Modeling graphs using dot product representations

Given a simple (weighted) graph, or a collection of graphs on a common vertex set, we seek an assignment of vectors to the vertices such that the dot products of these vectors approximate the weight/frequency of the edges. By transforming vertices into (low dimensional) vectors, one can bring geometric methods to bear in the analysis of the graph(s). We illustrate our approach on the Mathematicians Collaboration Graph [Grossman (1996) The ErdAs number project, http://www.oakland.edu/enp ] and the times series of Interstate Alliance Graphs (Gibler and Sarkees in J Peace Res 41(2):211-222, 2004).