Generating hierarchial scale free graphs from fractals

TL;DR

This paper introduces a deterministic method to generate hierarchical, scale-free networks from self-similar fractals, capturing key features of real-world networks such as scale-freeness, high clustering, and logarithmic diameter growth.

Contribution

The authors present a novel deterministic fractal-based model for hierarchical scale-free networks, with rigorous proofs of key network properties and a method to generate similar random graphs.

Findings

Networks exhibit scale-free degree distribution

High clustering coefficients are observed

Diameter grows logarithmically with network size

Abstract

Motivated by the hierarchial network model of E. Ravasz, A.-L. Barabasi, and T. Vicsek, we introduce deterministic scale-free networks derived from a graph directed self-similar fractal $\Lambda $. With rigorous mathematical results we verify that our model captures some of the most important features of many real networks: the scale free and the high clustering properties. We also prove that the diameter is the logarithm of the size of the system. Using our (deterministic) fractal $\Lambda $ we generate random graph sequence sharing similar properties.