  Exchangeable random graphs serve as an important probabilistic framework for
the statistical analysis of network data. In this work we develop an
alternative parameterization for a large class of exchangeable random graphs,
where the nodes are independent random vectors in a linear space equipped with
an indefinite inner product, and the edge probability between two nodes equals
the inner product of the corresponding node vectors. Therefore, the
distribution of exchangeable random graphs in this subclass can be represented
by a node sampling distribution on this linear space, which we call the graph
root distribution. We study existence and identifiability of such
representations, the topological relationship between the graph root
distribution and the exchangeable random graph sampling distribution, and
estimation of graph root distributions.
