example is presented by Allan Graham (1977). His
counterexample demonstrates that a pure positive loop can generate oscillations.
This note applies an extension of the formal definition of loop polarity to introduce
the concept of implicit loops or pathways, as components of a major loop. It demonstrates
that the presence of a negative loop (explicit or implicit) around a positive loop (explicit or
implicit) is necessary in order for a system to oscillate. As will be defined later, the implicit
loops help people to discern the behavioral characteristics of a dynamic system from its
feedback structure. Although, the note deals with Graham's counterexample, the reader
can apply this procedure to understand how a negative major loop with two level variables
and no minor loop oscillates!, or how a negative major loop with three level variables and
1. The equations of such an example are:
dX/dt=-Y
dY/dt=X
It is obvious that there is only one major loop with negative polarity in this system.
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no minor loop generates exploding oscillation?. Graham's counterexample uses a linear
fourth order system with no minor loop and only one major loop. Since the parameters of
the model are assumed to be positive (and equal to one), one may conclude that the polarity
of the major loop is positive. A STELLA diagram of the model is depicted below.
dP/dt
Figure 2: STELLA Diagram of a Major Positive Loop
The equations for the system are:
dX/dt = P fa
dY/dt = X 2]
aZ/dt = B]
dP/dt = Z 4]
The eigenvalues of such a system are +1, -1, +j, -j, where J = v1, The eigenvalue
of +1 implies that the system grows exponentially. The eigenvalue of -1 indicates how
quickly the system moves to its steady growth. The eigenvalues of +] and -j suggest that
the system can oscillate. The trajectory of such a system looks like:
X(t) = ay*et + ap*et + a3*sin(t+9) [5]
where aj, a2, a3, and @ depend on the right and left eigenvectors and initial conditions.
Figure 3 shows the behavior of the model, when the initial values for X and Y are -1 and
for Z and P are +1. Since no other loops are present, the oscillation then is presumably a
2, The equations for a negative major loop with three level variables and no minor loop are:
dX/dt = -Z
dY/dt =X
dZ/dt = Y
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result of the major positive loop. If all initial values are positive, the system will create
exponential growth with a growth rate of one. In other initial values, the model will grow
exponentially with the growth rate of one, after a transition period.
Figure 3: The Oscillatory Mode of a Fourth Order System with no Minor Loop
It is reasonable that the model generates pure exponential growth. However, because of
the absence of a negative loop around the major loop shown in Figure 1, one does not
expect the model to overshoot or oscillate.
The oscillation raises the question of whether it is enough to say that the loop
shown in Figure 1 is positive. What do we mean by a positive loop? There are several
definitions in the system dynamics literature of a positive loop. One of the more elaborated
definitions is suggested by Graham (1977). He defines a loop as positive if its open-loop
steady state gain is greater than one. This definition addresses the steady state characteristic
of the system, while we may be interested in the transient regimes of system behavior
rather than its long term behavior. Furthermore, what does a gain greater than one mean
intuitively? Richardson (1995) suggests that a loop is positive if the sign of the partial
derivative of the rates (inflow, outflow, or net flow) with respect to its corresponding level
variable is positive. Therefore, a loop is positive if,
Sign podxiad
ox 1? 0
[6]
where X is a level variable, and dX/dt is the rate of change in the level variable. A loop is
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negative if;
Sign Aces 120
7]
The last definition is consistent with the intuitive characterization. Richardson
argues: ""9X" can be thought of as "a small change in X" which is traced around the loop
until it results in "a small change in dX/dt" in the inflow rate. If the change 9(dX/dt) is in
the same direction as the change gx, then they have the same sign." Richardson uses this
definition to determine the polarity of major loops as well. Suppose a major loop consists
of n level variables; X1, X2, ..., Xp. The polarity of the major loop is defined as:
ao te 0(dX2/dt) * « 0(dX,,/dt) 1
Sign a
ent aX, aXna is]
Form the above definition one can see that the polarity of the links coming from the
major loop into a rate variable are determined, while the links coming from minor loops are
ignored. The polarity of the major loop is then the sign of the production of the polarity of
the links of the major loop. It is obvious that the links which connect rates to a level
variable are missing. Thus, changes in inflows to outflows (or outflows to inflows) over
time which is likely to occur in a dynamic system cannot be captured. This definition tells
only part of the story about the behavior of the system. The limitation arises form the fact
that some links coming into level variables are assumed to be constant. Relaxing this
assumption eliminates the limitation mentioned above and leads to the extension of formal
definition of loop polarity.
Although the extension of the formal definition of loop polarity is more complex, it
provides a more precise statement about polarity of implicit and explicit loops. In this
framework, explicit loops are equivalent to the conventional definition of major feedback
loops. An implicit loop is a pathway that starts from a particular state variable, X1, and
traces around the major loop until it reaches to the corresponding rate variable, dX 1/dt.
Thus, in a linear system, the number of implicit loops, or pathways, in a major feedback
loop is equal to the number of state variables in the major loop. It is important to note that
pathways are not "intrinsic" but merely "descriptive". Their polarity is time dependent and
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sensitive to initial values. They expose a connection between the behavioral characteristics
of a system and its structural characteristics.
To understand the polarity of the major loop of the system shown in Figure 1, the
extension of the formal definition of loop polarity is applied as follows:
Taking the derivative of [1] with respect to X yields:
d(dX/dt) _ gp
dx «dX
P 19]
The polarity of the implicit loop which determines X (Figure 4) depends on the sign of Z/P.
If both Z and P are positive or negative, then the implicit loop is positive. This means that
a change in X leads to a change in the inflow rate of X in the same direction. Otherwise, if
either X or P is negative (and the other one is positive) then the polarity will be negative.
Thus, a change in the inflow rate of X as a result of a change in X is in the opposite
direction. The polarity of the implicit loop or pathway, shown in Figure 4, shifts from
positive to negative (or negative to positive) at point Z=0. The polarity of this implicit loop
is not defined at P=0.
x x
+ _
if Z/P is positive if Z/P is negative
Figure 4: Polarity of the loop which determines X
By taking the derivative of [2] with respect to Y we have:
d(@Yidt) _ p
dy x [10]
The polarity of the implicit loop which determines Y is positive if both P and X are positive
or negative, which means a change in the inflow rate of Y caused by a change in Y is in the
same direction. If either X or P is negative (and the other is positive) then the polarity will
be negative. This implies that a change in Y leads to a change in its inflow rate in the
opposite direction. The polarity of the implicit loop, shown in Figure 5, shifts from
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positive to negative (or negative to positive) at point P=0. The polarity of this loop is not
defined at X=0.
Y y
+ =
if P/X is positive if P/X is negative
Figure 5: Polarity of the loop which determines Y
By the same procedure, the polarity of the implicit loops which determine Z and P can be
determined. Therefore,
a@dZidt) _&
dZ Y fly)
Zz Z
eS ina
if X/Y is positive if X/Y is negative
Figure 6: Polarity of the loop which determines Z
The polarity of the implicit loop which determines Z is positive if both X and Y are positive
or negative. Thus a change in Z leads to a change in its inflow rate in the same direction.
If either X or Y is negative (and the other is positive) then the polarity will be negative,
which means a change in the inflow rate of Z as a result of a change in Z is in the opposite
direction. The polarity of the implicit loop, shown in Figure 6, shifts from positive to
negative (or negative to positive) at point X=0. The polarity of this implicit loop is not
defined at Y=0. For P we have:
d(dP/dt) _y
dP Z [12]
the polarity of the implicit loop which determines Z is positive if both X and Y are positive
or negative. If either X or Y is negative (and the other is positive) then the polarity will be
negative.
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Pp P
+ -
if Y/Zis positive if Y/Z is negative
Figure 7: Polarity of the loop which determines P
The polarity of the implicit loop which determines P is positive if both Z and Y are positive
or negative. This means a change in P leads to a change in the inflow rate of P in the same
direction. If either Z or Y is negative (and the other is positive) then the polarity will -be
negative which implies a change in inflow rate of P as a result of a change in P is in the
opposite direction. The polarity of the implicit loop, shown in Figure 7, shifts from
positive to negative (or negative to positive) at point Y=0. The polarity of this implicit loop
is not defined at Z=0.
As we have seen, a change in a state variable results in a change in its inflow rate in
the same or opposite direction. This simply means that the implicit loop which determines
a state variable can be positive or negative. Furthermore, the implicit loop that is
responsible for generating the behavior of a state variable may or may not have the same
polarity as an implicit loop which determines another state variable.
In the above example, when X, Y, Z, and P all are positive (or negative), the
implicit loop which determines any state variable has positive polarity. This implies that all
derivatives of each state variable are positive (or negative). For example for X we have:
dx _
dt P>0
@X _dP_ 7.09
d2 dt
and in general, since X, Y, Z, and P are always positive, then
@X 50 ; foralln
dt
In such a condition, the system will produce exponential growth. When one or more of the
state variables are negative and the rest of them are positive the system overshoots or
oscillates, but the growth mode will be dominated in the long run, unless the absolute value
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of all initial value problems are the same and sum to zero (see figure 3). It is worth noting
that under this circumstances some derivatives of a state variable are negative and some are
positive.
In any case, the polarity of the major loop in Figure 1 is positive. In this Figure,
those implicit loops that are responsible for generating the behavior of different state
variables (Figure 4 through 7) are only tacitly being taken into account. Indeed, they are
collapsed into one loop as shown in Figure 1. The polarity of the implicit loop that is
responsible for creating the behavior of a level variable over time can be thought of as one
of the components of the polarity of the aggregated loop. The polarity of the components
of the aggregated major loop (Figure 1) varies in terms of magnitude and sign as level
variables change. The apparent changes in the strength and sign of the polarity of the
components of the major loop may cause the system to oscillate.
The polarity of the aggregated major loop shown in Figure 1, is a function of the
polarity of its components. A simple function form, but not complete, may be a
multiplicative one:
Lp = dP/dt x dX/dt x dY/dt « dZ/dt
dx/dt dY/dt dZ/dt dP/dt [13]
where LP is a rough measure of the polarity of the aggregated major loop. Substituting the
equivalents of the elements of LP, which are presented in equations [9] through [12] gives:
=ZePxXx¥_
LP PX YZ 1 for X, Y,Z,andP>0O 14]
From the above equation, one can see the polarity of the aggregated major loop is
greater than zero. When any of the state variables (X, Y, Z, P) are zero, the polarity of the
loop is not defined. The apparent discontinuity in the polarity of aggregated major loop is
generated by the shifts in the polarity of its components. Notice that if somehow only one
state variable, say P, becomes negative while the other state variables are positive, the loop
which determines X becomes negative (Figure 4). At the same time, the polarity of the
implicit loop which determines Y shifts from positive to negative. As a result, LP, a
measure of the polarity of aggregated loop, remains positive. Because of the problem of
cancellation, changes in the polarity of implicit loops shown in Figure 4 through 7, are not
completely reflected on the measure of polarity of aggregated major loop.
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In summary, the aggregate measure of the polarity of major feedback loops is not
sufficient to characterize the behavior of complex systems. The application of the extended
formal definition of loop polarity to identifying the polarity of major loop implies that the
components of a major loop do not necessarily have a fixed and continuous polarity.
However, the multiplicative function form used in [13], which is rough, abstract and
incomplete, cannot reflect all changes in the polarity of the components of the major loop.
The polarity of the components of the major loop may change, while the polarity of the
major loop itself remains unchanged. The shifts in the polarity of the components of the
aggregated major loop create discontinuity in the polarity of the major loop and lead the
system to overshoot or oscillate. It should be noted that the extension of the formal
definition of loop polarity only elaborates the identification of the polarity of the
components of a major loop, not the major loop itself. The components of a major loop --
or implicit loops -- reveal the behavioral characteristic of the system. Therefore, they are
merely descriptive, time variant and sensitive to initial conditions. The explanatory power
of implicit loops enable us to understand how changes occur in the behavior of a variable of
the system over time. In general, a positive (negative) implicit loop causes the
corresponding level variable to grow exponentially (goal-seeking), unless it is dominated
by other positive (negative) loops.
References:
Forrester, Jay W. 1968. Principles of Systems. Cambridge, MA: The MIT Press, Wright-
Allen Series.
Forrester, Jay W. 1968. Market Growth as influenced by Capital Investment. Industrial
Management Review, 9, 2.Cambridge, MA: The MIT Press, Wright-Allen Series.
Graham, Alan K. 1977. Principles of the Relationship between Structure and Behavior of
Dynamic Systems, Ph. D. dissertation, MIT, MA, 02139
Richardson, G. P. 1995. Loop Polarity, Loop Dominance, and the concept of dominant
polarity. System Dynamics Review 11(1) :67-88
. 1986. Problems with causal-loop diagrams. System Dynamics Review
2(2) :158-170.
. 1986. Dominant Structure. System Dynamics Review 2(1) :68-75.
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