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

We study first- and second-order phase transitions of ferromagnetic lattice models on scale-free 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.