Emergence of Chaos and Complexity During System Growth

Abstract

The main topic of this article is the emergence of chaos in networksdescribing adaptive systems.We investigate this process mainly duringthe system growth in dependency on network size when other parametersof the network do not change. However, we also compare the degreeof chaos for different parameters and network types including randomErdős-Rényi and BA scale-free networks. We use Kauffman networks,and we follow Kauffmann using their parameters and the notion ‘chaos’for them. However, we use more than two signal variants which weassume to be equally probable, therefore the Kauffman networks consideredcan become different from the Boolean networks. The terms ‘complexsystem’ and ‘complex network’ are commonly used but they have nocommon established definitions. We find that chaotic properties ofnetworks well meet our intuition of complexity and that the appearanceof chaotic features during system growth can be treated as complexitythreshold. Crossing this threshold defines certain properties ofsystem and mechanisms which create ‘structural tendencies’. Theseinteresting phenomena, however, are described in another articlein this book.