Clustering unlabeled point clouds is a fundamental problem in machine learning. One classical method for constructing clusters on graph-based data is to solve for Cheeger cuts, which balance between finding clusters that require cutting few graph edges and finding clusters which are similar in size. Although solving for Cheeger cuts on general graphs is very challenging, when the graph is constructed by sampling from a continuum domain one suspects that the problem may be more tractable. I will discuss recent work with Nicolás García Trillos and Matthew Thorpe which establishes quantified convergence rates of discrete Cheeger cuts on graphs over point clouds towards continuum Cheeger sets. This is accomplished by using i) novel estimates of the consistency of total variation energies on such graphs, and ii) recent stability results for classical, continuum isoperimetric problems. This mirrors a guiding principle that the stability of continuum variational problems can often be translated into the point cloud or geometric graph setting.