What is complexity?

The Latin word complexux means entwined and twisted together. So complexity is always jointed with several components. There is a basic duality between parts which are at the same time distinct and connected. More parts and connections lead to more complexity.

Complexity

The aspects of distinction and connection, determines two dimensions characterizing complexity. Distinction corresponds to variety and the fact that different parts behave differently. Connection corresponds to constraints and to the fact that different parts are not independent and the knowledge about one part, allows determination of features of the other parts. Distinction leads in the limit to disorder, chaos or entropy (Like gas which the position of every part is completely independent of the position of others). Connection leads in the limit to order and negentropy. (Like in a perfect crystal where the positions of the molecules are completed determined by positions of the neighboring molecules).

Complexity can only exist if both aspects are present, neither perfect disorder nor perfect order. Because disorder can be described statistically through the rule of large numbers and order is described by traditional deterministic methods. The complexity appears when there is something between order and disorder, or using a fashionable expression on the edge of chaos.

The simplest way to model order is through the concept of symmetry, which means “invariance of a pattern under a group of transformations”. In symmetric patterns, one part of the pattern (seed) is sufficient to reconstruct the whole. It is interesting to note that maximal disorder too is characterized by symmetry, not of the actual positions of the components, but of the probabilities of being in a certain position.

So complexity can be characterized by lack of symmetry or symmetry breaking which leads to this fact that no part or aspect of a complex entity can provide sufficient information to actually or statistically predict the properties of the other parts. This is the reason for difficulties of modeling complex systems. It is said that definition of complexity depends on the level of representation or scale. What seems complex in one representation may seem ordered or disordered in different scales. Scale is just another dimension, characterizing space or time ( Havel, 1995). Havel calls a system scale-thin if its distinguishable structure extends only over one or a few scales. A fractal or self-similar shapes has infinite scale expansion. We may conclude that complexity increases when the variety (distinction) and dependency (connection) of parts or aspects increase in several dimensions. These dimensions include

• Geometrical structure
• Dimension of time or dynamics
• Dimension of temporal or dynamical scale

We can call increase of variety, differentiation and strength of connection, integration. The complexity produced by differentiation and integration in the geometric dimension may be called structured, in the temporal dimension functional in the spatial dimension structural hierarchical and in the temporal scale dimension functional hierarchical.