A box-covering algorithm for fractal scaling in scale-free networks

TL;DR

This paper investigates a box-covering algorithm for measuring fractal dimensions in scale-free networks, highlighting the importance of box-split allowance and overlapping perspectives for accurate fractal characterization.

Contribution

It introduces the role of box-split allowance and overlapping in the algorithm, revealing their significance in fractal scaling of scale-free networks.

Findings

Box-split allowance is crucial for fractal scaling in scale-free networks.

Overlapping boxes lead to heterogeneous box membership distribution in SF networks.

The algorithm's behavior differs between fractal networks and Euclidean objects.

Abstract

A random sequential box-covering algorithm recently introduced to measure the fractal dimension in scale-free networks is investigated. The algorithm contains Monte Carlo sequential steps of choosing the position of the center of each box, and thereby, vertices in preassigned boxes can divide subsequent boxes into more than one pieces, but divided boxes are counted once. We find that such box-split allowance in the algorithm is a crucial ingredient necessary to obtain the fractal scaling for fractal networks; however, it is inessential for regular lattice and conventional fractal objects embedded in the Euclidean space. Next the algorithm is viewed from the cluster-growing perspective that boxes are allowed to overlap and thereby, vertices can belong to more than one box. Then, the number of distinct boxes a vertex belongs to is distributed in a heterogeneous manner for SF fractal networks, while it is of Poisson-type for the conventional fractal objects.