**2.2 More than two signal variants** *s* ≥ **2**

According to my previous [21] suggestions, here I also study a larger number of *s* (>2, usually 4) of equally probable signal states, which in random networks for every sensible *K* (≥ 2) always gives chaos (**Figure 1**). In the range of sensible parameters *s* and *K,* the order appears only for *s* = 2 and *K* = 2, it is absolutely exceptional (**Figure 1b,d**). Attempts to introduce more signal states already exist [12, 16], but they assume the possibility of an ordered phase for the random network therefore these states cannot be equally probable.

I repeat here briefly my basic arguments given in ([21]; ch.2) for using *s* ≥ 2 in Kauffman networks for statistical investigations:


#### **Figure 3.**

*Simulations met5 (changes accumulation) in the presentation of* A(t)*. Except for red description* q *on the left, each drawing was created dynamically on the screen during the simulation of one full set of initiation without blocking of reverse initial changes. It is accurate to the pixel. Description of the presentation elements in Figure 2. a - Full typical image for the* M13 *met5c (met5 in other figures, model c from met4), network* sf*. Almost an immediate end of the explosions to the chaos can be seen. At the top - the state of chaos in the Derrida balance (short due to optimization by interrupting the counting after 70 steps, as in Figure 2). At the bottom a repeating pattern in accordance with the global attractor marked on the top frame (pattern network state as in* tmx *before the first initiation of the set). Here L and R under the lower frame is the sum from the beginning of the evolution simulation of this network. In this set 383 of initial changes were accumulated of 1200 tested, but accepted changes defining* q *(not exceeding the threshold = 150) were a little bit more (with global attractor < 7). b - Typical image of network* er *simulation in met5c. The upper part of the almost identical to a is cut. The level of* q(t) *is lower, the belt at the bottom - clearly thinner, the time of the latest explosion to chaos - shorter. c, d – The lower part of the image for met5b (with minimal regulation). Here the level of* q(t) *was much higher than in a. In the model b, the width of the lower belt is greater due to the possibility of regulation. Simulations slightly different model than in Figure 4d - here without blocking of reverse changes, but with the condition non-decreasing of global attractor and accumulation of changes not less than* A *= 3, the shift of beginning = 2, but not 50. In these simulations, a distribution of damage size for ordered cases (*A *< 150) was studied on the section from* t *= 600 to* tmx *for a given set of initiations (purple curve on the right frame) and the sum of the sets in the final set* M20 *(blue curve in c). It is one of several ways to look for proof of the in-ice-modules existence. As can be seen, in both (c,d) shown cases in these distributions the significant peaks are visible. They indicate an existence of one (in the c* M20*) or two (in d* M1*) hypothetical in-ice-modules. Under the scope of these peaks, there is a clear gap in the minimum of distribution. An interpretation of these peaks can vary, they are not proof of the in-ice-modules existence, which was shown later watching nodes states repeating, but they are a strong premise. The* q *level here is high: in c* q *= 0.46 and in d* q *= 0.55. In c the attractor was not found at the beginning of the set (attr* ≥*900), and because it could not decrease, no one accumulation happened (not.PAS saved = 0). It does not mean, however, that there is no here acceptable (*A *< 150) cases (there are 220), which indicates* q *and wide black belt below the* A *= 150.*

probability, resulting from the similar probability of each one, but description basing on *p* – the probability of signal variant leads to much different results (**Figure 1a**) than for the case, when we consider all variants. Adding the parameter *p* to the two-value description does not solve the problem here. Adoption of *s* ≥ 2 equally probable signal variants is an alternative method of model realignment. However, it seems to be often more adequate. Both methods give different results which significantly increases the importance of the correct choice of description.

3.Parameter *s* is more important in the description of damage spreading than *K* treated as the most important – see **Figure 1b**. Value *dmx* – mean maximal damage, i.e. Derrida equilibrium for chaotic behavior (**Figure 1c,d**), much stronger depends on *s* than on *K*.

#### **2.3 Criteria of chaos, coefficient of damage propagation** *w*

The main characteristic of the chaotic behavior of dynamic systems is high sensitivity to initial conditions, leading to maximally different effects for very similar initial conditions. It is original, theories using Lyapunov exponents or percolation are derivative. I use the term 'chaos' in such the meaning, similarly as Kauffman [6] does. For chaotic Kauffman networks a small initiation of damage

*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*

#### **Figure 4.**

*Increasing regulation or another factor - the point attractor. The primary result of the met4. In the met4 removing a presumed cause of the poor performance of the met2, we start with the non-random system with extremely short attractor – a point attractor: initially, all states are set to 0 and f(0) = 0 (f – node function). The models were tested in the sequence a, b, c (*s,K *= 4,3) and d (*s,K *= 2,4) starting from strong regulation and ending with the lack of regulation in the models c and d. Care was taken that each signal has the same probability in the function of each node. Model a contains a negative feedback with a positive (1) and negative (3) deflection from equilibrium (0) in each of the three input signals. It contains also the leaving of homeostasis into the area of randomness (when deflection is too great or one of the input signals = 2, then the node function is defined randomly). A more exact description of this formula can be found in the text of ch.4.3 and is available in [1]. Model b has a minimal regulation: the condition of the point attractor f(0,0,0) = 0 is supplemented only by condition f(0,0,1) = f(0,1,0) = f(1,0,0) = 0 that there is no in model c. Model d of Boolean network (*s, K *= 2,4) has only condition f (0,0,0,0) = 0 similar to model c. Each model is simulated for three combinations of* N,tmx *= 400,200; 400,2000; 4000,200 for networks* sf *and* er, *so as to always number of initiations was 48,000 in the series. The threshold of small change for* N *= 400 was set to 100, and for* N *= 4000 to 800. Each initiation by definition of met4 is made for node state = 0 and for input state = (0,0,0). So only in the model c 3 other function values may be used for initiation. For model a the only one value 2 remains, for b only two values: 2 and 3, which are new states of a node without the mandatory fade out of damage at the destination. a,c - The counts* #(A) *of processes ending in* tmx *with value* A *(changed states of nodes in* tmx*) are shown. Also, the scale of the* P(A) *or* P(d) *are added. The results showed here in the linear plot a (*N *= 400) for models c and d are also in Figure 6 in log scale. The series showed in c contains 10 times fewer networks, which gave peaks much narrower (in damage* d *scale instead of* A*) than in a. The right peak for models b, a is becoming smaller due to increased regulation, which is reflected in the diagram d as less participation of chaos. Place of the right peak in a and c are well designated by Derrida balance (Figure 1d) (different for* s *= 2 and* s *= 4), which is the property of a mature chaos. b - The table of results* #(A) *for* tmx *= 200 for the same networks as in a for which* tmx *= 2000. The counts differ only for af by 140 and for df by 2 (less for left peak). d - A complementary for Figure 8 juxtaposition of a fraction of ordered cases (*q*) and chaotic cases (*1-q*) for minor experiments discussed in the article. While Figure 8 lists only the study of impact of small attractor, it is here - the impact of increasing the share of regulation in met2 (only* sf *2,4 can be considered in met2 as entry into half-chaos, see Figure 5a,b); of modularity in met3; assembling of met3 and met2 (Figure 9); assembling of point attractor and regulations in met4ab and met5b. Among them only met5b examined the evolutionary stability included in the definition of half-chaos. As can be seen, the assembling is more effective than approach alone and should be expected of such a strategy in biological evolution. The case af shows that the way evolution can lead to a state where the half-chaotic system may seem as ordered. Evolution met5b decreased* q *comparing met4b when met5 (Figure 8) worked in the opposite direction relative to met4c (these are uncertain trends), but the expected strategies of biological evolution its creative aspect is important, not modeled in the presented simulations, too simplified to such a task.*

#### **Figure 5.**

*Ordered fraction (*q*) as a function of time (*t*) after raising the share of negative feedbacks (met2) and the classic modularity (met3). The upper row of all part -* s,K *= 2,4 (Boolean network), lower -* s,K *= 4,3. A – For some moments* t *the shares of mechanisms: Wild - without interference met2; function narrowing as a side effect of the method; the increased participation of negative feedback by met2. For network* er, *the level of q resulting from participation* k *= 0 (nodes without outputs) is indicated by the green line. In the right column as a wild the modular system resulting from met3 is used, further described in (c) as a curve a. the type of networks* sf, ss, er *is described by a second letter. As can be seen, the results for the simulation parameters* s,K *= 2,4 and 4,3, and network types, differ significantly. For* s,K *= 2,4 the function narrowing is of utmost importance to increase* q*, but for* s,K *= 4,3 the importance of feedback turns out to be essential. For small* t *the effect of increase* q *is significant. From these data it can be suspected to achieve half-chaos for:* sf *2,4 - the result of functions narrowing and increase of the share of regulatory feedback, and for the assembly of modularity met3 with met2 using nets* er *- for 2,4 mainly due to the functions narrowing, but for 4,3 due to the met2. In the remain 5 presented cases the effect practically disappears already for* tmx *= 1000, the use of it by living entities require very rapid multiplication in comparison to the transformation of the construction and metabolism, which seems unattainable. Here evolutionary stability (included in the definition of half-chaos in the result of further studies restricting fundamental factors to a short attractor Figures 6–8) was not examined. The degree of entry into the plateau can be better assessed in b and c. the network ss gives a similar effect to the network* er*, but without the confounding effect of* k *= 0. b – Net* sf *4,3 (350 nets) not reached a plateau even at* t *= 20,000, where* q *is negligible, but* sf *2,4 (700 nets) is almost on plateau* q *at* t *= 5000, and this level is high (compare Figure 4d). c – The result of modularity (met3) and assembling it with met2. Result of met2 for network* er *is added, such as in b, omitting, however, the share of function narrowing enough presented in a. it can be seen that the wild system (without forced modularity, 700 nets for* s *= 2, 350 nets for* s *= 4) of network* er *very quickly descends to the level of* q *resulting only from* k *= 0. Also curve b - the result of the met2 quickly closer to that level, which can also be seen in a. forced modularity (curve a, 100 nets) gives a clear stable increase of* q*, and met2 help it (curve ab) to radically increase* q*, but for* s,K *= 4,3 appears to fall within the plateau above* t *= 20,000. For* s,K *= 2,4, almost all large and stable met2 effect results from the function narrowing only (curve afb). The network has* N *= 400 nodes assembled of* N2 *= 50 modules each of* N1 *= 8 nodes.*

typically causes a large avalanche of damage which spreads onto a big part (percolates) of the discrete and finite system and ends at a Derrida equilibrium level *dmx* [18, 21, 26] (**Figure 1c,d**), which is a maximal loss of information about the previous system. Such distance cannot be infinite in the finite network witch finite discrete *s*. The existence of this limitation is the main difference between this 'chaos' and the more commonly taken definition [27] used for continuous variables on infinite space, where Lyapunov description works. **The term 'chaos' is not reserved for one of those separate areas. The distribution of damage size is the experimental base to classify a particular system of Kauffman network as chaotic or ordered using levels of damage equilibrium calculated from Derrida's annealed approximation** (**Figure 1d**). In Derrida's model only case *s, K* = 2,2 (I use *s,K* as a vector) is ordered – it has no other cross of diagonal than in *dt+1 = dt* = 0. In any other cases, such cross called here *dmx* exists; it is Derrida equilibrium level of chaotic reaction for the disturbance.

*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 a typical case, the chaos is indicated by Lyapunov exponent, which describe the growth of distance for two, near, initial states. For finite discrete networks, it corresponds to "coefficient of damage propagation" *w* described in ([21] ch.2.2.1) and earlier, or eq. 4.8 in [23]. *w* = <*k* > (*s*-1)/*s*. It can be treated as damage multiplication coefficient on one node if only one input signal is changed. It indicates how many output signals of a node will be changed on average. For an autonomous network with fixed *K*, <*k* > = *K* and we can use *w=K*(*s*-1)/*s*. It is easy to see that for *w* > 1 damage grows, for *w* < 1 it disappears and *w* = 1 is critical – for *s* = 2 it gives known critical *Kc* = 2. In [7] similar eq. (6.2): *Kc*(*s*-1)/*s* = 1 is given which is a case for the condition *w* = 1. Coefficient *w* is a simplification for the beginning of damage growth, later a case of more than one changed input signal happens more and more often, but this first period is crucial (**Figure 1c**).

Note, we are going to know: is a particular network chaotic, ordered, or something else, therefore we test it by statistical experiment. We make small disturbance (perturbation) and look how great is a change of a function (damage *d*) of this deterministic network comparing to undisturbed network. Damage is an effect of this small disturbance. We make a lot of such small disturbances (see **Figure 2**), each in the same network being tested, and we get distribution *P(d)* for one, tested network. For a chaotic network, the *P(d)* contains one peak near *dmx*, for ordered – one narrow peak near *d* = 0. If there are both the peaks in the distribution for one particular network, then it is neither chaotic nor ordered network, it may be half-chaotic.

#### **2.4 Types of networks**

Several **types** of networks are considered. They differ in the rules of their creation (for *sf* and *ss* see **Figure 2** in [21]) and distributions of *k* (output links), (*K* – input links is fixed for all nodes of particular network): *sf* (scale-free [28]), *er* (classic Erdős-Rényi [29] "random"), and *ss* (single-scale). In the figures, the second letter of these shortcuts indicates the network type. In studies **met8** (denoted in figures by '8') the network grew - an addition or removal of the node was the disturbution. There networks *sh* and *si* are respectively *sf* and *ss* with 30% removal.

Parameters: network **type** together with *s,K* (treated as a vector) are the main variables in the simulations. In a wider description [1, 2] of here presented investigation, I used more network types.

#### **2.5 The main results**

At the beginning, in ch.2.1 there is the statement: 'the distribution of damage size at the time *tmx* as *P(d)* or *P(A)* is an especially important result'. It is shown in **Figure 6** for the main range of investigation and in **Figure 4** for mechanisms supporting half-chaos. However, it is the base for more important conclusions.

In obtained here distribution of damage size for the particular system there are two peaks: the left of small changes (ordered behavior) and the right of big changes (chaotic, near Derrida balance). Sharp boundaries of these peaks, supported by a clear gap between them define a "**small change**".

The main result, however, is a "**degree of order**"*q* – a fraction of effects (damage) of small perturbations which fit into the range of the "**small change**" of the functioning at the time *tmx*. It is summarized in **Figures 8** and **4d**. This *q* corresponds to the contents of the left peak or probability of acceptance of changes in the modeled evolution (lack of elimination).

The **degree of order** *q* is the base (see ch.2.3) to state, that we found half-chaos using definition given in the Introduction: **Half-chaos is a state of a system that is not fully random, with parameters that the random system make strongly**

#### **Figure 6.**

*The main result – distribution of damage size. Symbol of the method begins a signature. The methods: 'd', '5', '8' start from point attractor; 'd' (met4d, see ch.4.3) is the only with* s,K *= 2,4 and without evolution, remain* s, K *= 4,3 with evolution; '6' (met6) starts from small attractors; '7' (met7ea) starts from constructed in-icemodular system. After the method the second letter of network type ends signature. Results presented here (except 'd') are a sum from 4 already stabilized sets of initiation (see Figure 7). The gap between the peaks left (ordered) and right (chaotic, near Derrida balance, different for* s *= 2 and 4) is not empty only for not really small disturbations by adding or removing a node ('8' – met8). The share of the left peak as* q *– degree of order is summarized in Figure 8. It is the basic result of this study; it allows to introduce half-chaos. Collecting only permanent changes which give damage from the left peak (i.e. small changes) is sufficient to keep half-chaos in the evolution (Figure 7). The shape of the left peak is important for the modeling an evolution of adapted systems. It is shown (without '8') in more details on the left for variable* A = d\*N *where* N *is = 400. In the experiment '6' there is practically only* A *= 0 due to lack of in-ice-modularity. Network* sf *of '7' differs from the others in the left slope of the right peak, (see also Figure 7c) mechanism of this is unknown.*

**chaotic, but small disturbances give the ordered reaction with a similar probability to the chaotic reaction.** Such state is contrary to the current view, but the current view is based on the assumption of full randomness of the network which typically is not fitted.

The "**small change**" is a criterion of the acceptance of perturbing permanent changes creating the evolution, which is enough (**Figure 7**) to stay in half-chaos. It is the **evolutionary stability of half-chaos**. It **was included in the half-chaos definition**. **Acceptance of one** perturbing permanent change that gives a **big change** of the functioning at the time *tmx* (chaotic reaction) **leads to practically irreversible entry into normal chaos** (elimination)**.** Note that in such great change of behavior only states of network nodes differ before and after, but in both cases they have the same, random–look distribution. Nothing has changed for currently used methods to define: is this network chaotic or ordered, but the behavior is absolutely different.
