The research concerns the dynamics of complex autonomous Kauffman networks. The article defines and shows using simulation experiments half-chaotic networks, which exhibit features much more similar to typically modeled systems like a living, technological or social than fully random Kauffman networks. This represents a large change in the widely held view taken of the dynamics of complex systems. Current theory predicts that random autonomous systems can be either ordered or chaotic with fast phase transition between them. The theory uses shift of finite, discrete networks to infinite and continuous space. This move loses important features like e.g. attractor length, making description too simplified. Modeled adapted systems are not fully random, they are usually stable, but the estimated parameters are usually "chaotic", they place the fully random networks in the chaotic regime, far from the narrow phase transition. I show that among the not fully random systems with "chaotic parameters", a large third state called halfchaos exists. Half-chaotic system simultaneously exhibits small (ordered) and large (chaotic) reactions for small disturbances in similar share. The discovery of halfchaos frees modeling of adapted systems from sharp restrictions; it allows to use "chaotic parameters" and get a nearly stable system more similar to modeled one. It gives a base for identity criterion of an evolving object, simplifies the definition of basic Darwinian mechanism and changes "life on the edge of chaos" to "life evolves in the half-chaos of not fully random systems".
There are three main aims of this paper. 1-I explain reasons why I await life to lie significantly deeper in chaos than Kauffman approach does, however still in boundary area near 'the edge of chaos and order'. The role of negative feedbacks in stability of living objects is main of those reasons. In Kauffman's approach regulation using negative feedbacks is not considered sufficiently, e.g. in gene regulatory model based on Boolean networks, which indicates therefore not proper source of stability. Large damage avalanche is available only in chaotic phase. It models death in all living objects necessary for Darwinian elimination. It is the first step of my approach leading to structural tendencies which are effects of adaptive evolution of dynamic complex (maturely chaotic) networks. 2-Introduction of s ≥ 2 equally probable variants of signal (state of node in Kauffman network) as interpretively based new statistical mechanism (RSN) instead of the bias p -probability of one of signal variants used in RBN family and RNS. It is also different than RWN model. For this mechanism which can be treated as very frequent, ordered phase occurs only in exceptional cases but for this approach the chaotic phase is investigated. Annealed approximation expectations and simulations of damage spreading for different network types (similar to CRBN, FSRBN and EFRBN but with s ≥ 2) are described. Degree of order in chaotic phase in dependency of network parameters and type is discussed. By using such order life evolve. 3-A simplified algorithm called 'reversed-annealed' for statistical simulation of damage spreading is described. It is used for simulations presented in this and next papers describing my approach.
It is commonly accepted by those who consider macroevolution as a process decoupled from microevolution that its apparent jerkiness (and, hence, incompatibility with principles of population genetics) results from the structural complexity of epigenetic systems, since all complex cybernetic systems are expected to behave discontinuously. To analyse the validity of this assumption, the process of self-improvement has been analysed in a complex cybernetic system by means of computer simulations. It turns out that the investigated system tends to develop by accumulation of as small structural changes as possible, while larger changes are likely to result in the collapse of the system rather than in its persistence or improvement. This implies that cybernetic considerations alone cannot justify the claim that the very nature of epigenetic systems induces evolution by discrete steps rather than by gradual accumulation of small changes.
Up until now, studies of Kauffman network stability have focused on the conditions resulting from the structure of the network. Negative feedbacks have been modeled as ice (nodes that do not change their state) in an ordered phase but this blocks the possibility of breaking out of the range of correct operation. This first, very simplified approximation leads to some incorrect conclusions, e.g., that life is on the edge of chaos. We develop a second approximation, which discovers half-chaos and shows its properties. In previous works, half-chaos has been confirmed in autonomous networks, but only using node function disturbance, which does not change the network structure. Now we examine half-chaos during network growth by adding and removing nodes as a disturbance in autonomous and open networks. In such evolutions controlled by a ‘small change’ of functioning after disturbance, the half-chaos is kept but spontaneous modularity emerges and blurs the picture. Half-chaos is a state to be expected in most of the real systems studied, therefore the determinants of the variability that maintains the half-chaos are particularly important in the application of complex network knowledge.
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