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Funding Information: The simulations presented above were performed using computer resources within the Aalto University School of Science “Science-IT” project. A.F. acknowledges funding by Science Foundation Ireland Grant No. 16/IA/4470. Publisher Copyright: © 2022 American Physical Society. 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 to networks with varying degree distributions, numbers of application users, and probabilities of quarantine failures. Further, we study 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 susceptible-infectious-recovered (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 asthe application adoption rate. Finally, both strong homophily and especially heterophily with regard to application adoption can be detrimental. Overall, epidemic dynamics 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.