  Dense, stabilized, frictional particulate suspensions in a viscous liquid
undergo increasingly strong continuous shear thickening (CST) as the solid
packing fraction, $\phi$, increases above a critical volume fraction, and
discontinuous shear thickening (DST) is observed for even higher packing
fractions. Recent studies have related shear thickening to a transition from
mostly lubricated to predominantly frictional contacts with the increase in
stress. The rheology and networks of frictional forces from two and
three-dimensional simulations of shear-thickening suspensions are studied.
These are analyzed using measures of the topology of the network, including
tools of persistent homology. We observe that at low stress the frictional
interaction networks are predominantly quasi-linear along the compression axis.
With an increase in stress, the force networks become more isotropic, forming
loops in addition to chain-like structures. The topological measures of Betti
numbers and total persistence provide a compact means of describing the mean
properties of the frictional force networks and provide a key link between
macroscopic rheology and the microscopic interactions. A total persistence
measure describing the significance of loops in the force network structure, as
a function of stress and packing fraction, shows behavior similar to that of
relative viscosity and displays a scaling law near the jamming fraction for
both dimensionalities simulated.
