Exact statistical mechanics of the Ising model on networks

The Ising model is an equilibrium stochastic process used as a model in several branches of science including magnetic materials, geophysics, neuroscience, sociology and finance. Real systems of interest have finite size and a fixed coupling matrix exhibiting quenched disorder. Exact methods for the Ising model, however, employ infinite size limits, translational symmetries of lattices and the Cayley tree, or annealed structures as ensembles of networks. Here we show how the Ising partition function can be evaluated exactly by exploiting small tree-width. This structural property is exhibited by a large set of networks, both empirical and model generated.