Distinguishability of graphs: a case for quantum-inspired measures

The question of graph similarity or graph distinguishability arises often in natural systems and their analysis over graphical networks. In many domains, graph similarity is used for graph classification, outlier detection or the identification of distinguished interaction patterns. Several methods have been proposed on how to address this topic, but graph comparison still presents many challenges. Recently, information physics has emerged as a promising theoretical foundation for complex networks. In many applications, it has been demonstrated that natural complex systems exhibit features that can be described and interpreted by measures typically applied in quantum mechanical systems. Therefore, a natural starting point for the identification of network similarity measures is information physics and a series of measures of distance for quantum states. In this work, we report experiments on synthetic and real-world data sets, and compare quantum-inspired measures to a series of state-of-the-art and well-established methods of graph distinguishability. We show that quantum-inspired methods satisfy the mathematical and intuitive requirements for graph similarities, while offering high interpretability.