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Ising spin glass models versus Ising models: an effective mapping at high temperature II. Applications to graphs and networks

By applying a recently proposed mapping, we derive exactly the upper phase boundary of several Ising spin glass models defined over static graphs and random graphs, generalizing some known results and providing new ones.


Introduction
The science of graphs and networks is nowadays a well studied topic receiving more and more interest due to its large number of applications in the understanding of complex systems in, technological, biological and social structures. We refer the interested reader to the literature [1,2,3]. As a basic starting point in the analysis of the cooperative behavior typical of such complex systems, the study of the simplest dichotomy-interaction models, such as the Ising model, plays a crucial role and in recent years successful efforts have been made in solving this model [4,5], which, on networks, and especially on the so called "scale free networks", may manifest critical phenomena very different from those over regular lattice systems [6,7]. The Ising model with only ferromagnetic couplings, however, allows only a "friendly" interaction between the elements of the network, J b = +J, so that only a possible two-state "friendly" collective behavior is allowed. For taking into account that the interaction between the elements of the network can also be unfriendly, J b = ±J, the Ising spin glass model should instead be considered [8].
In the part I of this work [9], we have introduced a general mapping between a given Ising spin glass model and a related Ising one; the related Ising model, which, roughly speaking, is the model in which the couplings are uniform. The mapping turns out to be of immediate application when the solution of this related Ising model is known.
The mapping becomes exact when the dimensionality of the system goes to infinite. More precisely, we have seen that: for a system having a number of first neighbors per vertex proportional to D, the mapping applies exactly as D → ∞ (infinite dimensional in a strict sense) in all the paramagnetic region; whereas, more in general, the mapping becomes an exact method to find the upper critical surface when, in the thermodynamic limit, the total number of infinitely long paths per vertex goes to infinite and, choosing randomly two of them, the probability that they overlap each other for an infinite number of bonds goes to zero (infinite dimensional in a large sense).
In [9] we have applied the mapping to several models in which the above condition is satisfied, including, generalized Sherrington-Kirkpatrick models, which are infinite dimensional in a strict sense, and the spin glass model defined over the Bethe lattice, which is infinite dimensional in the large sense.
In this paper we will apply the mapping to general graphs and networks which are infinite dimensional in the large sense. First, we will consider generalized static tree-like structures allowing also for a finite number of loops per vertex. Second, we will analyze the case of random graphs where an ensemble of graphs G, together with a probability P (g), g ∈ G, are given and all the graphs g are infinite dimensional in the large sense, which, in particular, includes the so called "equilibrium random graphs" [2]. In both the cases, we find very general formulae for the exact upper critical surface: parmagnetic-spin glass and paramagnetic-disordered ferromagnetic/antiferromagnetic transitions. Whereas for the generalized tree-like structures these critical surfaces are here for the first time given, we observe that, for the random graphs, the critical surfaces have been just derived, but confined to a particular subclass of the equilibrium random graphs [8]. Furthermore, beyond the equilibrium random graphs, we provide a formula which applies exactly in the most general case for networks whose bonds are each other independent random variables. This formula gives us the upper critical surface of the Ising spin glass model in terms of the critical temperature of the Ising model over the network, which, in general, has to be determined numerically, but, as a non random model, its measure turns out to be hugely easier than a direct measure for the spin glass model. The paper is organized as follows. In Secs. 2 and 3, we recall briefly the models we are considering and the main result of the Ref. [9]. In Sec. 4, we apply the mapping to static generalized tree-like structures. In Sec. 5, a brief introduction for defining the random graphs is given. In the subsection 5.1, we recall the critical temperature of an Ising model defined over an ensemble of random graphs, which, by using the mapping, will be used in the following subsection 5.2 to easily find the upper critical surface of a generic Ising spin glass over an ensemble of random graphs. Finally, some conclusions are reported in Sec. 6.

Models
In this paper we will apply the mapping to systems defined over a graph g of N vertices whose set of links Γ g will be defined through the adjacency matrix of the graph, g i,j = 0, 1: The fully connected graph will be indicated with Γ f : The Hamiltonian of these spin glass models can be written as where the J b 's are quenched couplings, andσ b stays for the product of two Ising variables, where Z ({J b }) is the partition function of the quenched system and dP ({J b }) is a product measure given in terms of normalized measures dµ b ≥ 0 (we are considering a general measure dµ b allowing also for a possible dependence on the bonds) We take the Boltzmann constant K B = 1. A generic inverse critical temperature of the spin glass model, if any, will be indicated with β c .

Mapping
In the part I of this work we have provided a mapping between an Ising spin glass model and a related Ising one. The mapping is exact on and above the critical temperature when, in the thermodynamic limit, the dimensionality is infinite. For the aims of the present work, we will need to recall only the following definitions and rules of the mapping.
Given a spin glass model trough Eqs. (1-6), we define, on the same set of links Γ g , its related Ising model trough the following Ising Hamiltonian where the Ising couplings J In the following, a suffix I will be referred to the related Ising system with Hamiltonian (7). Finally, with z b (z

Case of a uniform measure (same disorder for any bond)
If dµ b ≡ dµ for any bond b of Γ g , the related Ising model corresponds to a uniform Ising model having a single coupling J ≡ J (I) and its critical behavior will be characterized by, at most, two points w AF are the critical ferro-and antiferromagnetic temperatures of the related Ising model, respectively. If Γ g is infinite dimensional in the large sense, the critical inverse temperature of the spin glass model, β c , is given by where β where F or AF , in the l.h.s and r.h.s. of Eq. (12), are in correspondence.

Generalization
In the case of an arbitrary (not uniform) measure dµ b , useful for example for anisotropic models, in which we have to consider even a given number of bond dependencies, the related Ising model is defined by a set of, typically few, independent couplings {J (I) b }, trough Eqs. (8-9) and its critical behavior will be fully characterized by the solutions of an equation of the type G I ({z b }) = 0. If Γ g is infinite dimensional in the large sense, Eqs. (10)(11)(12) are generalized as follows. The critical inverse temperature of the spin glass model β c is given by where β (SG) c and β (F/AF ) c are, respectively, solutions of the two set of equations

Phase diagram
Equations (10)(11) and their generalizations (13)(14)(15) give the exact critical paramagnetic-spin glass (P − SG), β (SG) c , and paramagnetic-F/AF (P − F/AF ), β (F/AF ) c , surfaces, the stability of which depend on which of the two ones is the minimum. In the case of a uniform measure, the suffix F and AF stay for ferromagnetic and antiferromagnetic, respectively. In the general case, such a distinction is not possible and the symbol F/AF stresses only that the transition is not P − SG.

Spin glass over generalized tree-like graphs (static graphs)
In the part I of this work [9], as application of the mapping (11)(12), starting from the critical value of the universal parameter of the Ising model over a Bethe lattice of coordination number q, we have recovered the critical surface of the corresponding Ising spin glass model: a spin glass (SG) and a disordered ferromagnetic (F ) transitions take place at β c , respectively solutions of the two following equations which, by using Eq. (10), give the upper phase boundary. We now generalize Eqs. (17) and (18) to a spin glass defined over a given generalized tree-like graph g. Here g is said to be a generalized tree-like graph if for any vertex there is at most a finite number of loops (see Fig. 1). This application provides a new non trivial result. We will exploit the concepts and the conclusions of R. Lyons [13].
Let us choose a vertex as the root 0. If v is a vertex, |v| will indicate the number of links of the shortest path connecting 0 to v. A cutset Π will indicate a finite set of vertices having the property that every infinite path starting from 0 intersects Π. In other words a cutset generalizes the concept of shell used in the Bethe lattice. Finally, to generalize the concept of branching, the branching number of Γ g , denoted by br Γ g , is introduced: It is easy to check that in the case of a regular lattice of coordination number q, one recovers br Γ g = q − 1. Note however that, in general, br Γ g does not coincide with the average of the number of branches per vertex.
In [13] is proved that in the uniform Ising model with positive coupling J (I) defined over a generalized tree like-graph, the ferromagnetic phase transition occurs at w (I) which generalizes Eq. (16). Correspondingly, by observing that a generalized tree-like graph is infinite dimensional in the large sense (see Sec. 3 of [9]), and by using the general rule (11)(12) holding for an Ising spin glass with a uniform measure, dµ b ≡ dµ, we immediately find that, a spin glass and a disordered ferromagnetic transitions, take place at β (SG) c and β (F ) c , respectively solutions of the two following equations Equations (21) and (22) generalize (17) and (18) to tree-like structures.
We can generalize even more the above result as follows. In [13] is proved that for an arbitrary, i.e. non uniform, Ising model with positive couplings {J given by where, for any vertex v ∈ Π, the product in the r.h.s. is extended to all vertices τ belonging to the shortest path from 0 to v (τ ≤ v), and b(τ ) denotes the link connecting τ and its previous point along the shortest path for v. Formally, in terms of the universal quantities {z We can now, correspondingly, apply the rules (14-15) to a spin glass with a measure, dP = b∈Γ f dµ b , and we find that a spin glass and a disordered ferromagnetic transitions take place at β (SG) c and β (F ) c , respectively solutions of the two following equations

Random graphs
In the case of random graphs, an ensemble of graphs G, together with a probability distribution P (g) for the elements g of G, are given. The free energy of a spin glass model defined over the ensemble G is then evaluated as where Z g ({J b }) is the partition function of the quenched system in the graph g with couplings {J b }.
In the following, we will apply explicitly the mapping to the so called "equilibrium random graphs" which are graphs maximally random under the constraint that the degree distribution of a vertex is a given one, p(k); k being the number of connections of the vertex. For this family of graphs (but not only) the corresponding sets of links Γ g are infinite dimensional in the large sense. In fact, even if the number of infinitely length loops per vertex is infinite (see, g.e., [11] and [12]), due to the fact that the bonds are randomly distributed with the only constrain that they have to generate the given degree distribution p(k), the probability that two of these loops overlap each other for an infinite number of bonds is zero. More in general, the elements of any ensemble G are infinite dimensional in the large sense under the very weak condition for the distribution P (g) that the bonds are, each other, independent random variables. Rather, the fact that the number of finite loops is infinite in certain scale free networks, has been recently raised as a possible doubt about the validity of the random tree approximation for the solution of the Ising model on the equilibrium random graphs [11] that we are about to recall in the following subsection.

Ising model over equilibrium random graphs
Before to consider the Ising spin glass over a random graph, we need to recall the general solution for the critical temperature of the Ising model over equilibrium random graphs [4,5]. Let us indicate with f (k) p ≡ ∞ k=0 p(k)f (k) the average of the function degree f (k) with respect to the degree distribution p(k). The critical inverse temperature β c of the uniform Ising model with a coupling J is given by the following relation It is interesting to see that this result can be approximately derived even by using directly our mapping. Let us consider a random graph with N vertices and L bonds so that the average degree is k p = 2L/N . A given couple of vertices among the total of N (N − 1)/2 couples, will be connected by one of the L bonds with a probability p = 2L/(N (N − 1)) = k p /(N − 1). Therefore, if we neglect the correlations among the bonds, an Ising model with coupling J defined over the graph will have an effective distribution of couplings described by where According to Eqs. (7)(8)(9), the related Ising model of this disorder model, whose disorder is given by Eqs. (29) and (30), is the Ising model defined over the fully connected graph Γ f for which a ferromagnetic or antiferromagnetic phase transition occurs at β c J (I) ) → 1/N for N large, we find that the critical inverse temperature β c of the Ising system over the random graph, is given by tanh(β c J) = 1/ k p , which equals Eq. (28) only in the case of the classical random graph, for which k 2 p − k p = k 2 p [10]. The error is due to having neglected the correlations among the bonds or, in other words, in assuming for dP({J b }) a product measure. On the other hand, in a random graph, the distribution of the bonds do not follow the degree distribution of the vertices. In fact, as is known, the probability that a bond points to a node of degree k is not p(k), but p(k)k/ k p . Correspondingly, one expects that, as an effective calculation, in Eq. (30) the degree average of a vertex k p , should be substituted with the degree average of the end of a bond: k 2 p / k p . By implementing this substitution one arrives at tanh(β (I) c J (I) ) = k p / k 2 p , which constitutes a good approximation to the exact one (28) [14].

Spin glass over random graphs
As pointed out above, in a random graph it is not possible to use an effective product measure dP({J b }) which takes into account rigorously also of the degree distribution. However, for an exact treatment we can proceed as follows. Let be given some disorder dµ(J b ) (we consider here only the case of a same disorder for any bond). We observe that, in Eq. (27), the averages with respect to dP({J b }) and to P (g) are interchangeable, so that, if for any fixed graph g we apply our mapping, (see Eqs. (67) of Ref. [9]), for the free energy F we find where ϕ(g) is proportional to the non trivial part of the high temperature expansion of the free energy density over the graph g and, at least in the limit β → β − c (g), it can be evaluated by ϕ ef f (g) through the mapping equations (24-25) of Ref. [9], β c (g) being the critical inverse temperature of the spin glass model defined over the (static) graph g and has to be determined according to Eqs. (10)(11)(12). In other words, for any fixed graph g, the mapping says that the critical behavior of the system is encoded in the effective density ϕ ef f (g) which is nothing else that the non trivial part of the high temperature expansion of the uniform Ising model defined over the graph g and in which, for any b ∈ Γ g , the bond parameter tanh(βJ) is simply substituted by dµ(J b ) tanh(βJ b ) or dµ(J b ) tanh 2 (βJ b ), depending on which transition we are interested in: disordered ferromagnetic or spin glass, respectively. Since this is true for almost any graph g ∈ G (with respect to the measure P (g)), we have that, in the limit β → β − c (g), the ensemble G of the Ising spin glass models can be effectively read as an ensemble G of uniform Ising models in which the parameters of the high temperature expansion are given by dµ(J b ) tanh(βJ b ) or dµ(J b ) tanh 2 (βJ b ), depending on which transition we are interested in. Due to this Ising-representation, by using Eq. (28) (which holds for an ensemble of uniform Ising models and where now β c and J must be meant as quantities of the related Ising model, i.e., β (I) c and J (I) , respectively) in Eqs. (11) and (12) applied to an ensemble of random graphs with weights P (g) and bond parameters dµ(J b ) tanh(βJ b ) or dµ(J b ) tanh 2 (βJ b ), we find immediatly the following disordered ferromagnetic or spin glass critical temperatures, respectively Finally, by applying Eq. (10), the upper critical surface of the phase diagram follows, whereas the multicritical point P − F − SG is the simultaneous solution of Eqs. (32) and (33) (system of equations). Equations (32-33), together with their consequences, have been just derived in the Ref. [8] in the framework of the replica approach, where an analysis of the order parameters has also been done. We stress however that, unlike [8], where the random graphs considered were confined to the particular class generated by the so called "static model" [15], our derivation shows that Eqs. (32) and (33) hold for the most general equilibrium random graph. More in general, by using the same above scheme, given an ensemble of random graphs G distributed with some P (g), if the set Γ g is infinite dimensional in the large sense for almost any g ∈ G with respect to P (g), as happens when the bonds are independent random variables, the critical disordered ferromagnetic and spin glass critical temperatures are the solutions of the two following equations, respectively where w (I) (G; P ) is the (universal) Ising critical parameter in the ensemble G with distribution P of the uniform Ising model having coupling J As anticipated in the introduction, we stress that, in general, a possible direct numerical experiment to measure the Ising critical parameter w (I) (G; P ), turns out to be hugely easier rather than a direct measure for the critical temperature β c of the spin glass model.

Conclusions
A general mapping presented in the part I of this work [9], allows to derive the upper critical surface of an Ising spin glass model starting from the critical temperature of a related Ising model. When the dimensionality of the set of links Γ is infinite, the mapping is exact. In the present work, we have applied such a mapping to the case of static generalized tree-like structures and networks.
Since in a generalized tree-like structure the number of loops per vertex is finite, these structures turns out to be infinite dimensional. Therefore, by using the known solution for the Ising model over a generalized tree-like structure, we have derived the upper critical surface. We stress that, the result found for the ferromagnetic disordered transition, holds in the most general case, regardless of the measure dµ b of the disorder, whereas the formulae for the spin glass transition were still unpublished.
Concerning networks, which are ensembles of random graphs characterized by some distribution P (g), the dimensionality is infinite as far as P (g) does not generate infinite long paths with an infinite overlapping of bonds, as happens in the so called equilibrium random graphs and, more in general, when P (g) is such that the bonds turns out to be each other independent random variables. For the case of equilibrium random graphs, whose P (g) can be completely described in terms of the only degree distribution of the vertices p(k), the known solution for the Ising model allowed us to find exactly the upper critical surface of the Ising spin glass model over the network. In this case, as in the Ising model, the critical surface is expressed in terms of the first and second moment of p(k). We stress that this result is very general, holding for any kind of equilibrium random graph and not only for the so called "static graph" [8]. Yet, we have provided a more general formula which, in terms of the critical temperature of the Ising model, applies exactly to any network as far as almost (with respect to the measure P (g)) any graph g is infinite dimensional (as happens when the bonds are each other independent random variables). Unlike the case of the equilibrium random graphs, however, now the critical temperature of the Ising model, and, therefore, of the Ising spin glass model as well, will be expressed in terms of more complicated averages than the simple moments of the degree distribution p(k). We point out however, that, even if in these cases the analytical knowledge of the critical temperature of the Ising model is absent, a direct numerical measure for it, turns out to be hugely easier than a direct measure for the critical temperature of the spin glass model.
By looking back at: Eqs. (17) and (18) for the Bethe lattice case; Eqs. (21) and (22) or (25) and (26) for the generalized tree-like graphs; Eqs. (32) and (33) for the equilibrium random graphs; and Eqs. (34) and (35) for more general random graphs, we recognize a sort of common structure for the critical surface and the critical behavior. On the other hand, this is not surprising considering that the related Ising models of these models are essentially reduced to an Ising model over some tree-like graph, if necessary, distributed according to some distribution whose fluctuations has to be taken into account. In this sense, we can say that the mapping establishes a sort of universality among different Ising spin glass models.
We conclude by observing that, as already mentioned in the part I, our mapping does not seem to be peculiar of the Ising spin glass models. For more general models, a mapping at high temperature "random system" → "non random system" seems possible; so that a known solution of a given non random model can be used to find the upper critical surface of the corresponding random model over infinite dimensional graphs or infinite dimensional networks.