  We consider the structure of anisotropic exchange interactions in
ytterbium-based insulating rare-earth magnets built from edge-sharing
octahedra. We argue the features of trivalent ytterbium and this structural
configuration allow for a qualitative determination of the different
anisotropic exchange regimes that may manifest in such compounds. The validity
of such super-exchange calculations is tested through comparison to the
well-characterized breathing pyrochlore compound Ba$_3$Yb$_2$Zn$_5$O$_{11}$.
With this in hand, we then consider application to three-dimensional pyrochlore
spinels as well as two-dimensional honeycomb and triangular lattice systems
built from such edge-sharing octahedra. We find an extended regime of robust
emergent weak anisotropy with dominant antiferromagnetic Heisenberg
interactions as well as smaller regions with strong anisotropy. We discuss the
implications of our results for known compounds with the above structures, such
as the spinels AYb$_2$X$_4$ (A = Cd, Mg, X = S, Se), the triangular compound
YbMgGaO$_4$, which have recently emerged as promising candidates for observing
unconventional magnetic phenomena. Finally, we speculate on implications for
the R$_2$M$_2$O$_7$ pyrochlore compounds and some little studied honeycomb
ytterbium magnets.
