  Complex network analysis methods have been widely applied to nonlinear
systems, but applications within fluid mechanics are relatively few. In this
paper, we use a network for the Lagrangian dynamics of the velocity gradient
tensor (VGT), where each node is a flow state, and the probability of
transitioning between states follows from a direct numerical simulation of
statistically steady and isotropic turbulence. The network representation of
the VGT dynamics is much more compact than the continuous, joint distribution
of a set of invariants for the tensor. We focus on choosing optimal variables
to discretize and classify the VGT states. To this end, we test several
classifications based on topology and various properties of the background flow
coherent structures. We do this using the notion of "community" or "module",
namely clusters of nodes that are optimally distinct while also containing
diverse nodal functions. The best classification based upon VGT invariants
often adopted in the literature combines the sign of the principal invariants,
$Q$ and $R$, and the sign of the discriminant function, $\Delta$, separating
regions where the VGT eigenvalues are real and complex. We further improve this
classification by including the relative magnitude of the non-normal
contribution to the dynamics of the enstrophy and straining stemming from a
Schur decomposition of the VGT. The traditional focus on the second VGT
principal invariant, $Q$, implies consideration of the difference between the
enstrophy and strain-rate magnitude without the non-normal parts. The fact that
including the non-normality leads to a better VGT classification highlights the
importance of unclosed and complex terms contributing to the VGT dynamics,
namely the pressure Hessian and viscous terms, to which the VGT non-normality
is intrinsically related.
