Profile and scaling of the fractal exponent of percolations in complex networks

TL;DR

This paper introduces a new finite size scaling method to analyze percolation transitions in complex networks, effectively identifying transition points and critical exponents for both infinite and second order transitions.

Contribution

It develops a novel finite size scaling analysis tailored for complex networks' percolation transitions, including infinite order cases with BKT-like singularities.

Findings

Validates the scaling method with Monte-Carlo simulations

Accurately determines transition points and critical exponents

Applies successfully to different network models

Abstract

We propose a novel finite size scaling analysis for percolation transition observed in complex networks. While it is known that cooperative systems in growing networks often undergo an infinite order transition with inverted Berezinskii-Kosterlitz-Thouless singularity, it is very hard for numerical simulations to determine the transition point precisely. Since the neighbor of the ordered phase is not a simple disordered phase but a critical phase, conventional finite size scaling technique does not work. In our finite size scaling, the forms of the scaling functions for the order parameter and the fractal exponent determine the transition point and critical exponents numerically for an infinite order transition as well as a standard second order transition. We confirm the validity of our scaling hypothesis through Monte-Carlo simulations for bond percolations in some network models: the decorated (2,2)-flower and the random attachment growing network, where an infinite order transition occurs, and the configuration model, where a second order transition occurs.