Monday, October 13, 2008

Causal Maps

As discussed in the previous posting, we determine the stability of a causal loop by offsetting plus and minus segments in the loop. A Causal Map is a hierarchy of interacting, interlocking causal loops. A map represents the overall model. An important question for such a model is will it reach equilibrium based on the nature of casual loops within it.

There are several strategies for determining the stability of a map that can be generalized into two categories: 1). Loops within a map are of unequal importance; 2.) Loops are of equal importance.

If loops are of unequal importance, one strategy for predicting equilibrium is to assume the fate of system is determined by most important loop. An issue arises that the judgment of loop importance may be arbitrary. We can address this with knowledge of the process being modeled to argue about the most important inner dynamic.

On the other hand, we can also neutrally look at which loop has the most variables of interest and asssume such a loop is most important. The greater the number of inputs to and outputs from a variable, the greater its importance. The loop with the greatest number of important variables is the most important loop. This is also known as the degree of the system.

In the example of urbanization that was modeled in the previous posting, variables P, M, S all have more than one output, as do variables P, D, B. So P,M,S,D,B are most important variables. We look for the loop that contains greatest number of these: P-M-S-B-D-P is that loop.

Causal Loop P-M-S-B-D-P has an even number of negative signs so it is deviation amplifying. We would conclude the system represented by the entire Causal Map is unstable, using this strategy. It will eventually destroy itself if something is not dramatically changed.

Now let’s assume a different strategy category for P-M-S-B-D-P –that the loops are of equal importance. To predict system stability, count the number of negative loops. If this is an even number, then the system is deviation amplifying, if odd then stable. In the urbanization case, we have one of the four loops negative, P-G-B-D-P. If the relationships are of equal importance, in our example the result is a stable system. The system will ultimately reach an equilibrium.

Another tactic for a strategy that assumes the causal loops in a map are of equal importance is to count the number of negative relationships between variables, making sure to count a relationship more than once if is in multiple loops. Relational algebras can be developed for such calculations for maps of any size.

It’s possible both strategy categories are applicable to a map, albeit at different times. Urbanization, in our example, may be stable for some period of time when all causal relationships are more or less equal in effect on overall system. Imbalances can, however, accumulate over time to change the egalitarian inter-relationships.

A differential speed that cycles are completed, or a growth in the number of times a particular loop is activated across periods of time can result in an aristocratic set of relationships – some holding more power and control over the system. Then we would change from a strategy assuming equal importance of all loops to analyzing the most important loop.

In sum, Causal Maps are a means to portray a complex interdependence so one can better question the situation. They help to impose some order to the domain being analyzed.

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