Fractal Boundaries of Complex Networks

TL;DR

This paper explores the fractal nature of boundary nodes in various complex networks, revealing power-law distributions and fractal cluster structures with implications for epidemic spreading.

Contribution

It introduces the concept of network boundaries, demonstrating their fractal properties across different network models and real-world networks.

Findings

Boundary nodes follow a power-law distribution with exponent -2.

Boundary clusters are fractals with a dimension approximately 2.

Results apply to both theoretical models and real networks.

Abstract

We introduce the concept of boundaries of a complex network as the set of nodes at distance larger than the mean distance from a given node in the network. We study the statistical properties of the boundaries nodes of complex networks. We find that for both Erd\"{o}s-R\'{e}nyi and scale-free model networks, as well as for several real networks, the boundaries have fractal properties. In particular, the number of boundaries nodes {\it B} follows a power-law probability density function which scales as $B^{-2}$. The clusters formed by the boundary nodes are fractals with a fractal dimension $d_{f} \approx 2$. We present analytical and numerical evidence supporting these results for a broad class of networks. Our findings imply potential applications for epidemic spreading.