Dynamics of Scale Free Random Threshold Network

TL;DR

This paper investigates the dynamics of Random Threshold Networks on scale-free networks, revealing the absence of phase transitions under certain conditions and exploring how parameters like temperature and threshold influence ordered and chaotic regimes.

Contribution

It introduces a detailed analysis of RTN dynamics on scale-free networks, including the effects of temperature, threshold, and degree distribution, and connects RTN to RBN in high-temperature limits.

Findings

No phase transition at T=0, h=0, D=0 for RTN.

Moderate thermal noise extends ordered regimes.

RTN reduces to RBN as T approaches infinity.

Abstract

We study the dynamics of Random Threshold Network (RTN) on scale free networks, with asymmetric links, some interaction rules where propagation of local perturbations depends on in-degree $k$ of the nodes. We find that there is no phase transition with respect to average connectivty independently of network topology for the case temperature T=0, threshold $h=0$ and the probability distribution of indegree $P(k)$ satisfies $P(0)=D=0$. We have investigated the emergence of phase transition involving three parameters, i.e. $T,h$ and $D$. RTN can be continuously connected to Random Boolean Network (RBN) in $T\to \infty$, and we find moderate thermal noise extends the regime of ordered dynamics, compared to RTN in T=0 regime and RBN. Furthermore, we discuss the dynamic properties from another point of view, dynamical mean field reaction rate equation.