This chapter surveys recent work in applying ideas from graphical models and message
passing algorithms to solve large-scale regularized regression problems. In particular, the …

“Approximate message passing”(AMP) algorithms have proved to be effective in
reconstructing sparse signals from a small number of incoherent linear measurements …

The natural habitat of most Bayesian methods is data represented by exchangeable
sequences of observations, for which de Finetti's theorem provides the theoretical …

We investigate unimodular random networks. Our motivations include their characterization
via reversibility of an associated random walk and their similarities to unimodular quasi …

Statistical network modelling has focused on representing the graph as a discrete structure,
namely the adjacency matrix. When assuming exchangeability of this array—which can aid …

We introduce and develop a theory of limits for sequences of sparse graphs based on $ L^ p
$ graphons, which generalizes both the existing $ L^\infty $ theory of dense graph limits and …

These expository notes are centered around the circular law theorem, which states that the
empirical spectral distribution of an× n random matrix with iid entries of variance 1/n tends to …

We give new formulas for the asymptotics of the number of spanning trees of a large graph.
A special case answers a question of McKay [Europ. J. Combin. 4 149–160] for regular …

In certain problems in a variety of applied probability settings (from probabilistic analysis of
algorithms to statistical physics), the central requirement is to solve a recursive distributional …

This book focuses on percolation on high-dimensional lattices. We give a general
introduction to percolation, stating the main results and defining the central objects. We …