Upper Bound of Relative Error of Random Ball Coverage for Calculating Fractal Network Dimension

TL;DR

This paper establishes an upper bound for the relative error in random ball coverage when calculating fractal network dimensions, and proposes an improved algorithm that becomes highly accurate for large networks, simplifying the computational complexity.

Contribution

The paper introduces a strict upper bound for the relative error of random ball coverage and proposes a twice-random ball coverage algorithm that reduces complexity for large fractal networks.

Findings

Upper bound of relative error tends to 0 for large network diameters.

Twice-random ball coverage algorithm improves accuracy.

Dimension calculation becomes a P problem for sufficiently large networks.

Abstract

Least box number coverage problem for calculating dimension of fractal networks is a NP-hard problem. Meanwhile, the time complexity of random ball coverage for calculating dimension is very low. In this paper we strictly present the upper bound of relative error for random ball coverage algorithm. We also propose twice-random ball coverage algorithm for calculating network dimension. For many real-world fractal networks, when the network diameter is sufficient large, the relative error upper bound of this method will tend to 0. In this point of view, given a proper acceptable error range, the dimension calculation is not a NP-hard problem, but P problem instead.