**4. Supports for stability**

#### **4.1 More of negative feedbacks in a random system, function narrowing**

It is generally believed that the stability of the various systems results from homeostasis based on regulation by negative feedbacks. Kauffman pointed instead to the property of the ordered phase (order for free) [10] as the most important reason, but for it extremely small *K* should be expected. **The regulatory feedbacks** are generally considered the basis for the stability of living entities and their **concentration is considered to be significantly increased** in relation to the random one. **However, the complex structure of the feedbacks for this statistical surplus has been replaced in the Kauffman model by their proper effect (ice)** and it remains only in random share. So much simplified model is not able to give a proper statistical picture of a system failure and conclusions for a stability mechanism can (and seem) significantly differ from reality.

This doubt was the main reason for undertaking the research, which initially aimed to strong raise the share of regulatory mechanisms.

In the presented study **we transform part of the feedbacks in random structure into negative feedback**. It is done by changing the random function when the state on the inputs was not used yet. It was the first method (**met1**) of correction of a random chaotic system. The similar, stronger **met2** has iterative change the pattern. Network *s,K* = 2,4 and 4,3 were investigated. **Figure 1c** suggests that Derrida chaotic balance is achieved even before the 15-th time step. **Initial research for** *tmx* **= 60 steps yielded very promising results** (**Figure 5a**) - *q* was significantly increased (especially for *s,K* = 4,3), the distribution of damage size already

contained two peaks separated by a gap. A large part of this effect (especially for *s, K* = 2,4) was the result of deviation from the randomness of node functions (function narrowing), which also may be included [11] to evolution tools. But it turned out (**Figure 5**) that obtained in met2 stability of *q* **usually significantly decreases with the elongation of** *tmx*, practically disappears already for *tmx* = 1000, only in the case of Boolean networks *sf* 2,4 this method could be considered to be effective to achieve half-chaos (not tested for evolutionary stability). As it can be seen in **Figure 5b,c**, *tmx* = 20,000 was used. Simulation series contains 700 nets for *s* = 2 and 350 nets for *s* = 4.

These studies demonstrated a **high range of results dependence on the network type** - the network *sf* is more ordered [9]; network *ss* and *er* are more chaotic, similar to the reaction, but *er* has part *k* =0(**Figures 4**, **5** and **8**) obstructing observation. **The parameters** *s,K* **= 2,4 and 4,3 also give a very different picture**. The simulation allowed for a deeper look at the process and its determinants, which **pointed to the short attractor** (ch.3.1).

#### **4.2 Modularity**

**It seemed that the most natural way to get short attractors is modularity. In met1 and 2 no modular effects were observed**, although modules exist in practically every network. It was assumed that in a random network the modules are too 'weak', then it was pre-checked, what stronger modularity gives for stability (**met3**). Here it turned out that sufficiently small spontaneous attractors can be expected only in so small modules that the consideration a state of chaos in them losing meaning. Consideration of chaos in the modules network has been postponed.

In the study, the network has *N* **= 400 nodes. It was assembled of** *N2* **= 50 modules, each of** *N1* **= 8 nodes**. Connectivity *K1* between nodes inside modules *K1 = K2* connectivity between modules. The rule of connection is taken like in type *er*. Simulation series consists of 100 nets.

**The modularity also gave raise** *q* (**Figure 5c**), **especially when met2**, which increased the share of negative feedback, **is used at the same time**, however, evolutionary stability was not checked. In the distribution of damage size, the typical for the half-chaos radical **gap between peaks was not observed, only the clear minimum**. An increase of *q* in the experiment met3 + met2 with *s,K* = 2,4, almost entirely resulted from non-randomness of functions (**function narrowing**). Both of these methods and their associated factors (such as function narrowing) belong to the most important methods of producing desired stability by biological evolution, but in both the short attractor is an important factor.

As was described in ch.3.3, classic modules cooperate with in-ice-modules. The theme of classic modularity and its role in system stability was here recognized only provisionally and requires much deeper research. However, it is one more source of modularity, than was found in [30], where the role of modularity is studied in depth in evolution.

#### **4.3 Regulation in system with point attractor**

Lack of expected radical effect of regulatory mechanisms in the met2 was found in the system starting from a random network, then we introduced strong regulation in a system with a radically short attractor – point attractor (**met4a**). This time the result was surprisingly strong (**Figure 4**), so we decreased the regulation to the minimum (**met4b**, see also **met5b**, **Figure 4d**) and next, regulation was rejected at all (**met4c,d** and later), which showed that the point attractor is sufficient to achieve half-chaos.

*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*

In met4 point attractor starts with all node states equal 0. It was not permitted in met8, where states are random. Model **met4c** for *s,K* = 4,3 used later in met5 is defined as f(0,0,0) = 0. **Model d** for *s,K* = 2,4 – as f(0,0,0,0) = 0. They are based only point attractor, without regulation. **Model b** with minimal regulation (*s,K* = 4,3) also used later in met5, has in addition to c also: f(0,0,1) = f(0,1,0) = f(1,0,0) = 0. For signal value 1 interpretation was taken: deviation from proper state '0', but still in the range of homeostasis. For **model a** (*s,K* = 4,3) also f(0,0,0) = 0, but description is much more complicated. Here direction of deviation in homeostasis range: 1 – positive; 3 – negative. The deviation of one of 3 input signals gives 0. The function also gives 0 if 2 signals are deviated, but in the opposite direction and third is 0. If 2 signals deviate in the same direction but third is 0 or 3 signals deviate, but they are not equal, then function result is deviated, i.e., is 1 or 3. If 3 signals deviate and are equal or at least one is 2, then the result of the base function is 2, but such value for a particular node is converted into random value in the way that share of each function value be equal. Other parameters of simulation in met4 are described in ch.3.3.

**The result of met4a shows how strong may be the effect of the regulation in the half-chaotic system** – right peak almost disappears, that is the probability of entry into chaos as a result of a small system failure (internal cause) is small. **This gives a deceptive picture of the ordered phase** [14, 23]. There remain external causes, which model of the autonomous network does not take into account from assumption. However, adaptation is to the environment, which can vary and the evolution should be tested using open systems as in [31].
