Complexity measures in decomposable structures

In this paper we present some measures of complexity of decomposable structures. The literature on complexity is reviewed and then the distinction among different measures of complexity is discussed. At first, a broad definition of complex systems is provided and the main characteristics of such systems are described briefly. In particular, a complex system is defined as a network of many highly interactive and interrelated elements, each performing its own functions. The elements are combined in such a way that each contributes to the behaviour of the structure. Moreover, the system components are characterised by interdependencies that make the contribution of each element to the overall performance (or behaviour) dependent on the contribution of the others. In general, systems are characterised by different degrees of complexity and also by different degree of decomposability. We also discuss the possibility of decomposing such complex systems, recalling the distinction that Simon (1969) made between decomposable, nondecomposable and nearly decomposable structures. Then, two particular kinds of measures are considered. The first group is based on information theory and deals with the concept of entropy. This index refers to the amount of information that we need in order to understand what is happening within a system. We claim that the entropy measure focuses only on the information complexity of a system without taking account of the interdependencies existing among the components. The second group of indicators, instead, explicitly considers this feature. Indeed, the group is composed of indexes which we call fitness measures based on the number of relevant relationships within a system. The basic features of Kauffman’s NK model are briefly reviewed. Finally, we discuss the usefulness of both entropy and Kauffman’s K value as appropriate measures of the complexity of decomposable structures. 1 We thank Michele Berardi, Roberto Gabriele, André Lorentz and Sandro Sapio for useful comments and suggestions on earlier drafts. All remaining errors are the author’s responsibility.