Mapping Koch curves into scale-free small-world networks

TL;DR

This paper introduces a novel method to transform Koch fractals into deterministic networks that exhibit properties similar to real-world complex networks, including scale-free degree distribution, high clustering, and small-world characteristics.

Contribution

The authors propose a new mapping technique that creates deterministic networks from Koch fractals, capturing key features of real-world complex systems with exact enumeration of network substructures.

Findings

Networks have power-law degree distribution with exponent 2-3

Networks exhibit high clustering coefficient

Exact counts of spanning trees and subgraphs are provided

Abstract

The class of Koch fractals is one of the most interesting families of fractals, and the study of complex networks is a central issue in the scientific community. In this paper, inspired by the famous Koch fractals, we propose a mapping technique converting Koch fractals into a family of deterministic networks, called Koch networks. This novel class of networks incorporates some key properties characterizing a majority of real-life networked systems---a power-law distribution with exponent in the range between 2 and 3, a high clustering coefficient, small diameter and average path length, and degree correlations. Besides, we enumerate the exact numbers of spanning trees, spanning forests, and connected spanning subgraphs in the networks. All these features are obtained exactly according to the proposed generation algorithm of the networks considered. The network representation approach could be used to investigate the complexity of some real-world systems from the perspective of complex networks.