How to calculate the fractal dimension of a complex network: the box covering algorithm

TL;DR

This paper compares algorithms for calculating the fractal dimension of complex networks through box covering, mapping the problem to graph coloring, and demonstrating the efficiency of established methods.

Contribution

It introduces a mapping of the box covering problem to graph coloring and evaluates multiple algorithms, highlighting the efficiency of well-established solutions.

Findings

Graph coloring algorithms are effective for box covering.

The proposed algorithms are near-optimal for calculating fractal dimensions.

Additional improvements to the algorithms are unlikely to be significant.

Abstract

Covering a network with the minimum possible number of boxes can reveal interesting features for the network structure, especially in terms of self-similar or fractal characteristics. Considerable attention has been recently devoted to this problem, with the finding that many real networks are self-similar fractals. Here we present, compare and study in detail a number of algorithms that we have used in previous papers towards this goal. We show that this problem can be mapped to the well-known graph coloring problem and then we simply can apply well-established algorithms. This seems to be the most efficient method, but we also present two other algorithms based on burning which provide a number of other benefits. We argue that the presented algorithms provide a solution close to optimal and that another algorithm that can significantly improve this result in an efficient way does not exist. We offer to anyone that finds such a method to cover his/her expenses for a 1-week trip to our lab in New York (details in http://jamlab.org).