Jürgen Klüver, Christina Stoica and Jörn Schmidt (2003)
Formal Models, Social Theory and Computer Simulations: Some Methodical Reflections
Journal of Artificial Societies and Social Simulation
vol. 6, no. 2
<http://jasss.soc.surrey.ac.uk/6/2/8.html>
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Received: 18-Sep-2002 Accepted: 17-Feb-2003 Published: 31-Mar-2003
f^{n} (S_{1}) = S_{ n+1}. | (1) |
A point attractor S_{a} of the trajectory is now simply defined as
f^{n} (S_{a}) = S_{a}, | (2) |
with the corresponding definitions of simple attractors with periods k > 1.
S = (C, St). | (3) |
Ro = (r,k). | (4) |
Figure 1. a Toynbee development with low EP-values: the culture is caught in an attractor |
Figure 2. a modern development: the culture transcends attractors |
(1,1) = 1 (1,0) = 0 (0,1) = 0 (0,0) = 0, |
(5) |
then obviously this rule is nothing else than the logical conjunction.
v = |(OD - OD_{min})| / |(OD_{max} - OD_{min})| and 0 ≤ v ≤ 1. | (6) |
OD_{min} is the outdegree vector, i.e. the vector for the outputs of the units, that contains the maximal possible homogeneous distribution of outputs; OD_{max} is the outdegree vector with the minimal possible homogeneous distribution and OD is the factual outdegree vector for the BN.
^{2} To be sure, the use of differential or difference equations does not necessarily mean a top down approach. It is quite possible to model systems with differential equations while going bottom up. For example, Kepler's equations are a typical top down model of the planetary system; Newton's theory of gravitation is strictly speaking a bottom up model.
^{3} Our translation from the German version
^{4} We have to make a caveat to this statement: of course there are also theoretical approaches in the social sciences that try to start with the basics of social behaviour and go on from there by enlarging their models. Such a procedure is, e.g., characteristic for Garfinkel and his school of ethnomethodology, the role theory of Goffman or other micro sociological approaches. Therefore our remark is to be understood as a remembrance of methodical procedures that are sometimes neglected.
^{5} We assume that the determinant of the pay off matrix is the significant parameter. Unfortunately neither Nowak and May nor Fogel did investigate this question.
^{6} According to a personal remark of Michael Cohen they were inspired to do this by the research of our group.
^{7} Readers who are acquainted with the famous "limit theorems" of Gödel, Church and Turing may be a bit surprised by our statement, that this procedure has no limits. But it is rather easy to demonstrate that for every single modelling it is possible to construct a sufficient powerful formal system, although for this formal system the limit theorems are also valid (cf. Klüver 2000).
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