  Over the years, plenty of classical interaction potentials for water have
been developed and tested against structural, dynamical and thermodynamic
properties. On the other hands, it has been recently observed (F. Martelli et.
al, \textit{ACS Nano}, \textbf{14}, 8616--8623, 2020) that the topology of the
hydrogen bond network (HBN) is a very sensitive measure that should be
considered when developing new interaction potentials. Here we report a
thorough comparison of 11 popular non polarizable classical water models
against their HBN, which is at the root of water properties. We probe the
topology of the HBN using the ring statistics and we evaluate the quality of
the network inspecting the percentage of broken and intact HBs. For each water
model, we assess the tendency to develop hexagonal rings (that promote
crystallization at low temperatures) and pentagonal rings (known to frustrate
against crystallization at low temperatures). We then introduce the
\emph{network complexity index}, a general descriptor to quantify how much the
topology of a given network deviates from that of the ground state, namely of
hexagonal or cubic ice. Remarkably, we find that the network complexity index
allows us to relate, for the first time, the dynamical properties of different
water models with their underlying topology of the HBN. Our study provides a
benchmark against which the performances of new models should be tested
against, and introduces a general way to quantify the complexity of a network
which can be transferred to other materials and that links the topology of the
HBN with dynamical properties. Finally, our study introduces a new perspective
that can help in rationalizing the transformations among the different phases
of water and of other materials.
