  Understanding the dynamics of the turbulent velocity gradient tensor (VGT) is
essential to gain insights into the Navier-Stokes equations and improve
small-scale turbulence modeling. However, characterizing the VGT dynamics
conditional on all its relevant invariants in a continuous fashion is extremely
difficult. In this paper, we represent the VGT Lagrangian dynamics using a
network where each node represents a unique flow state. This approach enables
us to discern how the VGT transitions from one state to another in a simplified
fashion. Our analysis reveals intriguing features of the resulting network,
such as the clustering of the commonly visited nodes where the eigenvalues of
the VGT are real, in the proximity of the Vieillefosse tail. We then relate our
complex network approach to the well-established VGT discretization based on
the sign of its principal invariants, $Q$ and $R$, and its discriminant,
$\Delta$. To this end, we separate the shortest paths on the network
(geodesics) based on the $Q$-$R$ region to which their starting and arrival
nodes belong. The distribution of the length of intra-region geodesics, with
starting and arrival nodes belonging to the same $Q$-$R$ region, exhibits a
distinct bimodality in two regions of the $Q$-$R$ plane, those in which the
deviatoric part of the pressure Hessian introduces complexity to the VGT
dynamics. Such bimodality is associated with infrequently visited nodes having
to follow a long, low probability path to drastically change the state of the
VGT compared to other flow states that can acquire the necessary
characteristics without changing their sign for $Q$ or $R$. We complement the
geodesics approach by examining random walks on the network, showing how the
VGT non-normality and the associated production terms distinguish the shortest
commuting paths between different $Q$-$R$ regions.
