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

Role is a fundamental concept in the analysis of the behavior and function of interacting entities in complex networks. Role discovery is the task of uncovering the hidden roles of nodes within a network. Node roles are commonly defined in terms of equivalence classes. Two nodes have the same role if they fall within the same equivalence class. Automorphic equivalence, where two nodes are equivalent when they can swap their labels to form an isomorphic graph, captures this notion of role. The binary concept of equivalence is too restrictive, and nodes in real-world networks rarely belong to the same equivalence class. Instead, a relaxed definition in terms of similarity or distance is commonly used to compute the degree to which two nodes are equivalent. In this paper, we propose a novel distance metric called automorphic distance, which measures how far two nodes are from being automorphically equivalent. We also study its application to node embedding, showing how our metric can be used to generate role-preserving vector representations of nodes. Our experiments confirm that the proposed automorphic distance metric outperforms a state-of-the-art automorphic equivalence-based metric and different state-of-the-art techniques for the generation of node embeddings in different role-related tasks.