Complexity Threshold for Functioning Directed Networks in Damage Size Distribution

TL;DR

This paper identifies a complexity threshold in random structured networks, marking the transition to chaotic behavior, based on damage distribution analysis during network growth across various types and parameters.

Contribution

It introduces a practical criterion for the complexity threshold in RSN based on damage size distribution, expanding understanding of network chaos transition beyond traditional phase transition points.

Findings

Two peaks in damage size distribution indicate the complexity threshold.

Zero frequency area between peaks serves as a practical complexity criterion.

Transition to chaos differs from known phase transition near K=2 for s=2.

Abstract

A certain complexity threshold is proposed which defines the term `complex network' for RSN, e.g. Kauffman networks with s>=2 - more than two equally probable state variants. Such Kauffman networks are no longer Boolean networks. RSN are different than RWN and RNS. This article is the second one of three steps in description of `structural tendencies' which are an effect of adaptive evolution of complex RSN. This complexity threshold is based on the appearances of chaotic features of a network during its random growth and disappearance of small network effects. Distribution of damage size (after small disturbance) measured in a fraction of damaged nodes, or in number of damaged external outputs and degree of chaos is investigated using simulation. It is done during growth (up to N=4000 nodes) for different: network types (including scale-free), numbers of node inputs (K=2,3,4, fixed for a network) and numbers of signal variants (s=2,3,4,16). In this distribution two peaks emerge and in-between them there appears an area of zero frequency - this is the best practical criterion of complexity threshold found in the investigation. No critical points are found in the area of emerging complexity. A special simplified algorithm (`reversed-annealed') is used which omits the problem of circular attractors. The investigated `transition' to chaos in respect to N is different from the known phase transition near K=2 for s=2.