A new space for comparing graphs

摘要

Finding a new mathematical representation for graphs, which allows direct comparison between different graph structures, is an open-ended research direction. Having such a representation is the first prerequisite for a variety of machine learning algorithms like classification, clustering, etc., over graph datasets. In this paper, we propose a symmetric positive semidefinite matrix with the (i, j)-th entry equal to the covariance between normalized vectors Ae and Ae (e being vector of all ones) as a representation for a graph with adjacency matrix A. We show that the proposed matrix representation encodes the spectrum of the underlying adjacency matrix and it also contains information about the counts of small sub-structures present in the graph such as triangles and small paths. In addition, we show that this matrix is a “graph invariant”. All these properties make the proposed matrix a suitable object for representing graphs.