Planar unclustered scale-free graphs as models for technological and biological networks

TL;DR

This paper introduces a family of deterministic, planar, scale-free graphs with modular and self-similar structures, providing exact formulas for key properties to model complex technological and biological networks.

Contribution

The paper presents a new family of deterministic, planar, scale-free graphs with exact analytic expressions for their properties, useful for modeling and understanding complex networks.

Findings

Graphs exhibit logarithmic average path length with network size.

Graphs have a power-law degree distribution.

Exact formulas for degree distribution, diameter, and average distance are derived.

Abstract

Many real life networks present an average path length logarithmic with the number of nodes and a degree distribution which follows a power law. Often these networks have also a modular and self-similar structure and, in some cases - usually associated with topological restrictions- their clustering is low and they are almost planar. In this paper we introduce a family of graphs which share all these properties and are defined by two parameters. As their construction is deterministic, we obtain exact analytic expressions for relevant properties of the graphs including the degree distribution, degree correlation, diameter, and average distance, as a function of the two defining parameters. Thus, the graphs are useful to model some complex networks, in particular several families of technological and biological networks, and in the design of new practical communication algorithms in relation to their dynamical processes. They can also help understanding the underlying mechanisms that have produced their particular structure.