Universality and non-universality in behavior of self-repairing random networks

TL;DR

This study investigates the universal and non-universal behaviors in self-repairing random networks, revealing universal properties in some measures but model-dependent fractal dimensions at the phase transition.

Contribution

It numerically analyzes a family of random networks, identifying universal and non-universal critical properties across different models.

Findings

Correlation radius index $ u_B$ is universal in the net-like phase.

Graph dimensions $d_{ ext{min}}$ and $D_{ ext{min}}$ are universal.

Backbone fractal dimension $D_B$ varies with model parameters.

Abstract

We numerically study one-parameter family of random single-cluster systems. A finite-concentration topological phase transition from the net-like to the tree-like phase (the latter is without a backbone) is present in all models of the class. Correlation radius index $\nu_B$ of the backbone in the net-like phase; graph dimensions -- $d_{\min}$ of the tree-like phase, and $D_{\min}$ of the backbone in the net-like phase appear to be universal within the accuracy of our calculations, while the backbone fractal dimension $D_B$ is not universal: it depends on the parameter of a model.