Network efficiency of spatial systems with fractal morphology: a geometric graphs approach

TL;DR

This paper investigates how the geometric and topological features of spatial fractal networks influence their efficiency and robustness, revealing that simple modifications can improve their performance to match regular lattices.

Contribution

It introduces a geometric graphs approach to analyze spatial fractal networks, highlighting how fractal morphology affects efficiency and robustness compared to regular structures.

Findings

Fractal trees offer advantages in structural cost across different fractal dimensions.

These networks are scale-free in space but have bounded degree distributions.

Efficiency of fractal trees can be improved to match regular lattices with minimal connectivity changes.

Abstract

The functional features of spatial networks depend upon a non-trivial relationship between the topological and physical structure. Here, we explore that relationship for spatial networks with radial symmetry and disordered fractal morphology. Under a geometric graphs approach, we quantify the effectiveness of the exchange of information in the system from center to perimeter and over the entire network structure. We mainly consider two paradigmatic models of disordered fractal formation, the Ballistic Aggregation and Diffusion-Limited Aggregation models, and complementary, the Viscek and Hexaflake fractals, and Kagome and Hexagonal lattices. First, we show that complex tree morphologies provide important advantages over regular configurations, such as an invariant structural cost for different fractal dimensions. Furthermore, although these systems are known to be scale-free in space, they have bounded degree distributions for different values of an euclidean connectivity parameter and, therefore, do not represent ordinary scale-free networks. Finally, compared to regular structures, fractal trees are fragile and overall inefficient as expected, however, we show that this efficiency can become similar to that of a robust hexagonal lattice, at a similar cost, by just considering a very short euclidean connectivity beyond first neighbors.