Self-Organised Criticality

Let us now look in more detail at just one of the above options:

Now, let the set of actors be ; the parameter under investigation be for some at time ; and be a metric on the parameter space.

Define the average:

Then we say is in a critical state if:

In order to calculate or approximate such a limit for a system, we need to know the rules. In this case all we need is a rule to give us each from some local neighbourhood , at time . We formalise this as follows:

Here, in full generality, is dependent on all properties of the previous state. For specific systems, the variation in may be less general, or even constant.

As an example, let us consider a one dimensional system, with constant in time, and a unit neighbourhood.

This is the update equation for a cellular automaton (the derivation of the second line requires that the actors are not uniquely identifiable). Thus CA will fit within this framework, with many other systems, although perhaps very few will satisfy the critical property. This is the kind of phenomenon which is likely to be emergent for many .

Current research[6] dealing with one particular type of rule - describing the interactions between predictive agents in an artificial economy - has demonstrated the existence of robust self-organised dynamic equilibria. The equilibria are found in the space defined by a metric which isolates the complexity of the predictive algorithms used by the agents. Simulations have shown this is a critical state, with power-law scaling of adaptive changes to the predictive models. Such results will be presented in a forthcoming paper of a less philosophical nature.