Structural tendencies - Effects of adaptive evolution of complex (chaotic) systems

TL;DR

This paper explores how complex systems modeled by extended Kauffman networks exhibit structural tendencies during adaptive evolution, including a complexity threshold and phenomena like terminal modifications, through simulations of network growth.

Contribution

It introduces the concept of structural tendencies in complex networks, extends Kauffman networks beyond Boolean models, and identifies a complexity threshold during network growth.

Findings

Identification of a complexity threshold during network growth.

Observation of structural tendencies like terminal modifications.

Simulation results on various network types including scale-free networks.

Abstract

We describe systems using Kauffman and similar networks. They are directed funct ioning networks consisting of finite number of nodes with finite number of discr ete 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 netw orks, to more than two signal variants and we find a phenomenon during network g rowth 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 sens e) 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. E.g. 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. T hey 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.