Finite-Size Scaling in Complex Networks

A finite-size-scaling (FSS) theory is proposed for various models in complex networks. In particular, we focus on the FSS exponent, which plays a crucial role in analyzing numerical data for finite-size systems. Based on the droplet-excitation (hyperscaling) argument, we conjecture the values of the FSS exponents for the Ising model, the susceptible-infected-susceptible model, and the contact process, all of which are confirmed reasonably well in numerical simulations.

Figure 1

Data collapse of $m$ for the static network with $\lambda =3.87$, using $\beta /\overline{\nu }=0.37$ and $\overline{\nu }=3.5$. Insets: Double logarithmic plots of the critical decay of $m$ and ${\chi }^{\prime }$ against $N$ with slopes $\beta /\overline{\nu }=0.37(5)$ and ${\gamma }^{\prime }/\overline{\nu }=0.26(4)$.

Figure 2

Data collapse of $\rho$ for the UCM network with $\lambda =2.75$, using $\beta /\overline{\nu }=0.58$ and $\overline{\nu }=2.4$. Inset: Double logarithmic plots of the critical decay of $\rho$ and ${\chi }^{\prime }$ against $N$ with slopes $\beta /\overline{\nu }=0.58(1)$ and ${\gamma }^{\prime }/\overline{\nu }=-0.16(2)$.