Sharing or publishing social network data while accounting for privacy of individuals is a difficult task due to the interconnectedness of nodes in networks. A key question in k-anonymity, a widely studied notion of privacy, is how to measure the anonymity of an individual, as this determines the attacker scenarios one protects against. In this paper, we systematically compare the most prominent anonymity measures from the literature in terms of the completeness and reach of the structural information they take into account. We present a theoretical characterization and a distance-parametrized strictness ordering of the existing measures for k-anonymity in networks. In addition, we conduct empirical experiments on a wide range of real-world network datasets with up to millions of edges. Our findings reveal that the choice of the measure significantly impacts the measured level of anonymity and hence the effectiveness of the corresponding attacker scenario, the privacy vs. utility trade-off, and computational cost. Surprisingly, we find that the anonymity measure representing the most effective attacker scenario considers a greater node vicinity yet utilizes only limited structural information and therewith minimal computational resources. Overall, the insights provided in this work offer researchers and practitioners practical guidance for selecting appropriate anonymity measures when sharing or publishing social network data under privacy constraints.