The problem of optimizing the edge weights of large networks arises in many contexts, for example, machine learning, extremal graph theory, and large deviations of exponential random graphs. Finding the exact minimizers in these problems is often very difficult. In practice, various dynamic optimization schemes such as gradient descent or stochastic gradient descent are used to find the approximate minimizers. From a practical viewpoint, it is often important to study these algorithms or optimization problems in large dimensions. These algorithms yield a process of graphs/matrices and it is natural to investigate the limits of these processes as the dimension goes to infinity. This is done via taking the graphon limit of these processes.

In the first part of this talk, we give several motivating examples where such optimization problems arise and the algorithms that are used in practice. We give a brief introduction to the theory of graphons and recast these problems in the framework of graphons. In the second part of the talk, we introduce the notion of gradient flow on graphons and argue the existence and uniqueness of gradient flow for a rich class of functions. We then show that a large class of algorithms on large graphs does have a graphon limit and the limit can be described as the gradient flow or regularization. Time permitting, we will discuss some applications and limitations of this approach and end with several problems that remain to be investigated.