The thorough survey of measures for network similarity makes for a fine paper. This paper aims to assess and dissect the main assumptions of a network role similarity metric. This is a metric that does not just match the topological properties of a network onto another one or achieve the same goal through graph theory; it finds nodes that may have similar roles to others in the way they are connected. Think of two families where the roles are clear: there are almost always two parents, yet the two families may have a different number of children and/or related family members. When comparing two networks, one may want to find the nodes playing the role of parent in both networks. Little by little, the authors explore and build the intuition for defining ranking metrics beyond SimRank that are up to the task.

The paper is technically precise, highly understandable, and well written. Accessible to readers with only a little background in network science, this is essential reading for network scientists, whether they are new or in need of similarity metrics beyond more traditional ones such as the Jaccard index or link similarity, to mention two more traditional examples.

The authors go on to introduce their own axiomatic role similarity metric, with full understanding of the current and previous literature on the subject. They aim to provide a sound measure with nothing but the essential properties for optimal network role similarity. Then they return to more established algorithms and test them against their axiomatization. They show, for example, that SimRank is not admissible because automorphism confirmation does not hold; this is also true for MatchSim. The authors claim that their RoleSim similarity measure, however, is optimal. They later proceed to experimental evaluation and concerns related to the time complexity of algorithms and other aspects, even testing on real-world networks (the Internet). The appendix is full of details, including theorems and proofs that support the claims in the main text.