Semi-Adaptive Evolution with Spontaneous Modularity of Half-Chaotic Randomly Growing Autonomous and Open Networks

Abstract

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.

Cancer and Chaos and the Complex Network Model of a Multicellular Organism

“…The assumption of autonomy is the assumption that the environment is so constant that its influence can be neglected. The study that takes into account the influence of the environment on half-chaos was initiated by the publication [ 3 ], but it only studied the ability of a system to maintain the half-chaos based on the assessment of the size of the change at the system’s outputs. This is only an introduction to the study of adaptive evolution, still with established signals from the environment.…”

“…In [ 3 ], large modules were observed that fall into chaos after disturbance, but they are smaller than thresholds of small change and, therefore, such cases of permanent initiations were accepted as evolutionary changes. After such changes, half-chaos remains, which should be interpreted as the fact that the modeled cell survives.…”

“…This study intends to present the interpretation mentioned above on the basis of which searches will be easier and closer to intuition. Half-chaos introduced and demonstrated in [ 2 , 3 ] significantly changes and expands the range of parameters acceptable in modeling living organisms in comparison to the preceding RBN model (RBN (Random Boolean Network) is constructed as classic “random” (er) [ 4 ] (see Section 4.1 ). “Boolean” means two variants of signals values, but they may not be equally probable.…”

“…Genome chaos (also known as chromosomal or karyotypic chaos) is more close to the carcinogenesis problem [ 10 , 11 , 12 ]. Current differences between deterministic chaos used in the mathematical theory of chaos and chaos used in “genome chaos” may be decreased by using observations in the simulation of half-chaotic, growing, open systems described in [ 3 ], where large modules falling into chaos compete with half-chaos.…”

“…A more detailed but still simplified description of models, networks and experiments of half-chaos investigations is provided below in Section 4 . Full existing descriptions contain documentation [ 47 , 48 ] and articles: longer manuscripts [ 2 , 3 ] and shorter article [ 23 , 49 ].…”