  Large ensembles of stochastically evolving interacting particles describe
phenomena in diverse fields including statistical physics, neuroscience,
biology, and engineering. In such systems, the infinitesimal evolution of each
particle depends only on its own state (or history) and the states (or
histories) of neighboring particles with respect to an underlying, possibly
random, interaction graph. While these high-dimensional processes are typically
too complex to be amenable to exact analysis, their dynamics are quite well
understood when the interaction graph is the complete graph. In this case,
classical theorems show that in the limit as the number of particles goes to
infinity, the dynamics of the empirical measure and the law of a typical
particle coincide and can be characterized in terms of a much more tractable
dynamical system of reduced dimension called the mean-field limit. In contrast,
until recently not much was known about corresponding convergence results in
the complementary case when the interaction graph is sparse (i.e., with
uniformly bounded average degree). This article provides a brief survey of
classical work and then describes recent progress on the sparse regime that
relies on a combination of techniques from random graph theory, Markov random
fields, and stochastic analysis. The article concludes by discussing
ramifications for applications and posing several open problems.
