Scale-free networks embedded in fractal space

TL;DR

This paper introduces a model for scale-free networks embedded in fractal space, showing how spatial embedding strength influences degree distribution and network phase transition, with real-world soil network validation.

Contribution

The study presents a new model linking fractal spatial embedding with scale-free network properties and analyzes the phase transition related to edge length fluctuations.

Findings

Degree exponent decreases with fractal dimension under strong embedding

Weakly embedded networks maintain a degree exponent of 2

Network transition from non-compact to compact phase is linked to edge length divergence

Abstract

The impact of inhomogeneous arrangement of nodes in space on network organization cannot be neglected in most of real-world scale-free networks. Here, we wish to suggest a model for a geographical network with nodes embedded in a fractal space in which we can tune the network heterogeneity by varying the strength of the spatial embedding. When the nodes in such networks have power-law distributed intrinsic weights, the networks are scale-free with the degree distribution exponent decreasing with increasing fractal dimension if the spatial embedding is strong enough, while the weakly embedded networks are still scale-free but the degree exponent is equal to $\gamma=2$ regardless of the fractal dimension. We show that this phenomenon is related to the transition from a non-compact to compact phase of the network and that this transition is related to the divergence of the edge length fluctuations. We test our analytically derived predictions on the real-world example of networks describing the soil porous architecture.