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.PACS numbers: 05.50.+q, 89.75.Hc, 89.75.Da, 64.60.Ht Critical phenomena in complex networks have attracted much attention recently and traditional models in statistical physics have been examined on diverse networks [1,2,3,4,5,6,7]. Interesting questions in such probes would be the existence of phase transitions and the network-dependent critical behavior near the transition. Up to now, various equilibrium systems such as Ising, Potts, and XY models [1,2,3,4] as well as nonequilibrium systems such as percolation [5], directed percolation [6], and synchronization models [7] have been studied by means of the mean-field (MF) approach [2], the replica method [3], and the thermodynamic potential hypothesis [1]. It is predicted that phase transitions in complex networks exhibit mostly the standard MF critical behavior except in highly heterogeneous scale-free (SF) networks where the interesting heterogeneity-dependent (but still MF-type) critical behavior appears [1,2,3].The MF prediction has been brought up to the test by extensive numerical simulations and passed it reasonably well in cases of less heterogeneous networks like random, small-world, and even some SF networks [4,5,6,7]. However, for the highly heterogeneous networks (like the most of SF networks found in nature), the asymptotic scaling regime could not be reached easily in numerical tests due to huge finite size effects and consequently there is no reasonably solid numerical analysis reported as yet. Therefore it is essential to understand the finite-size-scaling (FSS) behavior analytically in networks, not only to analyze numerical data for finite-size networks but also to explore the physics of correlated size scales in networks.In the present Letter, we propose a FSS theory for various models in complex networks, based on a dropletexcitation (hyperscaling) argument. Our conjecture for the FSS exponent values is confirmed via numerical simulations for the Ising model and the contact process [8].We first start with the standard FSS theory in low dimensional systems. As a typical example, consider the ferromagnetic Ising model. Its critical behavior near the transition is characterized by the singular behavior in the magnetization m ∼ ǫ β , the susceptibility χ ∼ |ǫ| −γ , and the correlation length ξ ∼ |ǫ| −ν , where ǫ is the reduced temperature defined by ǫ ≡ (T c − T )/T c .The standard FSS theory for the singular part of the free energy f reads [9]where b is the scale factor, d the spatial dimension, L the system linear size, and h the external field. The two scaling dimensions, y T ...

Stochastic driven flow along a channel can be modeled by the asymmetric simple exclusion process. We confirm numerically the presence of a dynamic queuing phase transition at a nonzero obstruction strength, and establish its scaling properties. Below the transition, the traffic jam is macroscopic in the sense that the length of the queue scales linearly with system size. Above the transition, only a power-law shaped queue remains. Its density profile scales as δρ ∼ x −ν with ν = 1 3, and x is the distance from the obstacle. We construct a heuristic argument, indicating that the exponent ν = 1 3 is universal and independent of the dynamic exponent of the underlying dynamic process. Fast bonds create only power-law shaped depletion queues, and with an exponent that could be equal to ν = 2 3, but the numerical results yield consistently somewhat smaller values ν ≃ 0.63(3). The implications of these results to faceting of growing interfaces and localization of directed polymers in random media, both in the presence of a columnar defect are pointed out as well.

Although variations exist between the individual cancer sites, obesity was associated with an overall increased risk of cancer in postmenopausal Korean women. To reduce the risk of cancer, active strategies to prevent obesity should be implemented in postmenopausal women.