  We develop a graph-based model of the hydrogen bond network in water, with a
view towards quantitatively modeling the molecular-level correlational
structure of the network. The networks are formed are studied by the
constructing the model on two infinite-dimensional lattices. Our models are
built \emph{bottom up}, based on microscopic information coming from atomistic
simulations, and we show that the predictions of the model are consistent with
known results from ab-initio simulations of liquid water. We show that simple
entropic models can predict the correlations and clustering of
local-coordination defects around tetrahedral waters observed in the atomistic
simulations. We also find that orientational correlations between bonds are
longer ranged than density correlations, and determine the directional
correlations within closed loops and show that the patterns of water wires
within these structures are also consistent with previous atomistic
simulations. Our models show the existence of density and compressibility
anomalies, as seen in the real liquid, and the phase diagram of these models is
consistent with the singularity-free scenario previously proposed by Sastry and
co-workers (Sastry et al, PRE 53, 6144 (1996)).
