A New Mathematical Space for Social Networks

Finding a new mathematical representation for graphs, which allows direct comparison between different graph structures, is still an open-ended research direction. Having such a representation in a common, well-understood metric space is the first prerequisite for a variety of machine learning algorithms like classification, clustering, etc, over graph datasets. In this study, we propose a symmetric positive semidefinite matrix with the (i, j)-th entry equal to the covariance between normalized vectors Ae and Ae as a representation for graph with adjacency matrix A. We argue that the covariance between vectors of the form Ae and Ae, given some i and j, is an informative feature. We present theoretical results supporting this argument. Our representation, being a covariance matrix in a fixed dimensional metric space, can be directly compared across different graph structures. This naturally provides a measure of similarity on graph objects. In the task of social network classification, our proposal outperforms the state-of-the-art methodologies. In addition, the computation can be performed in operations linear in the number of edges, which makes the proposed approach faster and scalable.