  We have synthesized single crystals of Na$_2$(Ir$_{1-x}$Ti$_x$)O$_3$ and
polycrystals of Li$_2$(Ir$_{1-x}$Ti$_x$)O$_3$ and studied the effect of
magnetic depletion on the magnetic properties by measurements of the magnetic
susceptibility, specific heat and magnetocaloric effect at temperatures down to
0.1~K. In both systems, the non-magnetic substitution rapidly changes the
magnetically ordered ground state into a spin glass, indicating strong
frustration. While for the Li system the Weiss temperature $\Theta_{\rm W}$
remains unchanged up to $x=0.55$, a strong decrease $|\Theta_{\rm W}|$ is found
for the Na system. This suggests that only for the former system magnetic
exchange beyond nearest neighbors is dominating. This is also corroborated by
the observation of a smeared quantum phase transition in
Li$_2$(Ir$_{1-x}$Ti$_x$)O$_3$ near $x=0.5$, i.e. much beyond the site
percolation threshold of the honeycomb lattice.
