Various modes of convergence can be defined to control the asymptotic `global' properties of a growing sequence of sparse graphs. Variants of these that allow directed and labeled edges have appeared independently in the study of sofic entropy in ergodic theory. In both settings, the theory has run into fundamental open questions about the convergence of some basic examples, especially random regular graphs. In the first part, I will sketch some of these questions with their background. In the second part, I will describe some analogs for unitary representations that have answers in terms of operator algebras. I will also indicate some applications to the study of random matrices if time allows.

(A more elementary account of some of these topics will occupy the Mordell lecture on Thursday.)