Features of Complex Systems

This chapter uses the representative examples of complex systems discussed in the previous chapter to arrive at a list of the distinctive features of complex systems. These features include numerosity; disorder and diversity; feedback; and non-equilibrium. The interesting thing about complex systems is that these conditions can give rise to the following products: spontaneous order and self-organisation; nonlinearity; robustness; nested structure and modularity; history and memory; and adaptive behaviour. Not all these features are present in all complex systems. Whenever any of the products are found in a system, they are the collective result of the conditions, but not all the products are found in all complex systems. Often products help produce other products — for example, memory is impossible without a degree of robustness, and adaptive behaviour can build nested structure and modularity. The chapter considers each of them in turn in more detail and assesses whether each is necessary and/or sufficient for complexity on any or some conceptions of what complex systems are.

Measuring Features of Complex Systems

This chapter offers a guide to quantifying complexity based on the fruits of the analysis of the previous chapters. Many measures of complexity have been proposed since scientists first began to study complex systems, and the list is still growing. If complexity is a collection of features rather than a single phenomenon, then all quantitative measures of complexity can quantify only aspects of complexity rather than complexity as such. The chapter demonstrates the truism of complexity science that it is computational and probabilistic. It also further explains some of the new kinds of invariance and forms of universal behaviour that emerge when complex systems are modelled as networks and information-processing systems. The chapter then looks at a few, by now classic, measures of complexity from the 1980s and 1990s, including effective complexity, effective measure complexity, statistical complexity, and logical depth.

Introduction

This introductory chapter provides an overview and a brief history of complexity science, which is the study of complex systems. All living systems and all intelligent systems are complex systems. Complexity science is relatively new but already indispensable. Many of the most important problems in engineering, medicine, and public policy are now addressed with the ideas and methods of complexity science. However, there is no agreement about the definition of 'complexity' or 'complex system', nor even about whether a definition is possible or needed. The conceptual foundations of complexity science are disputed, and there are many and diverging views among scientists about what complexity and complex systems are. Even the status of complexity as a discipline can be questioned given that it potentially covers almost everything. The origins of complexity science lie in cybernetics and systems theory, both of which began in the 1950s. Complexity science is related to dynamical systems theory, which matured in the 1970s, and to the study of cellular automata, which were invented at the end of the 1940s. By then computer science had become established as a new scientific discipline.