Finite-size scaling of percolation on scale-free networks

Critical phenomena on scale-free networks with a degree distribution pk∼k−λ exhibit rich finite-size effects due to its structural heterogeneity. We systematically study the finite-size scaling of percolation and identify two distinct crossover routes to mean-field behavior: one controlled by the degree exponent λ, the other by the degree cutoff K∼Vκ, where V is the system size and κ∈[0,1] is the cutoff exponent. Increasing λ or decreasing κ suppresses heterogeneity and drives the system toward mean-field behavior, with logarithmic corrections near the marginal case. These findings provide a unified picture of the crossover from heterogeneous to homogeneous criticality. In the crossover regime, we observe rich finite-size phenomena, including the transition from vanishing to divergent susceptibility, distinct exponents for the shift and fluctuation of pseudocritical points, and a numerical clarification of previous theoretical predictions.