A combinatorial approach to role discoveryMaster's Thesis Espoo, December 8, 2016 Supervisor:Professor Aristides Gionis, Aalto University Advisor:Prof. Aristides Gionis Nikolaj Tatti, D.Sc. Professor Aristides Gionis Advisor: Aalto UniversityProf. Aristides Gionis Nikolaj Tatti, D.Sc. We provide a new formulation for the problem of role discovery in graphs. Our definition is structural and recursive: two vertices should be assigned to the same role if the roles of their neighbors, when viewed as multi-sets, are similar enough. An attractive characteristic of our approach is that it is based on optimizing a well-defined objective function, and thus, contrary to previous approaches, the role-discovery task can be studied with the tools of combinatorial optimization.We demonstrate that, when fixing the number of roles to be used, the proposed role-discovery problem is NP-hard, while another (seemingly easier) version of the problem is NP-hard to approximate. On the positive side, despite the recursive nature of our objective function, we can show that finding a perfect (zero-cost) role assignment with the minimum number of roles can be solved in polynomial time. We do this by connecting the zero-cost role assignment with the notion of equitable partition. For the more practical version of the problem with fixed number of roles we present two natural heuristic methods, and discuss how to make them scalable in large graphs.