Spin-glass phase transitions on real-world graphs
  We use the Bethe approximation to calculate the critical temperature for the transition from a paramagnetic to a glassy phase in spin-glass models on real-world graphs. Our criterion is based on the marginal stability of the minimum of the Bethe free energy. For uniform degree random graphs (equivalent to the Viana-Bray model) our numerical results, obtained by averaging single problem instances, are in agreement with the known critical temperature obtained by use of the replica method. Contrary to the replica method, our method immediately generalizes to arbitrary (random) graphs. We present new results for Barabasi-Albert scale-free random graphs, for which no analytical results are known. We investigate the scaling behavior of the critical temperature with graph size for both the finite and the infinite connectivity limit. We compare these with the naive Mean Field results. We observe that the Belief Propagation algorithm converges only in the paramagnetic regime. 
J. M. Mooij, H. J. Kappen