In this paper, we revisit the notion of perimeter on graphs, introduced in [19], and we extend it to so-called inner and outer perimeters. We will also extend the notion of total variation on graphs. Thanks to the co-area formula, we show that discrete total variations can be expressed through these perimeters. Then, we propose a novel class of curvature operators on graphs that unifies both local and nonlocal mean curvature on an Euclidean domain. This leads us to translate and adapt the notion of the mean curvature flow on graphs as well as the level set mean curvature, which can be seen as approximate schemes. Finally, we exemplify the usefulness of these methods in image processing, 3D point cloud processing, and high dimensional data classification.

In this paper, we first introduce a new family of operators on weighted graphs called p-bilaplacian operators, which are the analogue on graphs of the continuous p-bilaplacian operators. We then turn to study regularized variational and boundary value problems associated to these operators. For instance, we study their well-posedness (existence and uniqueness). We also develop proximal splitting algorithms to solve these problems. We finally report numerical experiments to support our findings.

In this paper we study continuum limits of the discretized [[EQUATION]] -Laplacian evolution problem on sparse graphs with homogeneous Neumann boundary conditions. This goes far beyond known results by handling much more general class of kernels, possibly singular, and graph sequences whose limit are the so-called [[EQUATION]] -graphons. More precisely, we derive a bound on the distance between two continuous-in-time trajectories defined by two different evolution systems (i.e. with different kernels, second member and initial data). Similarly, we provide a bound in the case that one of the trajectories is discrete-in-time and the other is continuous. In turn, these results lead us to establish error estimates of the full discretization of the [[EQUATION]] -Laplacian problem on sparse random graphs. In particular, we provide rate of convergence of solutions for the discrete models to the solution of the continuous problem as the number of vertices grows.

In this Element, the authors consider fully discretized p-Laplacian problems (evolution, boundary value and variational problems) on graphs. The motivation of nonlocal continuum limits comes from the quest of understanding collective dynamics in large ensembles of interacting particles, which is a fundamental problem in nonlinear science, with applications ranging from biology to physics, chemistry and computer science. Using the theory of graphons, the authors give a unified treatment of all the above problems and establish the continuum limit for each of them together with non-asymptotic convergence rates. They also describe an algorithmic framework based proximal splitting to solve these discrete problems on graphs.