Abstract

There are numerous examples of sparse massive networks, in particular the Internet, WWW and online social networks.  How do we model and learn these networks?  In contrast to conventional learning problems, where we have many independent samples, it is often the case for these networks that we can get only one independent sample.  How do we use a single snapshot today to learn a model for the network, and therefore be able to predict a similar, but larger network in the future?  In the case of relatively small or moderately sized networks, it’s appropriate to model the network parametrically, and attempt to learn these parameters.  For massive networks, a non-parametric representation is more appropriate.  In this talk, we first review the theory of graphons, developed over the last decade to describe limits of dense graphs, and the more the recent theory describing sparse graphs of unbounded average degree, including power-law graphs.  We then show how to use these graphons as nonparametric models for sparse networks.  Finally, we show how to get consistent estimators of these non-parametric models, and moreover how to do this in a way that protects the privacy of individuals on the network.   

Part I of this talk reviews the theory of graph convergence for dense and sparse graphs.  Part II uses the results of Part I to model and estimate sparse massive networks.