When correlations matter - response of dynamical networks to small perturbations

TL;DR

This paper investigates how small perturbations affect the damage spreading in random Boolean and threshold networks, revealing a new characteristic connectivity and the limitations of average-based predictions in finite biological networks.

Contribution

It introduces the concept of a characteristic connectivity $K_s$ where damage becomes size-independent and shows that internal correlations increase the likelihood of large damage events.

Findings

Identification of a new characteristic connectivity $K_s$

Finite size effects cause deviations from theoretical predictions

Damage size distribution is broad and fat-tailed

Abstract

We systematically study and compare damage spreading for random Boolean and threshold networks under small external perturbations (damage), a problem which is relevant to many biological networks. We identify a new characteristic connectivity $K_s$, at which the average number of damaged nodes after a large number of dynamical updates is independent of the total number of nodes $N$. We estimate the critical connectivity for finite $N$ and show that it systematically deviates from the annealed approximation. Extending the approach followed in a previous study, we present new results indicating that internal dynamical correlations tend to increase not only the probability for small, but also for very large damage events, leading to a broad, fat-tailed distribution of damage sizes. These findings indicate that the descriptive and predictive value of averaged order parameters for finite size networks - even for biologically highly relevant sizes up to several thousand nodes - is limited.