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 gives that while the degree distribution converges to a pure power law when Delta -> 8, 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 approximate to 2, which concerns many of the real-world networks reported in the literature. Copyright (C) EPLA, 2015