Spin models on random graphs with controlled topologies beyond degree constraints
Conrad Perez-Vicente and Anthony Coolen
Journal of Physics A
We study Ising spin models on finitely connected random interaction graphs
which are drawn from an ensemble in which not only the degree
distribution p(k) can be chosen arbitrarily, but which allows for further fine-tuning of the topology
via preferential attachment of edges on the basis of an arbitrary function Q(k,k') of the degrees of the
vertices involved. We solve these models using finite connectivity equilibrium replica theory, within the replica symmetric ansatz.
In our ensemble of graphs, phase diagrams of the spin system are found to depend no longer
only on the chosen degree distribution, but also on the choice made for Q(k,k').
The increased ability to control interaction topology in solvable models beyond prescribing only the degree distribution
of the interaction graph enables a more accurate modeling of
real-world interacting particle systems by spin systems on suitably defined random graphs.