What Is Complexity?
It would require many different concepts to capture all our notions of the meaning of complexity. The concept that comes closest to what we usually mean is effective complexity (EC). Roughly speaking, the EC of an entity is the length of a very concise description of its regularities. A novel is considered complex if it has a great many scenes, subplots, characters, and so forth. An elaborate hierarchy can contribute to complexity, as in the case of nested industrial clusters each composed of a great variety of firms and other institutions. In general, though, what are regularities? We encounter in many different situations the interplay between the regular and the random or incidental: music and static on the radio, specifications and tolerances in manufacturing, etc. But ultimately the distinction between the regular and the incidental depends on a judgment of what is important, although the judge need not be human or even alive. For instance, in the case of songs of a male bird in the nesting season, the identification of regularities is perhaps best left to the other birds of the same species — what features are essential in repelling other males from the territory or attracting a suitable female? A technical definition of EC involves the quantity called algorithmic information content (AIC). The description of an entity is converted to a bit string and a standard universal computer is programmed to print out that string and then halt. The length of the shortest such program (or, in a generalization, the shortest that executes within a given time) is the AIC. The AIC is expressed as the sum of two terms, one (the EC) referring to the regularities and the other to the random features. The regularities of a real entity are best expressed by embedding it conceptually in a set of comparable things, the rest of which are imagined. The EC can then be related to the AIC of the set, the choice of which is restricted by the conditions imposed by the judge. Theorists like to study highly simplified models of complex systems, often by computer modelling. What can be claimed for such models? Overall agreement with observation is hardly to be expected. However, in many cases simple regularities can be found in both the observational data and the model, which may then be helpful in understanding those regularities. Examples are given, involving scaling laws and also “implicational scales”.