Self-similar planar graphs as models for complex networks

TL;DR

This paper introduces a family of self-similar planar graphs that exhibit small-world and scale-free properties, serving as mathematical models for complex networks despite having zero clustering coefficient.

Contribution

The paper presents a new family of planar, self-similar graphs with properties similar to complex networks, expanding modeling options beyond traditional hierarchical networks.

Findings

Graphs are planar, modular, self-similar with small-world and scale-free features.

Clustering coefficient of these graphs is zero, differing from typical hierarchical networks.

Parameters of the graphs align with those of real-world complex systems.

Abstract

In this paper we introduce a family of planar, modular and self-similar graphs which have small-world and scale-free properties. The main parameters of this family are comparable to those of networks associated to complex systems, and therefore the graphs are of interest as mathematical models for these systems. As the clustering coefficient of the graphs is zero, this family is an explicit construction that does not match the usual characterization of hierarchical modular networks, namely that vertices have clustering values inversely proportional to their degrees.