Fig. 1. The scaling relations of the average occupation number m_k versus kρ^δ at δ = 0.2 for different SF network sizes: (a) L = 1000, (b) L = 2000, (c) L = 4000, (d) L = 8000, at various densities ρ = 0.5 (purple open diamonds), ρ = 2 (blue open triangles), ρ = 10 (red open squares), ρ = 50 (green open circles), respectively.

Fig. 2. The presence of the finite density effect in the crossover degree ratios k_c(ρ)/k_c(ρ = 2) versus the parameter δ with an SF network size L = 4000, for various densities ρ = 1 (black open diamonds and solid line), 5 (blue open triangles and solid line), and 10 (red open squares and solid line), respectively. The solid lines correspond to the theoretical calculation and the symbols to the MC simulation data. Note that k_c’s do not exist in regime δ > δ_c (δ_c = 0.5 is the vertical red dotted dash line) for ρ = 5 and ρ = 10 due to the lower bound of degree k_min, two theoretical dashed lines are only formally drawn in that region.

Fig. 3. Data collapse plot of (a) the average occupation number m_k versus kρ^δ_c at the transient time t = 2¹⁰, (b) the evolution of ratio of IPR to its steady state limit, I_t/I_∞, versus t/ρ^ν, with ν = 1 – δ being the dynamic exponent in the relaxation dynamics, at δ = 0.2 for network size L = 4000, and various densities ρ = 0.5 (purple open diamonds), ρ = 2 (blue open triangles), ρ = 10 (red open squares), and ρ = 50 (green open circles), respectively.

Table 1. The scaling relations of the crossover degree k_c and the average degree occupation number m_k w.r.t. the finite density ρ and the parameter δ.