Multiscaling in an YX model of networks

We investigate a Hamiltonian model of networks. The model is a mirror formulation of the XY model (hence the name)--instead of letting the XY spins vary, keeping the coupling topology static, we keep the spins conserved and sample different underlying networks. Our numerical simulations show complex scaling behaviors with various exponents as the system grows and temperature approaches zero, but no finite-temperature universal critical behavior. The ground-state and low-order excitations for sparse, finite graphs are a fragmented set of isolated network clusters. Configurations of higher energy are typically more connected. The connected networks of lowest energy are stretched out giving the network large average distances. For the finite sizes we investigate, there are three regions--a low-energy regime of fragmented networks, an intermediate regime of stretched-out networks, and a high-energy regime of compact, disordered topologies. Scaling up the system size, the borders between these regimes approach zero temperature algebraically, but different network-structural quantities approach their T=0 values with different exponents. We argue this is a, perhaps rare, example of a statistical-physics model where finite sizes show a more interesting behavior than the thermodynamic limit.