Solvable Spin Model on Dynamical Networks

We consider an Ising model in which spins are dynamically coupled by links in a network. In this model there are two dynamical quantities which arrange towards a minimum energy state in the canonical framework: the spins, si , and the adjacency matrix elements, cij . The model becomes exactly solvable without recourse to the replica hypothesis or other assumptions because micro-canonical partition functions reduce to products of binomial factors as a direct consequence of the cij s minimizing energy. We solve the system for finite sizes and for the two possible thermodynamic limits and discuss the phase diagrams. The model can be seen as a model for social systems in which agents are not only characterized by their states but also have the freedom to choose their interaction partners in order to maximize their utility.