Contact tracing via digital tracking applications installed on mobile phones is an important tool for controlling epidemic spreading . Its effectivity can be quantified by modifying the standard methodology for analyzing percolation and connectivity of contact networks . We apply this framework for networks with varying degree distribution, the number of application users and the probability of quarantine failure . Further, we include structured populations with homophily and heterophily and the possibility of degree-targeted application distribution . Our results are based on a combination of explicit simulations and mean-field analysis . They indicate that there can be major differences in the epidemic size and epidemic probabilities which are equivalent in the normal SIR processes . Further, degree heterogeneity is seen to be especially important for the epidemic threshold but not as much for the epidemic size . The probability that tracing leads to quarantines is not as important as the application adaption rate . Finally, both strong homophily and especially heterophily with regards to application adoption can be detrimental . Overall, epidemics are very sensitive to all of the parameter values we tested out, which makes the problem of estimating the effect of digital contact tracing an inherently multidimensional problem.