Statistical Analysis of Sets of Graphs: From Unlabelled Graphs to Spatial Graphs
Sets of graphs (or networks) arise in many different fields, from medicine to finance, from sport to the social sciences. The analysis of such sets of graphs is far from trivial due to the highly non-Euclidean and discrete nature of graph data. In this talk, we focus on two specific types of graphs: unlabelled graphs - i.e. graphs with different sets of nodes that needs to be matched – and spatial graphs – i.e. graphs with spatial coordinates on the nodes. We give an overview of how to embed such data in a natural and geometrically manageable space and how to define data analysis methods in such geometrical context. For the unlabelled graph case, we embed graphs in a discrete quotient space and define statistical methods like PCA using tools from length metric spaces. For the spatial graph case, we embed graphs in a space equipped with a Gromov-Wasserstein metric and we study the graphs distribution using tools from metric statistics.