Ising model on networks with an arbitrary distribution of connections

We find the exact critical temperature $T_c$ of the nearest-neighbor ferromagnetic Ising model on an “equilibrium” random graph with an arbitrary degree distribution $P\left(k\right).$ We observe an anomalous behavior of the magnetization, magnetic susceptibility and specific heat, when $P\left(k\right)\mathrm{}$ is fat tailed, or, loosely speaking, when the fourth moment of the distribution diverges in infinite networks. When the second moment becomes divergent, $T_c$ approaches infinity, the phase transition is of infinite order, and size effect is anomalously strong.