**2.1 Variables** *K, k, t, N, A* **and** *d* **in Kauffman networks**

The considerations concern the statistical stability of the deterministic discrete Kauffman networks [5, 6, 22] (a little bit extended). The network consists of *N* nodes. A node in such a network receives signals at the *K* inputs, converts them uniquely using its function to the output signal called the state of the node, and then sends it to other nodes by *k* output links. States of all *N* nodes together creates a state of the network. The calculation of function takes a time step. Up to now, 2 (logical) signal states (variants) have been used. In the simplest case, it was assumed the same probability of signal variants and full randomness of connections, functions, and initial states of each node, such networks were called **RBN** (Random Boolean Networks). Here, deviation from this full randomness is made2 by assuming **short attractor** (a small number of time-steps until meeting the same network state), especially – **point attractor** (next network state is the same). In other here described investigations (met7, ch.3.5) – by controlled construction of in-ice-modular network (ch.3.3) or (met1-4b, ch.4) – by an increase of the fraction of negative feedbacks or classic modularity. *K* (called "connectivity", see [19]) was the basic variable for Kauffman.

Synchronous computing is used, i.e., the states of nodes from the discrete time *t* are input signals and arguments of the function of other nodes, and the results of these functions are nodes states at the next moment (*t* + 1). Variable *t* – is the number of time steps from a disturbance initiation. As the disturbance a permanent change in the value of the function of the node for its input state is used at the time *t* = 0; in method '8' (**met8**) it was an addition or removing a node. Parameter *tmx* the maximum number of calculated time steps is chosen arbitrarily, but it is checked whether its increase does not change the results (**Figures 2**, **3** and **5**).

Considerations have been limited to autonomous systems – they do not take signals from the environment. Determining the states and functions of all nodes and the connections between nodes uniquely determines the trajectory - consecutive states of the whole network (sets of states of all nodes). We simulated the process of transformation of the disturbed system on the section *tmx*, then we compared the resulting state of the system with the undisturbed system. It is also looked after the node functions are correctly random, but this assumption cannot always be fully met, so the impact of the derogations is checked.

The size of a change in a network function at time *t* after a small disturbance is measured by the number *A* (from Avalanche [23]) of the nodes, which have a different state in the pattern network – identical network, but without disturbance. The value *d = A/N* is called damage. The distribution of damage size at the time *tmx* as *P(d)* or *P(A)* is an especially important result (**Figure 6**).

For random networks, this result creates two system states – ordered and chaotic. In system parameters space they occupy areas which Kauffman calls 'solid' and 'gas' respectively. Between them, there is a fairly quick transition (near *K* = 2, if Boolean signals are equally probable) treated as a phase transition. Only in systems in the vicinity of this transition (Kauffman calls it 'liquid' - the area between 'solid' and 'gas') changes in the system function (damage) often enough are small, therefore suitable for biological evolution. This is the main basis for the Kauffman's hypothesis: life on the edge of chaos. However, this conclusion aroused doubts [21], therefore, it has been subjected to a deeper analysis presented here.

<sup>2</sup> It is made using few method, in short: 'met'. Each of them is called using digit on the end and, if need be, some letter for its variant. In this case they are: met4c, met4d, met5, met6, met8, described in ch.3.1–4.

#### **Figure 2.**

*Half-chaos and chaos in the presentation of* A(t) *for a full set of initiations on the example of met7b* J *and* X *for network* ss*. This is a presentation observed dynamically during a simulation on the screen pixels. The details should be watched in enough magnification. In met7b* N *= 800,* tmx *= 2000 was used. A rectangle has the dimension of 400\*1000 pixels, so on each axis, one pixel shows 2 values. In Figure 3, for which this figure is a description of form,* N *= 400 and* tmx *= 1000 is used, so the there unit on the axes corresponds to a pixel. The vertical axis is originally scaled in the* A *- number of the nodes states different than in the pattern. The horizontal axis is the number of steps* t *of simulation of network functioning. After each initiation by small permanent change, the state* A(t) *was drawn with a continuous line on the screen after every step of the calculation. In case of initiation of a node in the in-ice-module black color is used and for initiation in the walls between in-ice-modules - purple. In met5 shown in Figure 3, this distinction was not known and always black was used. To optimize the simulation a counting after 70 steps from the explosion to chaos (crossing over the threshold, here = 300, marked in red on the left) was stopped - there the process has no chance to return. As can be seen, the transition to chaos in the vicinity Derrida balance is not slow, but rapid in several to over a dozen steps, where A increases drastically, so - "explosion." after deflection from a small value to say* A *= 80 no longer the returns happened (as checked without optimization, see [1]). After the end of initiation set, the red curve*

## *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*

q(t) *was added to the figure. In met7 it is originally scaled by the A as the number of initiation, which does not exceed the threshold = 300, but there are 3* N *= 2400 of initiations. In met5 in a Figure 3* q(t) *is divided by the number = 3 of initiation in node, so that* q *= 1 for* A=N*. The red description in the left has been added for readability and here* q(t) *is the share of processes that in the time* t *did not pass the threshold. a – Half-chaos, experiment* J *for network* ss, *model b. There were 600 of such simulations for each type of networks* sf, ss, er *and models a and b of met7. The red curve* q(t) *quickly stabilizes at a high level* q *= 0.22. In the lower part of the graph, many trajectories are visible (there are L = 532 of 2400) that a little over* t *= 200 no longer explode. So R = 1868 processes from the very beginning went to chaos - a Derrida balance. b - Chaos on the example of experiment* X *performed immediately after the measurement of the* J *illustrated above in the a. There were 300 of such simulations for each type of networks* sf, ss, er *and models a and b of met7 and for each experiment of* X, S, T, F*. Here,* q(t) *is steadily decreased until all the processes are not 'exploded'. At the end there is exact LX = 0 of them, means* q *= 0. Blue points describe the number of processes that currently have* A *= 0, i.e., damage fade out, but for the* X *the secondary initiations lead to their explosion.*

The conflict [7, 8] of a size of *K* in the Kauffman model and *K* estimated from nature [19] is a problem solved here. Kauffman postulates that the natural property of the random ordered systems (order for free [10]) is the source of stability, but then *K* should be extremely small (*K* ≤ 2) [18]. The attempts to prove that the real genetic network (using model **GRN** – Gene Regulatory Network) is ordered [14, 15, 17, 23] assume such a source of stability. Different circumstances allowing system with greater *K* to be in the ordered phase were indicated (p.48 in [7]), such as a significant difference in probabilities of logical states [18], or deviation from the randomness of the function (canalizing [11]), but these and other suggestions are not satisfactory for many reasons [21]. The model GRN has disappointed many expectations, mainly due to restrictions arising from the range of 'liquid region', it was replaced by the more attractive Banzchaf model [24], but GRN is still being studied [25].

For investigation shown here, as typically, the same *K* for all *N* nodes of the network are taken.
