**5. False assumptions of Kauffman's model – summary**

The Kauffman's widely known hypothesis "life on the edge of chaos and order" [5, 6], pointed out an important factor in modeling of biological evolution, processes in social organizations, and technical constructions, however, it was based on too simple model, even – on few false assumptions:

1.Any network of conditions can be described as Boolean, then it is sufficient to study the Random Boolean Networks (RBNs). Such complex networks are finite, discrete, deterministic, and fully random.

The assumption that the statistical properties of Boolean networks are general is false [21]. The number *s* of equally probable signal variants should be also considered higher than only two.

2.RBNs can be either ordered or chaotic, which is observed and confirmed by the current mathematical theory of chaos

The current mathematical theory defines chaos by Lyapunov coefficients in **infinite, continuous space**. High sensitivity to initial conditions, leading to maximally different effects for very similar initial conditions is the main characteristic of the chaotic behavior of dynamical systems. Kauffman [6] uses such the term 'chaos' to describe **finite, discrete networks**. **The term 'chaos' is not reserved for just one of those separate areas**. This theory is used for finite discrete networks (e.g. [19]), but such a method **is an approximation**, **which loses a few important phenomena**, e.g., repeating the same argument for a function, the path length to the attractor and attractor length (in steps of a process). Analog of Lyapunov exponent for networks (coefficient of damage propagation [21], eq. 4.8 in [23] or eq. 6.2 in [7] in the case of half-chaos turns out to be misleading.

**Model-forced strong limitations on parameters** are not compatible with estimations from nature [7, 12, 16, 19]. For the evolution of life, the model allows only extreme *K* = 2 (connectivity, *K*—number of node inputs) and *s* = 2. Higher values of *K* or *s* lead to useless chaos.

3.Random networks contain all possible networks, then it is not important that **living organisms are not random** in the aspect of stability due to natural selection. Many works have assumed that this stability is explained by natural properties of the ordered system known as "order for free" [10]. These are false assumptions. Such a picture was not very consistent with the observed delicacy of living entities, not emphasized of regulatory structures and did not contain a model of death necessary for the Darwinian elimination. Kauffman [6] considered negative feedbacks, but practically [21] he left them on a random level.

**In this work, it is experimentally shown that among discrete and finite systems that are not fully random, with parameters** *s* **and** *K* **which for fully random system result in chaos, there is a third state of systems** I call **half-chaos**. The not fully random networks where half-chaos is found are obtained considering the specific correlation of parameters which Kauffman simplifying took as random. The analogy to the phase transition is more complex here - it is rather the "superheating".

The particular half-chaotic system exhibits small and large damage. Current theory does not foresee such the possibility, but it is easy to show examples using computer simulation – system with point attractor is half-chaotic. The modeled objects (like living or administrative units, technological processes, and technical constructions) are certainly neither infinite nor continuous. Half-chaotic systems better describe the modeled objects, **freeing modeling from difficult theoretical limitations** (see point 2 above), which until now are the typical basis of many considerations [7, 10, 12, 13, 16, 17, 23]. This opens the door to adequate models with complex networks.

The large gap between small and large damage defines in a natural way a small change, which is very important for interpretation. The peak of great changes (of functioning—damage) well model a death and elimination. After the great change, the system becomes forever simply chaotic, but a small change retains half-chaos and identity of the system, then evolution can go on. This feature as 'evolutionary stability of half-chaos' was included in the half-chaos definition. Half-chaos together with given initializing changeability completed by the multiplication of evolving system resulting from the demand of long evolution offers the full basic Darwinian mechanism.

The Kauffman model is trying to describe living systems and similar ones using several easy to show, and it would appear that the main parameters, the rest of them simplifies assuming their randomness, but natural selection works on all possible parameters, which may be easier and more important for selection and its effect. Indeed, it is difficult to imagine the possibility of the existence of half-chaotic systems from Kauffman point of view. In fact, after the system is drawn, it is either chaotic or ordered (ad. 'observed' in point 2 above) and the set of random systems contains all the possible ones (point 3 above). In the interpretation of the results of this approach, it has not been seen that the statistical absence of intermediate systems does not imply a small number of such systems. There are a lot of halfchaotic systems, but their share is negligible because there are radically more chaotic systems with given parameters (e.g., *K*) - for larger *N* not imaginable many. Model GRN based on RBN is not false, but its assumptions are too simple. Each

*Life Is Not on the Edge of Chaos but in a Half-Chaos of Not Fully Random Systems. Definition… DOI: http://dx.doi.org/10.5772/intechopen.93864*

model is a simplification, for some applications it can be useful, but if it gives a false expectation of important parameter, then some simplifications must be rejected which is the next step of approximation. It cannot be found without a previous step.

Regulatory feedbacks (misinterpreted and practically included only on the random level in the Kauffman model [21]) also the classic modularity and narrowing of the function significantly increase the stability, which was noticed, but the main and the new condition is the short attractor. They take over the role of explaining the experience [14, 15, 17] from "order for free", which in the half-chaos lost importance. The reached a deeper interpretation of Kauffman hypothesis gives a picture much more consistent with the observation and indicates systems more adequate to the modeling of biological evolution. This significantly alters the existing basis of many considerations and probably their conclusions. Likewise, the description of the systems from 'liquid' region [5], where Kauffman saw living objects - "small lakes of activity in the ice" (originally [5]: "unfrozen islands") remains valid for the primary and the most appropriate form of the half-chaos for the evolution - in-ice-modularity discovered in these studies. The base of the in-icemodularity is an activity of nodes (they change their states) in the ice (where nodes do not change their states), however, in-ice-modularity is supported by classic modularity, which is always present.
