Small world-Fractal Transition in Complex Networks: Renormalization Group Approach

TL;DR

This paper applies renormalization group theory to complex networks to classify their topologies, identify phase transitions between small-world and fractal structures, and analyze the distribution of shortcuts affecting information flow.

Contribution

It introduces a RG-based framework to distinguish universality classes of network topologies and explains the coexistence of fractal and small-world phases.

Findings

Identifies stable and unstable fixed points corresponding to different network phases.

Explains the coexistence of fractal and small-world structures.

Provides insights into shortcut distributions in real-world networks.

Abstract

We show that renormalization group (RG) theory applied to complex networks are useful to classify network topologies into universality classes in the space of configurations. The RG flow readily identifies a small-world/fractal transition by finding (i) a trivial stable fixed point of a complete graph, (ii) a non-trivial point of a pure fractal topology that is stable or unstable according to the amount of long-range links in the network, and (iii) another stable point of a fractal with short-cuts that exists exactly at the small-world/fractal transition. As a collateral, the RG technique explains the coexistence of the seemingly contradicting fractal and small-world phases and allows to extract information on the distribution of short-cuts in real-world networks, a problem of importance for information flow in the system.