Damage Spreading in Spatial and Small-world Random Boolean Networks

TL;DR

This paper investigates how different spatial and small-world topologies in Random Boolean Networks affect damage spreading, stability, and scaling behaviors, revealing key trade-offs for network robustness and wiring costs.

Contribution

It introduces an analysis of damage spreading in spatially extended and small-world RBNs, highlighting how topology influences stability and scaling, which was not previously explored.

Findings

Spatial local connections alter damage scaling at low connectivities.

Critical stability connectivity $K_s$ differs from random networks.

Relevant component scales with system size with different exponents for local and small-world networks.

Abstract

The study of the response of complex dynamical social, biological, or technological networks to external perturbations has numerous applications. Random Boolean Networks (RBNs) are commonly used a simple generic model for certain dynamics of complex systems. Traditionally, RBNs are interconnected randomly and without considering any spatial extension and arrangement of the links and nodes. However, most real-world networks are spatially extended and arranged with regular, power-law, small-world, or other non-random connections. Here we explore the RBN network topology between extreme local connections, random small-world, and pure random networks, and study the damage spreading with small perturbations. We find that spatially local connections change the scaling of the relevant component at very low connectivities ($\bar{K} \ll 1$) and that the critical connectivity of stability $K_s$ changes compared to random networks. At higher $\bar{K}$, this scaling remains unchanged. We also show that the relevant component of spatially local networks scales with a power-law as the system size N increases, but with a different exponent for local and small-world networks. The scaling behaviors are obtained by finite-size scaling. We further investigate the wiring cost of the networks. From an engineering perspective, our new findings provide the key design trade-offs between damage spreading (robustness), the network's wiring cost, and the network's communication characteristics.