Graph is one of the most common representations of complex data and plays a crucial role in various research areas and many practical applications. Recently many different approaches to find similarity between two graphs are proposed for efficient algorithms. However, these methods are not scalable to large-scale graphs containing millions of edges, which are common in today’s applications.
As a result, effective and scalable methods for large-scale graphs comparison are urgently needed.
In this talk, a new distance to compare two large-scale graphs based on the theory of quantum probability is introduced. Our proposed distance between two graphs is defined as the distance between the corresponding moment matrices of their spectral distributions. An explicit form for the spectral distribution of the corresponding adjacency matrix of a graph is established. It is shown that the spectral distributions of their adjacency matrices in a vector state includes information not only about their eigenvalues, but also about the corresponding eigenvectors. Also, such distance includes enough information of the structure of a graph. Moreover, we prove that such distance is graph invariant and substructure invariant. Various examples are given, and the proposed distances between graphs with few vertices are shown. Computational results for real large-scale graphs and random graphs show that its accuracy is better than any existing methods and time cost is extensively cheap.