Applications that can be described by variational models profit from all the advantages those models bring along. Both on the functional level as on the level of the associated differential equations, powerful techniques have been developed over the years to study these models. Up until fairly recently, such models were typically formulated in a continuum setting, i.e. as the minimization of a functional over an admissible class of functions whose domains are subsets of Euclidean space or Riemannian manifolds. The field of variational methods and partial differential equations (PDEs) on graphs aims to harness the power of variational methods and PDEs to tackle problems that inherently have a graph (network) structure.
In this talk we will encounter the graph Ginzburg--Landau model, which is a paradigmatic example of a variational model on graphs. Just as its continuum forebear is used to model phase separation on a continuum domain —it assigns to each point of the domain a value from an (approximately) discrete set of values— the graph Ginzburg–Landau model describes phase separation on the nodes of a graph. This makes it extremely well suited for applications such as data clustering, data classification, community detection in networks, and image segmentation.
Theoretically there are also interesting questions to ask, often driven by the properties that have already been established for the continuum Ginzburg–Landau model, such as Gamma-convergence properties of the functional and relationships between its associated differential equations. This presentation will give an overview of some recent developments.