Giant component sizes in scale-free networks with power-law degrees and cutoffs

TL;DR

This paper analyzes how the size of the giant component in scale-free networks with finite degree cutoffs depends on the cutoff and the power-law exponent, revealing slow convergence to the infinite limit especially near 2.

Contribution

It provides a detailed analysis of the asymptotic behavior of the giant component size in finite scale-free networks with degree cutoffs, highlighting the slow convergence near 2.

Findings

Giant component size increases rapidly once the cutoff exceeds the threshold for existence.

Convergence to the infinite network limit is slow, especially for 22 exponent.

The slowest convergence occurs near 2, common in real-world networks.

Abstract

Scale-free networks arise from power-law degree distributions. Due to the finite size of real-world networks, the power law inevitably has a cutoff at some maximum degree $\Delta$. We investigate the relative size of the giant component $S$ in the large-network limit. We show that $S$ as a function of $\Delta$ increases fast when $\Delta$ is just large enough for the giant component to exist, but increases ever more slowly when $\Delta$ increases further. This makes that while the degree distribution converges to a pure power law when $\Delta\to\infty$, $S$ approaches its limiting value at a slow pace. The convergence rate also depends on the power-law exponent $\tau$ of the degree distribution. The worst rate of convergence is found to be for the case $\tau\approx2$, which concerns many of the real-world networks reported in the literature.