The effect of disorder on the hierarchical modularity in complex systems

TL;DR

This paper investigates how introducing randomness affects the hierarchical modularity of complex systems, showing that sufficient randomness can destroy this structure, with implications for understanding real-world systems.

Contribution

It characterizes hierarchical modularity through property periodicity and examines how randomness impacts this structure in complex systems.

Findings

Sufficient randomness destroys hierarchical modularity.

Hierarchical modularity persists with limited randomness.

Experimental systems with algebraic clustering suggest limited randomness.

Abstract

We consider a system hierarchically modular, if besides its hierarchical structure it shows a sequence of scale separations from the point of view of some functionality or property. Starting from regular, deterministic objects like the Vicsek snowflake or the deterministic scale free network by Ravasz et al. we first characterize the hierarchical modularity by the periodicity of some properties on a logarithmic scale indicating separation of scales. Then we introduce randomness by keeping the scale freeness and other important characteristics of the objects and monitor the changes in the modularity. In the presented examples sufficient amount of randomness destroys hierarchical modularity. Our findings suggest that the experimentally observed hierarchical modularity in systems with algebraically decaying clustering coefficients indicates a limited level of randomness.