Table Of ContentANNALS OF SYSTEMS RESEARCH
VOL. 5
In the ANNALS OF SYSTEMS RFSEARCH are published original papers in the field of general systems
research, both of a mathematical and non-mathematical nature. Research reports on special
subjects which are of importance for the general development of systems research activity as a
whole are also acceptable for publication. Accepted languages are English, German and French.
Manuscripts in three-fold should be typewritten and double spaced. Special symbols should be
inserted by hand. The manuscripts should not contain directions to the printer, these have to be
supplied on a separate sheet. The author must keep a copy of the manuscript.
The title of the manuscript should be short and informative. An abstract and a mailing address
of the author must complement the manuscript.
Illustrations must be added in a form ready for reproduction.
Authors receive 25 offprints free of charge. Additional copies may be ordered from the publisher.
All manuscripts for publication and books for review should be sent to:
H. Koppelaar
Associate Editor Annals of Systems Research
Institute for Methodology and Statistics
State University Utrecht
Oudenoord 6
Utrecht, the Netherlands
ANNALS
OF SYSTEMS RESEARCH
VOLUME 5, 1976
PUBLIKA TIE VAN DE SYSTEEMGROEP NEDERLAND
PUBLICATION OF THE NETHERLANDS SOCIETY FOR SYSTEMS
RESEARCH
EDITOR
B. V AN ROOTSELAAR
Social Sciences Division
tMartinus~ijhoff
CLeiden 1976
ISBN-IJ: 978-90-207-0657-4 e-ISBN-IJ: 978-1-4613-4243-4
001: 10.10071978-1-4613-4243-4
© 1976 H. E. Stenfert Kroese B.V./Leiden - The Netherlands
PREFACE
The Netherlands Society for Systems Research was founded on 9 May 1970
to promote interdisciplinary scientific activity on basis of a systems approach.
It has its seat in Utrecht, The Netherlands.
Officers for the years 1975/1976:
President: G. Broekstra, University of Delft
Secretaries: G. De Zeeuw, University of Amsterdam (acting secretary)
G. R. Eyzenga, University of Groningen
Treasurer: J. N. Herbschleb, Computer Laboratory, Department of Cardio
logy, University Hospital, CatharijnesingellOl, Utrecht.
All information about the society can be obtained from the acting secretary.
The editor is happy to announce that H. Koppelaar from the State University
Utrecht will act as associate editor of the Journal.
Moreover, the following scientists have declared to be willing to act as member
of the editiorial board:
Professor G. Klir, State University of New York, Binghamton, New York,
U.S.A.
Professor S. Braten, Institute of Sociology, University of Oslo, Blindern,
Norway
Professor B. R. Gaines, Department of Electrical Engineering Science, Univer
sity of Essex, Colchester, U.K.
Professor Maria Nowakowska, Department of Praxiology, Polish Academy of
Sciences, Warszawa, Poland.
Professor F. Pichler, Department of Systems Theory, Johannes Kepler Univer
sity, Linz-Auhof, Austria.
Professor B. Zeigler, Department of Applied Mathematics, Weizmann Institute
of Science, Rehovot, Israel.
The editor
ADDRESSES OF AUTHORS
Broekstra, G., Graduate School of Management, Poortweg 6-8, Delft, The
Netherlands.
Dalenoort, G. J., Institute for experimental psychology, State University
Groningen, Biological Centre, Section D, Kerklaan 30, Haren (Gr.), The
Netherlands.
Klir, G. J., School of Advanced Technology, State University New York,
Binghamton, N.Y. 13901, U.S.A.
Kooijman, S. A. L. M., Institute for theoretical biology, Stationsweg 25,
Leiden, The Netherlands.
Koppelaar, H., Institute for Methodology and Statistics, State University
Utrecht, Oudenoord 6, Utrecht, The Netherlands.
Masser, I., Institute for Urban and Regional Planning, Heidelberglaan 2,
Utrecht, The Netherlands.
Scheurwater, J., Institute for Urban and Regional Planning, Heidelberglaan 2,
Utrecht, The Netherlands.
Uyttenhove, Hugo J. J., School of Advanced Technology, State University
New York, Binghamton, N.Y. 13901, U.S.A.
CONTENTS
Koppelaar, Ho: Predictive Power Theory 1-5
0 0 0 0 0 0 0 0 0 0 0
Dalenoort, Go Jo: Collectivity in information-processing systems 7-28
Klir, Go, Uyttenhove, Ho Jo Jo: Computerized methodology for
structure modelling 29-65
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Broekstra, Go: Constraint analysis and structure identification 67-80
Masser, I., Scheurwater, Jo: Spatial interaction in the Amersfoort
region: a systems analysis 81-112
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Kooijman, So A. L. Mo: Some remarks on the statistical analysis of
grids, especially with respect to ecology 113-132
Editor's note 133
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
PREDICTIVE POWER THEORY
HENK KOPPELAAR
Summary
Mulder's 'Game for Power', published in [I], has been fully formalized in [2] according to a
method originated by Popper. We exploit this psychological theory further. Our results pertain
to an overall analysis of the theory, whereby encompassing computer-simulation.
1. Introduction
After publication of [2] by Hezewijk et al., some work was left to be done on
Mulder's Power Theory [1], because this publication provided a transformation
of Mulder's theory in terms of Forresters programming language DYNAMO
[3] and we expected it to be feasible to reformulate the DYNAMO - version
into differential equations.
Model formulations in terms of differential equations are most suitable
for an overall analysis of the model. With 'overall analysis' we mean the phase
plane method which facilitates the prediction of systems behaviour for any
point in time and any parameter value. Hence the name 'predictive power
theory' for our exposition in the sequel.
The reformulation of DYNAMO-statements in terms of differential equa
tions is quite straight-forward, as follows.
By definition:
d Xi() _ /. X(t+h) -X(t)
- t - 1m ---"---'-------'-'-
dt h~O h
In view of this definition, the D YNA M0 equation
X.K = X.J + DT * Y.K (1)
where DT > 0, is reformulated in
~ X(t) = Y(t) (2)
dt
So (1) is the straight-forward Euler discretization of (2).
Annals of Systems Research,S (1976), 1-5
2 H. KOPPELAAR
2. Reformulation of the theory
Hezewijk et al. [2], formulate the following (p. 56)
KOSWI.KL = (ljAT) (MA.K • PER VAl • REALI) (3)
BATWI.KL = (ljAT) (MA.K • GENI • REALI) (4)
+
KOBAI.K = KOBAI.J (DT) (KOSWI.JK - BATWI.JK) (5)
where pervai, geni and reali are personality constants.
We reformulate (3), (4) and (5) in one equation (6)
~ KOBA l(t) = _1 . [(PERVA I - GENI)REALI] MA(t) (6)
dt AT
The rest of the statements in [2] are
VERSTI.KL = (ljAT) (MA.K • PERVAI) (7)
+
MANST.K = MANST.J (DT) (VERSTI.JK) (8)
BATWE.KL = (ljAT) (MA.K • GENE. REALE) (9)
KOSWE.KL = (ljAT) (MA.K • PERVAE • REALE) (10)
+
KOBAE.K = KOBAE.J (DT) (KOSWE.JK-BATWE.JK) (11)
VERSTE.KL = (ljAR) (MA.K • PERVAE) (l2)
+
MAVST.K = MAVST.J (DT) (VERSTE.JK) (13)
MAN.KL = (lIAT) (MANST.K-KOBAI.K) (14)
MA V.KL = (ljAT) (MAVST.K - KOBAE.K) (15)
MA.K = MA.J + (DT) (MAV.JK-MAN.JK) (16)
The reformulation in terms of differential equations is
for (7), (8): ~ MANST(t) = (ljAT) (PERVAI)MA(t) (17)
dt
for (9), (10), (11): ~ KOBAE(t) = (ljAT) [(PER VAl -GENE)REALE]
dt
MA(t) (18)
for (12), (13): ~ MAVST(t) = (1IAT) (PERVA E)MA(t) (l9)
dt
3. The Formal Power Theory
Our excursion into previous work [2] on Mulder's theory yields equations:
(6), (14), (15), (17), (18), (19).
As a matter of fact these can be substituted directly into (16) without loss of
generality, pertaining to one second order equation:
PREDICTIVE POWER THEORY 3
d2
dt MA(t) = ac2.MA(t) (20)
2
where a = PERVAE-PERVAl-(PERVAE-GEN)REALE
(PER VAl - GEN/)REALI
c = ItAT, AT >0
In algebraic format (20) reads:
(0, ')
~
(21)
(xo) = (xo)
Xl ac 0 Xl
where: Xo == xo(t) = MA(t)
d
and Xl == xl(t) = - MA(t)
dt
From our substitution of (14) and (15), in (16) we know
~ +
MA(t)lr=o = (1tA]) (MAVST(O)-KOBAE(O)-MANST(O)
dt
+ro~~) ~
Formula (21) represents the whole model given in [2], where (22) represents
an initial condition of X I.
4. Predictions from the model
The model for Mulder's Power Theory [1] is
d
- x(t) = Ax(t)
dt
COc ~)
where x(t) is a vector (xo(t), XI(t»T and A = 2
with constants a and c according to (20) and initial conditions.
In the case det(A) = -ac2 > 0 we have in the phase-plane a centre point in the
origin, fig. 1. °
In the case det(A) = -ac2 < we have in the phase-plane a saddle point in the
origin, with asymptotes X I = ± Xo, fig. 2.
The case det(A) = 0, that is if a = 0, means that the system is in equilibrium,
psychologically this says that the power distance is constant, because of the
personality structure between the more and the less powerful person.
