Finite-Size Scaling in Complex Networks

Abstract

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 ...

Quasistationary simulations of the contact process on quenched networks

“…Such a claim was criticized by Park and collaborators [11], who later proposed an alternative FSS ansatz, based on a droplet excitation theory [12]. A discrimination between the two approaches turned out to be nontrivial: Even for simulations on annealed networks, which are expected to be described exactly by mean-field theory, numerical results did not satisfactorily conform to any of the two competing theories [13].…”

Percolation of spatially constrained Erdős-Rényi networks with degree correlations

“…For spatially embedded random networks, it has been shown that a useful definition of d, different from the embedding space dimension, can be given [50], and a connection to classical percolation scaling theory can be drawn based on this definition. A different case is that of the nonspatial complex networks, where it has been proposed that the product νd can be replaced byν in the case that the criticality belongs to the mean-field universality class [51]. In our scaling analysis below we will therefore consider only the combined expressions γ /νd and 1/νd.…”

Quasistationary analysis of the contact process on annealed scale-free networks

“…The usual QS analysis assumes a power law dependence of the order parameters with the size at criticality. Such assumption is commonly used as a criterion to determine the critical point of absorbing phase transitions in regular lattices [11] and has been extended to quenched complex networks [19,20]. Corrections to scaling in the form 1 − const × N −0.75 were already observed in QS simulations of the directed percolation universality class in hypercubic lattices, including the contact process [24,32].…”