Effective field theory for models defined over small-world networks: First- and second-order phase transitions

Abstract

We present an effective field theory to analyze, in a very general way, models defined over small-world networks. Even if the exactness of the method is limited to the paramagnetic regions and to some special limits, it provides, yielding a clear and immediate (also in terms of calculation) physical insight, the exact critical behavior and the exact critical surfaces and percolation thresholds. The underlying structure of the nonrandom part of the model-i.e., the set of spins filling up a given lattice L0 of dimension d_{0} and interacting through a fixed coupling J0 -is exactly taken into account. When J_{0}> or = 0 , the small-world effect gives rise, as is known, to a second-order phase transition that takes place independently of the dimension d_{0} and of the added random connectivity c . When J0<0 , a different and novel scenario emerges in which, besides a spin-glass transition, multiple first- and second-order phase transitions may take place. As immediate analytical applications we analyze the Viana-Bray model (d_{0}=0) , the one-dimensional chain (d_{0}=1) , and the spherical model for arbitrary d_{0} .

Communication and correlation among communities

“…However, as we have already showed in the Ref. [20], in our approach for any choice of T , dµ and c, independently of the signs of the couplings and on the fact that the corresponding order parameters are zero or not, for the free energy, and then for any observable, there are always two -and only two -stable solutions that we label as F and SG and that in the thermodynamic limit only one of the two survives. Therefore, an AF like phase in our approach is not represented by another solution; an AF like phase, if any, occurs in the solution with label F. In the Ref.…”

“…Therefore, an AF like phase in our approach is not represented by another solution; an AF like phase, if any, occurs in the solution with label F. In the Ref. [20] we showed that, for n = 1, for the solution with label F and SG there are two natural decoupled order parameters that we have indicated as m (F) and m (SG) , respectively. Similarly, now we have n coupled order parameters m (F;l) for the solution F and n other coupled order parameters m (SG;l) for the solution SG, l = 1, .…”

“…Notice in fact that, without any intention to be exhaustive in citing the large literature on the subject, the state of the art of analytical methods for disordered Ising models defined over Poissonian small-world graphs results nowadays as follows: i) in the case of no short-range couplings, J 0 = 0, and for one community, n = 1, modulo a large use of some population dynamics algorithm for low temperatures, the replica method and the cavity method [25,29,30,31] have established the base to solve exactly the model in any region of the phase diagram, even rigorously in the SK case [32,33] and in unfrustrated cases [34]; ii) for J 0 = 0 and n = 1 these methods have been successfully applied to the one-dimensional case [35,36] but a generalization to higher dimensions (except infinite dimensions [37]) seems impossible due to the presence of loops of any length [53]; on the other hand, even if it is exact only in the P region, the method we have presented in the Ref. [20], modulo solving analytically or numerically a non random Ising model, can be exactly applied in any dimension, and more in general to any underlying pure graph (L 0 , Γ 0 ); iii) for J 0 = 0 and n ≥ 2, the problem was solved only in the limit of infinite connectivity: exactly in the n = 2 CW case in its general form, which includes arbitrary sizes of the two communities, but with no coupling disorder [23]; and, within the replica-symmetric solution, in the generic n SK case, but only in the presence of a same mutual interaction among the n communities of same size [38,39]. Out of this range of models, no method was known to face analytically the general problem with finite connectivities, in arbitrary dimension d 0 , and with a general disorder, despite its relevance in network theory, as e.g., in social networks [54].…”

“…In the Ref. [20] (n = 1) we established the general scenario of the critical behavior coming from these equations, stressing the differences between the cases J 0 ≥ 0 and J 0 < 0, the former being able to give only second-order phase transitions with classical critical exponents, whereas the latter being able to give rise, for a sufficiently large connectivity c, to multicritical points with also first-order phase transitions. The same scenario essentially takes place also for n ≥ 2 provided that the J 0 's and the J's be almost the same for all the communities (and in fact in this case, i.e., near the homogeneous case, the partition in n communities does not turn out to be very meaningful and taking n = 1 would lead to almost the same result), otherwise many other situations are possible.…”

“…In [20] we established a new general method to analyze critical phenomena on small-world models: we found an effective field theory that generalizes the Curie-Weiss mean-field theory via the equation…”