Edge-exchangeable graphs and sparsity
Abstract
A known failing of many popular random graph models is that the Aldous-Hoover Theorem guarantees these graphs are dense with probability one; that is, the number of edges grows quadratically with the number of nodes. This behavior is considered unrealistic in observed graphs. We define a notion of edge exchangeability for random graphs in contrast to the established notion of infinite exchangeability for random graphs --- which has traditionally relied on exchangeability of nodes (rather than edges) in a graph. We show that, unlike node exchangeability, edge exchangeability encompasses models that are known to provide a projective sequence of random graphs that circumvent the Aldous-Hoover Theorem and exhibit sparsity, i.e., sub-quadratic growth of the number of edges with the number of nodes. We show how edge-exchangeability of graphs relates naturally to existing notions of exchangeability from clustering (a.k.a. partitions) and other familiar combinatorial structures.
Cite
@article{arxiv.1603.06898,
title = {Edge-exchangeable graphs and sparsity},
author = {Tamara Broderick and Diana Cai},
journal= {arXiv preprint arXiv:1603.06898},
year = {2016}
}
Comments
This paper appeared in the NIPS 2015 Workshop on Networks in the Social and Information Sciences, http://stanford.edu/~jugander/NetworksNIPS2015/. An earlier version appeared in the NIPS 2015 Workshop Bayesian Nonparametrics: The Next Generation, https://sites.google.com/site/nipsbnp2015/