Qualitative Behavior Associated to System
Dynamics Influence Diagrams
Javier Aracil* and Miguel Torot
April 28, 1992
Abstract
The paper introduces a simple dynamical system associated to the in-
fluence diagram which contains only qualitative information. It is analysed
how with the information in the influence diagram only limited conclusions
about the behavior of the system can be reached. However, with some extra
qualitative information regarding the relative weight of the influences those
limitations are overcome.
1 Introduction
Since its beginning system dynamics has claimed to have a strong qualitative com-
ponent. On the one side, system dynamics models are reelaborations of verbal
descriptions, where the qualitative aspects (in the sense of pre-quantitative) are
dominant. On the other, most of the conclusions got from a system dynamics model
are mainly qualitative. They refer to the modes of behavior: growth, oscillation,
decay,... and not to the quantitative detail of the trajectories.
In the system dynamics context the word qualitative has been used at least
with two senses. In the first of them, qualitative is synonymous of pre-quantitative
or poorly quantitative. In this sense it is said that the influence or causal diagram
contains only qualitative information. The qualitative analysis of a system dynamics
model can consist on the elucidation of the feedback loops, the determination of the
sign of these loops and of the charactor of self-regulation or of explosive behavior
associated with them. Wosthelholme has proposed to call the first use qualitative
system dynamics. (Wosthelholme, 1990).
“Escuela Superior de Ingenieros Industriales, Universidad de Sevilla, Avda. Reina Mercedes
s/n, 41012 Sevilla, Spain
tFacultad de Infromética, Universidad de Sevilla, Avda. Reina Mercedes s/n, 41012 Sevilla,
Spain
In the other of the senses, the word qualitative is used as in the qualitative theory
of dynamical systems (Abraham and Shaw, 1987; Guckenheimer and Holmes, 1983).
This use has deep topological and geometrical connotations and it is based on the
concept of qualitative that has its roots in the work of Poincaré, that has been
updated by Thom (Thom, 1977, 4-7) and Zeeman (Zeeman 1977, 319-329). The
relevance of these results for the system dynamics method has been emphasized
elsewhere (Aracil 1981, 1984, 1986, Aracil and Toro 1992).
In this paper our aim is to explore how the formal qualitative analysis techniques,
based on the second of the above senses of qualitative, can be used to solve the kind
of questions suggested hy the first of the uses. In this way a synthesis of both
senses can be reached. We will assume that the qualitative information about a
given concrete system comprises no numerical information beyond the classification
of the variables in a system as states, rates and auxiliaries, the signs of the influences
and the relative value (the weight) of these influences. With this information we
try to get as much knowledge as possible on the behavior modes of the system, in
the concrete meaing given to behavior mode in the qualitative theory of nonlinear
dynamical systems.
2 Influence diagrams and digragphs
The first step in system dynamics conceptualizing is to built the influence diagram
of the system to be modeled. Then we have to classify the variables appearing in it
as states, rates and auxiliaries. This classification is driven mostly by the experience
of the modelist. There are algorithms too help in it (Burns, 1979). It should be
noted that this classification involves a knowledge about the structure of the system
that is a further more involved that the one in the influence diagram. However it
is still of a qualitative nature: do not involves any quantitative knowledge yet. Our
goal in this paper is to analyse what can be said about the behavior of the system
from this knowledge.
A weighted digraph G is a mathematical object formed by a set N of points
called vertices or nodes, a set I’ of lines or couplings called edges or archs, where
each edge can de represented by an ordered pair of vertices, and a set W formed
by the weights associated to each edge. That is, G = (N,I,W). If every non-null
wiz = 1 then we have a digraph or directed graph (Fig. 1a). To a digraph it can be
associated a binary incidence matrix. If every non-null w;; = 1 or —1 then we have
a signed digraph (Fig. 1b) and a signed incidence matrix. For an arbitrary W we
have a weighted digraph (Fig. 1c). Lastly, if to each edge more than one weight can
be associated then we have a family of weighted digraphs G; = (N,T,W;). These
weights change as do the values taken by some variables x associated to the members
of the set N. These variables define the space 2. In this last space regions 2; € 2
where the weights are constant are defined. To each region 2; a G; can be associated.
A path is defined as a series of edges starting at one vertice and ending on another,
on ED =
A d—30B
Figure 1: Different digraphs: (a) digraph , (b) signed digraph and (c) weighted
digraph
rx
a
aw x— 32
r
Figure 2: System dynamics allowed couplings between states, rates, auxiliaries and
parameters.
without crossing any vertice twice. The path linking consecutive nodes 7, j and k
will be denoted by (ijk). A path that returns to its starting point in called a loop.
The weight of the path linking nodes i and j will be denoted by wi; or by w(ij)
and, when i and j are not contiguous, is defined as the product of the weights of
the edges that form it; that is,
w(ik...j) = w(ik)w(kl)...w(mj)
Classical system dynamics influence diagrams are signed digraphs. In these digraphs
the set N represents all the quantities in the model: variables and parameters. This
set N is partitioned in the subsets: states X, rates R, auxiliaries Z and parameters
P. This last set P includes all exogenuous variables, even if they are not constants.
According to the partition
N=XURUZUP
There are some restrictions about the couplings allowed between the variables, de-
pending on whether they are states, rates, auxiliaries or parameters. These rules
are summarized in Fig. 2.
Once the variables of the influence diagram have been classified into states, rates
and auxiliares it is easy to see that the mathematical form of the model can be
= 8 ty, ahe _ 3% _ at,
wae Bx, 3X woe aS Bg
x ————_———_ rm x
/ _ in ——-
wae OZ Bz,
2
Figure 3: Partial derivatives and weights of the digraph associated to the system.
written in the form:
& = Ar
= f.(x,2,p) (1)
z = f.(z,p)
Where z stands for the state variables, z € X, r for the rate variables, r € R, z for
the auxiliary variables, z € Z, and p for the parameters p € P. Matrix A isn x m,
with a;; = 1 ifr; influences positively on 2;, aj; = —1 if it influences negatively, and
ai; = 0 it there is no influence.
The functions f,; (resp. f.;) give the value of a rate variable r; (resp. of an
auxiliary z;) from the value of the state variables z, the auxiliary variables z and
the parameters p that influence r; (resp. 2;). In this preliminary stage of the
modeling process the concrete mathematical form of these functions is assumed not
to be known.
The Jacobian matrix of dynamical system (1) can be written
J =D.f = A[Dzf, + (Dz f-)(Defz)|
If B = AD, f, and C = A(D,f,)(Drfz), then J = B+C. To compute matrix J
the functions f, and f, are needed. Actually what are needed are the partial of
derivatives of these functions. But these partial derivatives can be, in some way,
identified with. the weights of the digraph associated to the system (Fig. 3).
With this identification of partial derivative and weight of the relationship, it is
easy to see that the element b;; of B is given by the weight of the path that links
the state j directly with a rate i that affects that state variable. In effect,
ye, Ofte
b= can 28;
where a,, is the weight of the influence r, — 2; and og is the weight of the influence
z;— r,. Then a, ee is the weight of the path 2; > r, — 2;. Therefore 6;; is the
- 44 -
sum of the weights of all the paths that link the state variable 2; with 2; through
rate variables.
In the same way, the element c;; of C is given by the sum of the intensities of
the different paths that link i to j throught any auxiliary variable. That is,
ag bls Bln
ae Yat Oy On;
where a,, has the same meaning as in B; fe is the weight of the influence 2; > rx;
and Ba is the weight of the influence x; > z. Then jy ee Sha is the weight of the
3
path x; + z — rx — 2;. Then c; is the sum of the weights of all the paths that
link the state variable x; with x; through at least an auxiliary variable. If more than
one auxiliary variable is in the path, then applying the chain rule a similar result is
obtained. With these results we can state the following rule:
The element J,; of the qualitative Jacobian matrix is given by the sum
of the weight s of all the paths that start in level variable j and end in
level variable 7, without crossing any other level variable, and it is zero
if there is no path from 7 to j.
According to this property the only information needed to get the qualitative
Jacobian matrix of a system dynamics model is supplied by the influence diagram
and a measure of the relative weight of the relations in that diagram. This is a
very remarkable property for qualitative analysis, as far as the Jacobian matrix
incorporates a huge amount of information on the qualitative behavior (specially,
on the stability properties) of a dynamical system.
According to conventional system dynamics multivariate influence is such that
the functions f, or f, are either single nonlinear relationships, or an arithmetic com-
bination of variables, or an arithmetic combination of single nonlinear relationships.
In any case there is a separation propertity that isolates” the nonlinearities in sin-
gle nonlinearities. For example, very often they are given a separable multiplicative
formulation. Then, it can be written:
u F(Yt) Ya. yn)
won) xe( 2) ona (2)
Where the functions f; are the well known system dynamics multipliers (Forrester
1969, p.22-30), y;,... ys are states, rates or auxiliary variables, and un, Yiny-- Ykn
stand for the normal values of variables u, yi,... yx respectively. It should be noted
that if:
1. f; is monotone,
wg ee
Figure 4: Two loops structure of a limits to growth model
2. Un > 0, yin > 0, and
3. fi > 0,Vys,
then du a
Fem Ex Siow Fo a (2)
and
we (5) = (55) ®
It is a remarkable fact that properties 2. and 3. above (f; being or increasing or
decreasing) are hold by most of the models in classical system dynamics literature.
Equation (3) can be generalized when f; do not holds the hypothesis above.
Having isolated the nonlinearities, the slope of the function f,(x) gives at least
the sign of the relation. THe weight can be obtained from (2). If fj = 1 and
Un/Yin = 1 the weight is given by the slope of function f;/ The weight can be
considered as a measure of the strength of the influence relation.
The question that raises this approach is if it is possible characterize the behavior
only from the information in the influence diagram. The answer is possitive in some
cases (Aracil and Toro, 1989). However, when ambiguities occur more information
is needed and then the weighted digraph should be used. But even in this last case
can happen that given a single weight to each arch is not enough. This happen,
for example, when the modelled system shows a behavior whith problems of loop
dominance. We shall see an example of this case to clarify this point.
3 Example
Consider the influence diagram of Fig. 4. This structure shows to loops: one
possitive and the other negative. The first one is responsible of a growth process,
that should grow for ever unless the action of the second loop counteracts, limiting
the growth. It is a well known structure that gives rise to the sigmoidal growth. It
has been recognized as one of the system archetypes (Senge, 1990).
- 46 -
What is interesting with this structure is that tha system shows a shift in loop
dominancy. When poputation is low, the positive loop dominates and a net growth
in the population is produced. When the population is high enough, the negative
loop dominates and the population tends to stabilizate.
This structure supplies a very good and elementary example of a model whose
qualitative analysis leads to ambiguities. The equations of the model can be written:
oP 2 HHP) — HHP) = 1) )
where P stands for the population, and f;(P) represent functions whose concrete
value is unknow, but that is know that they are monotonically increasing functions.
They represent what in system dynamics is known as a positive influence. In the
QSIM qualitative structural description they are denoted by Mj (Kuipers, 1989). In
this case f;(P) represents the births and f}(P) the deaths, and is is assumed that
it is only known that they grow with the population. It can be also assumed that
fi (0) = f# (0) =0 whitout breaking the qualitative nature of the knowledge.
The simplest way to formalize system (4) retaining these qualitative character-
istics is making f;*(P) = k;P, that is, making the functions f}*(P) linear. However,
that do not capturates the shift of dominancy effect that is so essential to this struc-
ture. This last is associted to the appearance of an ambiguity in the knowledge
capturated by this structure. In effect, if the jacobian is computed applying the
above rule, as there are two ways from node P to himself, one having positive sign
and the oter negative, then we have:
Sy = (+) + (-)
which leads to an ambiguity. To overcome this ambiguity more information is
needed. As a matter of fact, we know not only that the functions f;+(P) are mono-
tonically increasing, but we know too that the slope of f;*(P) is higher than the
one of f}(P), for small values of P; and that the reverse happens for high values
of P, This is shown in Fig. 5a. With this shape for f'(P) and f}(P), then f(P)
of (4) has the form shown in Fig. 5b. This form is very interesting as it shows two
equilibria, one unstable at the origin and other stable (the attractor of the system)
at A, in the figure. With that form for the functions f'(P) and fj'(P) we have
the possibly simplest system that capturates all the qualitative characteristics of the
structure and behavior of system in Fig. 4. It should be remarked that then we
have a piecewise linear system, that is a nonlinear system that is linear by pieces.
4 A qualitative modeling methodology
The previous results sugest a qualitative modeling methodology which should com-
prose the folowing steps:
-47-
P
P 4 2
Figure 5: Piecewise linear form for functions f*(P) and f}(P) (a); and resulting
form for f(P) (b).
abs
2.
5
Built the influence diagram (signed digraph).
Analyse the influence diagram applying the above rule to find ambiguities.
. Ask to the modeler to solve ambiguities giving relative weight to the influences
involved in every ambiguity (built a weighted digraph).
. Analyse eventual changes in the weight of every influence along the qualitative
range of the variables (built a multiweighted digraph).
. Built a piecevise linear system asociated to that multiweighted digraph.
. Develop the qualitative analysis (in the classical sense of qualitative theory of
nonlinear systems) of the piecevise linear system.
. Iterate the previous steps to reach an aceptable model.
Conclusions
We conclude that with the only information of the classical influence diagram (signed
digraph) only in limited cases general conclusions about the behavior of the model
can de reached. However, whith some aditional information, the weight of the influ-
ences, that is still of a qualitative nature, we can reach a more complete prespective
about those behavior modes.
6
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