We consider the privacy of interactions between individuals in a network. For many networks, while nodes are anonymous to outside observers, the existence of a link between individuals implies the possibility of one node revealing identifying information about its neighbor. Moreover, while the identities of the accounts are likely hidden to an observer, the network of interaction between two anonymous accounts is often available. For example, in blockchain cryptocurrencies, transactions between two anonymous accounts are published openly. Here we consider what happens if one (or more) parties in such a network are deanonymized by an outside identity. These compromised individuals could leak information about others with whom they interacted, which could then cascade to more and more nodes' information being revealed. We use a percolation framework to analyze the scenario outlined above and show for different likelihoods of individuals possessing information on their counter-parties, the fraction of accounts that can be identified and the idealized minimum number of steps from a deanonymized node to an anonymous node (a measure of the effort required to deanonymize that individual). We further develop a greedy algorithm to estimate the \emph{actual} number of steps that will be needed to identify a particular node based on the noisy information available to the attacker. We apply our framework to three real-world networks: (1) a blockchain transaction network, (2) a network of interactions on the dark web, and (3) a political conspiracy network. We find that in all three networks, beginning from one compromised individual, it is possible to deanonymize a significant fraction of the network ($>50$%) within less than 5 steps. Overall these results provide guidelines for investigators seeking to identify actors in anonymous networks, as well as for users seeking to maintain their privacy.