Critical Line in Random Threshold Networks with Inhomogeneous Thresholds

TL;DR

This paper analytically and numerically investigates the critical connectivity in Random Threshold Networks with inhomogeneous thresholds, revealing universal scaling laws, effects of threshold distribution on damage propagation, and the impact of correlations on network dynamics.

Contribution

It provides the first analytical calculation of the critical connectivity in RTN with inhomogeneous thresholds and explores how threshold distribution and correlations influence network behavior.

Findings

Critical connectivity $K_c$ scales as $h^2/(2\ln|h|)$ for large thresholds.

Inhomogeneous thresholds increase damage propagation in sparse networks.

Correlations between thresholds and in-degree can induce transitions between ordered and chaotic dynamics.

Abstract

We calculate analytically the critical connectivity $K_c$ of Random Threshold Networks (RTN) for homogeneous and inhomogeneous thresholds, and confirm the results by numerical simulations. We find a super-linear increase of $K_c$ with the (average) absolute threshold $|h|$, which approaches $K_c(|h|) \sim h^2/(2\ln{|h|})$ for large $|h|$, and show that this asymptotic scaling is universal for RTN with Poissonian distributed connectivity and threshold distributions with a variance that grows slower than $h^2$. Interestingly, we find that inhomogeneous distribution of thresholds leads to increased propagation of perturbations for sparsely connected networks, while for densely connected networks damage is reduced; the cross-over point yields a novel, characteristic connectivity $K_d$, that has no counterpart in Boolean networks. Last, local correlations between node thresholds and in-degree are introduced. Here, numerical simulations show that even weak (anti-)correlations can lead to a transition from ordered to chaotic dynamics, and vice versa. It is shown that the naive mean-field assumption typical for the annealed approximation leads to false predictions in this case, since correlations between thresholds and out-degree that emerge as a side-effect strongly modify damage propagation behavior.