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We perform simulations of random Ising models defined over small-worldnetworks and we check the validity and the level of approximation of a recentlyproposed effective field theory. Simulations confirm a rich scenario with thepresence of multicritical points with first- or second-order phase transitions.In particular, for second-order phase transitions, independent of the dimensiond_0 of the underlying lattice, the exact predictions of the theory in theparamagnetic regions, such as the location of critical surfaces and correlationfunctions, are verified. Quite interestingly, we verify that theEdwards-Anderson model with d_0=2 is not thermodynamically stable under graphnoise.

First- and second-order phase transitions in Ising models on small world networks, simulations and comparison with an effective field theory