Generalizations of the Concept of Complexity

Abstract

In this chapter, the new representation of complexity of an actual entity is proposed. The complex subject appears as a point within the “complexity space” in which the “coordinates” stand for the three components of complexity: synchronic, diachronic, and combinatorial.

The systems theory, as it has been formulated by von Bertalanffy, talks about the holistic integration of components. In this form, the system theory overlaps with the theory of complexity. Here I wish to argue that, by looking on the methods used in their epistemology, these two theories are on the first sight equivalent. However, the rigorous analysis which is represented in this book has provided the deeper insight. While complexity could be regarded as a property, the system is more or less complex subject. Systems are composed from particular elemental units, which are interrelated in such a way that they can perform specific function in the surroundings. These elemental units of the system are complex, and their complexity is defined by the contribution of the three components of complexity: synchronic, diachronic, and combinatorial (Fig. 5.1). From the discussion in this work, it is clear that the nature of synchronic (or, structural) complexity is essentially different from the nature of diachronic and combinatorial complexities. While the synchronic complexity has a level structure that originates from the different binding energies for the formation of entities of the higher level, the change in the diachronic and combinatorial complexities appears on the scale between the homogenous and chaotic limits. Actual entities, i.e., the basic building blocks for the formation of systems, are situated in such a “complexity space”. It is important to point out that such a complexity space is not based on any universal metrics. The complexity metrics is “local”, and it depends only on the nature of complex objects. Accordingly, the “coordinates” in the Fig. 5.1 are not quantitative, and they have no units.

Fig. 5.1

Components of complexity, the “complexity space”

To illustrate the proposed complexity space, let us analyse the structure shown in the Fig. 5.2. Within the frame of the synchronic complexity, this structure belongs to the molecular level of complexity. On the scale of the diachronic complexity, this molecule is in the regular, repetitive vibrational state. Since this is the stable molecule, all of its vibrational modes (the changes of bond-lengths and the changes bond angles) are repetitive, and its diachronic complexity is characterized with the regular oscillations typical for the thermodynamic equilibrium. Within the frame of combinatorial complexity, the structure is the typical example of the fractal.

Fig. 5.2

The structure of the dendrimer molecule of poly(glycerol-succinic acid). (Reproduced by permission from Ref. [1])

Let me now move from complexity to system. The system, as it has been defined by Bertalanffy, is composed from the elemental units of different inherent complexity. In the first sections of this work, I have called them actualities or actual entities. This term I have borrowed from the A. N. Whitehead’s metaphysics in which the Universe is principally composed from the temporal units, rather than from the spatial entities [2]. Every system is formed from actualities whose complexity is characterized with synchronic, diachronic, and combinatorial contributions (Fig. 5.3). For example, different systems could consist from a variety of actualities with different positions within the “complexity space” (Fig. 5.4).

Fig. 5.3

Actualities (actual entities) as elements in constructing systems (Reproduced by permission from Ref. [3])

Fig. 5.4

Construction of various systems from the actualities in the “complexity space” (Reproduced by permission from Ref. [3])

However, to form the systems, the actualities should be interconnected in the particular combinations. The interconnectivity between the actualities is a consequence of the pressure of the environment which is manifested in the requirement for accomplishing particular function. These interconnectivities between the actualities which constitute the system can be visualized by using convenient representation. In the next chapter, the systems will be represented as topological structures which can be mathematically formalized by using already discussed methods of the graph theory.