Finite-size scaling of percolation on scale-free networks

Abstract

Critical phenomena on scale-free networks with a degree distribution $p_k\sim k^{-\hat{I}»}$ 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 $\hat{I}»$, the other by the degree cutoff $K\sim V^{\hat{I}}º$, where $V$ is the system size and $\hat{I}º\in [0,1]$ is the cutoff exponent. Increasing $\hat{I}»$ or decreasing $\hat{I}º$ 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.

10 pages, 9 figures