Emergence of Matured Chaos During Network Growth, Place for Adaptive Evolution and More of Equally Probable Signal Variants as an Alternative to Bias p

Abstract

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.

Life Is Not on the Edge of Chaos but in a Half-Chaos of Not Fully Random Systems. Definition and Simulations of the Half-Chaos in Complex Networks

“…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. 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.…”

“…Both models give very different results, it means, that they cannot replace each other. (See also ( [21] Fig. 4)).…”

“…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).…”

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

“…c,d -theoretical damage spreading calculated using the Derrida's annealed approximation model. d -The change of damage in one step of the time in synchronous calculation known as the 'Derrida plot', extended [21] for the case s > 2. The crossing of curves d t+1 (d t ,s,K)with diagonal d t+1 =d t shows equilibrium levels dmx up to which damage can grow.…”