Order, Disorder and Complexity in Hierarchies

J.S. Shiner, Matt Davison and P. T. Landsberg

Whether we are dealing with physical, biological, social, economic or symbolic systems, the concepts of order, complexity, and emergence are intimately involved with hierarchical structures. [Baas] "... a large proportion of the complex systems we observe in nature exhibit hierarchic structure." [Simon] "A prodigious complexity emerges from the consideration of a 'hierarchical order' ..." [Needham] "We do not know what forms of life exist, but we can safely assume that whenever there is life, it is hierarchically organized." [Koestler] The most commonly used measure of disorder (and order, through an appropriate inverse relation) is the Boltzmann-Gibbs-Shannon or information entropy. However, for the study of hierarchies it is not an appropriate measure. In general a hierarchy will have different numbers of states available to it at its different levels. Entropy is an extensive quantity and in general will change from one level of a hierarchy to another simply because of the difference in number of available states. A better measure for disorder is given by normalizing the entropy to an appropriate maximum value. [Landsberg] We call this measure "disorder" (in quotes!) to distinguish it from the general concept of disorder. It lies between 0 and 1. We now take "order" to be 1 - "disorder". These measures also lead directly to one possible simple measure for complexity as the product of "order" and "disorder". (We again distinguish the measures "order" and "complexity" from the corresponding general concepts by enclosing their names in quotes.)

With these measures the development of "order" and "complexity" in hierarchies has been analyzed. We find that there are (at least) two different kinds of hierarchical structures with different properties. In combinatory hierarchies the states or elements of level i + 1 are n_i tuplets of the states or elements of level i. Here we find that the "disorder" of level m relative to the absolute maximum entropy (= logarithm of the number of available states) can be written as the product of the "disorders" of all levels 1 up through m relative to the entropy of the next lower level. This in turn implies that "order" cannot decrease in a combinatory hierarchy on going from one level to a higher level. However, simple counter examples show that the entropy can increase or decrease. In the second sort of hierarchy the next higher level is created from the next lower level by splitting the states. For example, state 1 splits into two states 1a, 1b; state 2 into three states 2a, 2b, 2c; state three remains a single state; etc. Here it is found that the entropy can not decrease, but "order" may increase or decrease in passing from one level to the next. In both classes of hierarchies, however, it can be shown that "disorder" is a better measure of randomness than is entropy.

The results for combinatory hierarchies emphasize the importance of the choice of maximum entropy in the definition of "order". Here we found a relation between the absolute "disorder" (that relative to the absolute maximum entropy) and the relative "disorders" of the levels, where relative implies that the maximum entropy for a given level is taken to be the entropy (not the maximum entropy) of the next lower level. In general, one can always distinguish between at least two possible maximum entropies for physical systems. One is the absolute maximum, that for the equiprobable distribution. The other is the entropy of the "equivalent random system", where "equivalent" means here that the values of all appropriate extensive quantities, other than the entropy itself, are held constant. This point will be discussed in general and for the example of simple spin systems (Ising models). This allows for a multiplicity of "order"-"complexity" relations, although the definition we use of "complexity" would seem to allow only one universal one.

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