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

We describe systems using Kauffman and similar networks. They are directed functioning networks consisting of finite number of nodes with finite number of discrete states evaluated in synchronous mode of discrete time. In this paper we introduce the notion and phenomenon of "structural tendencies". Along the way we expand Kauffman networks, which were a synonym of Boolean networks, to more than two signal variants and we find a phenomenon during network growth which we interpret as "complexity threshold". For simulation we define a simplified algorithm which allows us to omit the problem of periodic attractors. We estimate that living and human designed systems are chaotic (in Kauffman sense) which can be named — complex. Such systems grow in adaptive evolution. These two simple assumptions lead to certain statistical effects, i.e., structural tendencies observed in classic biology but still not explained and not investigated on theoretical way. For example, terminal modifications or terminal predominance of additions where terminal means: near system outputs. We introduce more than two equally probable variants of signal, therefore our networks generally are not Boolean networks. They grow randomly by additions and removals of nodes imposed on Darwinian elimination. Fitness is defined on external outputs of system. During growth of the system we observe a phase transition to chaos (threshold of complexity) in damage spreading. Above this threshold we identify mechanisms of structural tendencies which we investigate in simulation for a few different networks types, including scale-free BA networks.

Download Full-text

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

Symmetry ◽

10.3390/sym14010092 ◽

2022 ◽

Vol 14 (1) ◽

pp. 92

Author(s):

Andrzej Gecow ◽

Laszlo Barna Iantovics

Keyword(s):

Adaptive Evolution ◽

Network Structure ◽

Network Stability ◽

Network Growth ◽

Correct Operation ◽

Edge Of Chaos ◽

Open Networks ◽

Autonomous Networks ◽

Negative Feedbacks ◽

Small Change

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.

Download Full-text