Jürgen Klüver and Jörn Schmidt (1999)
Journal of Artificial Societies and Social Simulation vol. 2, no. 3, <http://jasss.soc.surrey.ac.uk/2/3/7.html>
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Received: 15-Jun-99 Accepted: 5-Oct-99 Published: 31-Oct-99
It is well known that the dynamics of Boolean networks and the logically similar cellular automata are governed by control parameters. Less well known is the fact that the geometry of these artificial systems, understood as their topology and metric, also contain specific control parameters. These "geometrical" control parameters can be expressed using graph theoretical concepts such as the density of graphs or geodetical properties. Further, the dynamics of those artificial systems depend on the values for the geometrical parameters.
These mathematical investigations are quite important for social research: On the one hand, social dynamics and social structure appear to be two closely related aspects of social reality; on the other hand, a general hypothesis may be drawn from our results, namely that social structural inequality yields simple dynamics whereas social equality gives rise to complex dynamics. Therefore the dynamical complexity of modern democratic societies may be in part due to their democratic structures.
Definition: For any pair of elements, e and f, define a distance measure, d, by:
Definition: The structure of a social system is a pair, (R, G), where R is the set of all general rules of a dynamical system and G is the set of all geometrical rules - that is, the rules governing the particular topology and metric of the system.
Sy = k/n.
2One of the advantages of modeling dynamical systems in this manner is that the geometry of a system can easily be taken into account. This can be seen even more clearly when modeling predator-prey systems where the geometry of the artificial system is the model of the geometry of the physical space. The Lotka-Volterra equations have no regard for the physical space of predator-prey interactions.
3The verification of this statement is left to the reader. When using the term "topology" we refer to the classical neighborhood axioms of Hausdorff, which can be found in any book on set theory or general topology.
4Strictly speaking, this definition is only valid for non-directed graphs because in directed graphs there need not be interactions between both a and b both. It may be necessary to introduce a "weak" metric, that is a metric where the axiom of symmetry is not always valid. It may be useful to define the distance, for example, between a superior (sup) and a subordinate (sub), with d(sup, sub) = 1 and d(sub, sup) = 1.5 in order to consider the fact that the influence of the superior to the subordinate is bigger than that of the subordinate to the superior. But these are questions for the future.
5In contrast to Kauffman we do not believe that K is an important parameter, though the results reported by him are, of course, valid. We think K to be a "derived" or secondary parameter insofar as the simple dynamics for K = 1 or 2 are just a consequence of the fact that in these cases most Boolean functions have high values of P or CF (see below) and vice versa. It is quite easy, by the way, to generate complex dynamics with K = 2. But these are problems for specialists and we shall not develop them further in this paper.
6For the sake of simplicity we skip the Z-parameter (Wuensche and Lesser 1992), which measures the probability of computing backward; that is, determining a system's state at time t-1 with known state at time t.
7When K is not equal throughout the net the definition of V is a bit more complicated but basically the same. We skip here the mathematical details for how V is measured. Readers who are interested in these details may contact the authors for further information.
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