Beyond rankings: comparing directed acyclic graphs

Defining appropriate distance measures among rankings is a classic area of study which has led to many useful applications. In this paper, we propose a more general abstraction of preference data, namely directed acyclic graphs (DAGs), and introduce a measure for comparing DAGs, given that a vertex correspondence between the DAGs is known. We study the properties of this measure and use it to aggregate and cluster a set of DAGs. We show that these problems are 𝐍𝐏 -hard and present efficient methods to obtain solutions with approximation guarantees. In addition to preference data, these methods turn out to have other interesting applications, such as the analysis of a collection of information cascades in a network. We test the methods on synthetic and real-world datasets, showing that the methods can be used to, e.g., find a set of influential individuals related to a set of topics in a network or to discover meaningful and occasionally surprising clustering structure.