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cess to them, as instead the RM or the CM could do, if they were applicable also to models with loops. 

In this paper we consider random Ising models defined over small-world networks having an underlying regular lattice L$_{0}$ of dimension d$_{0}$ = 1 , 2 , 3. Given the initial lat- tice L$_{0}$ with N sites, we build the small-world network by adding cN/ 2 links uniformly spread over L$_{0}$ . This im- plies that at each site, besides the 2 d$_{0}$ neighbors, there are additional long-range neighbors whose number is dis- tributed according to a Poisson distribution with average c . Interactions act via a coupling J$_{0}$ for the 2 d$_{0}$ short- range neighbors and via a further coupling J for the other long-range neighbors. This way of building a small-world network is different from the re-wiring method of Watts and Strogatz[7] (in which the number of shortcuts per site is also Poissonian distributed) and it is more conve- nient for analytical calculations. However, just by using the effective field theory at the base of this paper, it is possible to deal with the similar re-wiring small-world models as well, and to show rigorously that the critical behavior of the two kind of models is identical [15]. 

By using Monte Carlo (MC) simulations we check the predictions of the effective field theory for the critical sur- faces, the susceptibility, the average magnetization and the two point connected correlation function as a func- tion of the Euclidean distance r defined on L$_{0}$ . 

The ferromagnetic Ising model (both J and J$_{0}$ posi- tive) on small-world networks has been extensively stud- ied [8, 16–20]. However, of remarkable interest, for both its theoretical and practical implications, is the case with negative short-range antiferromagnetic coupling, J$_{0}$ < 0. The anti-ferromagnetic Ising model on small world net- works was studied in [21] for the case where there is only one antiferromagnetic coupling constant ( J$_{0}$ = J ), but, apart from the fully connected case [22], no attention has been paid to the fact that, in the more general situation, may exist multicritical points with first- and second-order phase transition. In fact, when J$_{0}$ < 0 and J > 0, the effective field theory predicts two critical temperatures with first or second-order phase transitions separating two P regions. In this case, simulation results show large fluctuations and a rather slow approach to the thermo- dynamic limit, confirming the slow dynamics of the frus- trated system and the importance to have analytical or semi-analytical frustrated-free tools to investigate these models. Quite interestingly, the model with given cou- plings J$_{0}$ and J , and slightly different values of the con- nectivity c , can show drastically different phase diagrams either showing two ferromagnetic (F) phase transitions or a single spin-glass (SG) phase transition. In the case of first-order phase transitions the magnetization disconti- nuity is not exactly predicted by the theory but simu- lation results show clearly the signature of a discontinu- ity for both the F and SG order parameters in corre- spondence of the theoretical P-SG critical temperature. Furthermore, we see good agreement of the susceptibil- ity predictions in the P phase. We also consider a two- dimensional modified Edwards-Anderson model[23] with 

added long-range shortcuts. This is a special case where besides the connectivity disorder there is also disorder on the value of the short-range coupling. As predicted by the theory, simulations confirm that any infinitesimal addition of shortcuts leads the system to have a finite temperature P-SG phase transition which coincides with the theoretical one. 

The paper is organized as follows. In Secs. II and III we recall the definition of the small-world models and the method, which mainly consists in finding the solution of the self-consistent equation (13) and minimizing the ef- fective free energy (25). In Sec. IV we report our MC simulations for several interesting cases selected for com- parison with the theoretical predictions. Finally, in Sec. V some conclusions are drawn. 

We consider random Ising models constructed by super-imposing random graphs with finite average con- nectivity onto some given lattice L$_{0}$ whose set of bonds ( i, j ) and dimension will be indicated by Γ$_{0}$ and d$_{0}$ , re- spectively. Given an Ising model - shortly the pure model - of N spins coupled over L$_{0}$ through a coupling J$_{0}$ and with Hamiltonian 

and given an ensemble C of unconstrained random graphs c , c ∈ C , whose bonds are determined by the adjacency matrix elements c$_{i,j}$ = 0 , 1, we define the correspond- ing small-world model - shortly the random model - as described by the following Hamiltonian 

the free energy F and the averages 〈 O 〉 l being defined in the usual (quenched) way as 

<img src='content_image/386520.jpg'>

and (in the following a bar notation ¯ · indicates the two independent averages over the graph and couplings real- izations) 

<img src='content_image/386521.jpg'>

where Z c ; J is the partition function of the quenched sys- tem 

## II. SMALL WORLD MODELS 

<img src='content_image/386518.jpg'>

<img src='content_image/386519.jpg'>

<img src='content_image/386522.jpg'>

