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Role is a fundamental concept in the analysis of the behavior and function ofinteracting entities represented by network data. Role discovery is the task ofuncovering hidden roles. Node roles are commonly defined in terms ofequivalence classes, where two nodes have the same role if they fall within thesame equivalence class. Automorphic equivalence, where two nodes are equivalentwhen they can swap their labels to form an isomorphic graph, captures thiscommon notion of role. The binary concept of equivalence is too restrictive andnodes in real-world networks rarely belong to the same equivalence class.Instead, a relaxed definition in terms of similarity or distance is commonlyused to compute the degree to which two nodes are equivalent. In this paper, wepropose a novel distance metric called automorphic distance, which measures howfar two nodes are of being automorphically equivalent. We also study itsapplication to node embedding, showing how our metric can be used to generatevector representations of nodes preserving their roles for data visualizationand machine learning. Our experiments confirm that the proposed metricoutperforms the RoleSim automorphic equivalence-based metric in the generationof node embeddings for different networks.

An Automorphic Distance Metric and its Application to Node Embedding for Role Mining