Gradient flows on graphons: existence, convergence, continuity equations

TL;DR

This paper establishes the convergence of gradient flows on large graphs to a continuum limit on graphons, extending the theory of Wasserstein gradient flows to graph-based structures.

Contribution

It introduces a novel continuum limit for gradient flows on graphons, bridging the gap between particle systems and graphon-based models.

Findings

Gradient flows on graphs converge to graphon curves as size increases.

The framework applies to functions like homomorphism counts and entropy.

Detailed examples demonstrate the theory's applicability.

Abstract

Wasserstein gradient flows on probability measures have found a host of applications in various optimization problems. They typically arise as the continuum limit of exchangeable particle systems evolving by some mean-field interaction involving a gradient-type potential. However, in many problems, such as in multi-layer neural networks, the so-called particles are edge weights on large graphs whose nodes are exchangeable. Such large graphs are known to converge to continuum limits called graphons as their size grow to infinity. We show that the Euclidean gradient flow of a suitable function of the edge-weights converges to a novel continuum limit given by a curve on the space of graphons that can be appropriately described as a gradient flow or, more technically, a curve of maximal slope. Several natural functions on graphons, such as homomorphism functions and the scalar entropy, are covered by our set-up, and the examples have been worked out in detail.