  We investigate a Hamiltonian model of networks. The model is a mirror
formulation of the XY model (hence the name) -- instead 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, but no finite-temperature critical behavior. The ground
state and low-order excitations for sparse, finite graphs is 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, and
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.
