First- and second-order phase transitions in scale-free networks

Abstract

We study first-and second-order phase transitions of ferromagnetic lattice models on scalefree networks, with a degree exponent γ. Using the example of the q-state Potts model we derive a general self-consistency relation within the frame of the Weiss molecular-field approximation, which presumably leads to exact critical singularities. Depending on the value of γ, we have found three different regimes of the phase diagram. As a general trend first-order transitions soften with decreasing γ and the critical singularities at the second-order transitions are γ dependent.Complex networks, which have more complicated connectivity structure than periodic lattices (PLs) have attracted considerable interest recently [1,2]. This research is motivated by empirical data collected and analyzed in different fields. Small-world (SW) networks [3], which can be generated from PLs by replacing a fraction p of bonds by new random links of arbitrary lengths, are suitable to model neural networks [4] and transportation systems [5]. On the other hand, scale-free (SF) networks [6] are realized among others in social systems [7], in protein interaction networks [8], in the internet [9] and in the world-wide web [10]. In a SF network the degree distribution P D (k), where k is the number of links connected to a vertex, has an asymptotic power-law decay P D (k) ∼ k −γ , thus there is no characteristic scale involved. In natural and artificial networks the value of the degree exponent is usually in the range 2 < γ < 3 [11].Cooperative processes such as spread of epidemic disease [12], percolation [13], Ising model [14,15], etc. have also been studied in the SW and the SF networks. For SW networks numerical studies show [16] that any finite fraction of new, long-range bonds, p > 0, brings the transition into the classical, mean-field (MF) universality class. It is understandable since for systems with longrange interactions the MF approximation is exact. In the SF networks, where links between remote sites exists, too, at first thought one could expect also a traditional MF critical behavior. In specific problems, however, it turned out that it is only true for losely connected networks, when the degree exponent γ is large enough. Otherwise the critical singularities of the transition are model independent, but nonuniversal; the critical exponents continuously depend on the value of the degree exponent. In particular, for 2 < γ ≤ 3, when k 2 is divergent the systems are in their ordered phase for any value of the control parameter (temperature, percolation probability, transition rate, etc.), and the critical properties can be investigated in the limit of infinitely strong fluctuations.Till now investigations on cooperative processes in the SF networks are almost exclusively limited to continuous phase transitions. However, in many problems the phase transitions on PLs are first order and it seems natural to ask what happens with these transitions on the SF networks? There is a general tendency that the discontinuities (e.g., the latent ...

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