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

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.