In the continuum, close connections exist between mean curvature flow, the Allen-Cahn (AC) partial differential equation, and the Merriman-Bence-Osher (MBO) threshold dynamics scheme. Graph analogues of these processes have recently seen a rise in popularity as relaxations of NP-complete combinatorial problems, which demands deeper theoretical underpinnings of the graph processes. The aim of this paper is to introduce these graph processes in the light of their continuum counterparts, provide some background, prove the first results connecting them, illustrate these processes with examples and identify open questions for future study.We derive a graph curvature from the graph cut function, the natural graph counterpart of total variation (perimeter). This derivation and the resulting curvature definition differ from those in earlier literature, where the continuum mean curvature is simply discretized, and bears many similarities to the continuum nonlocal curvature or nonlocal means formulation. This new graph curvature is not only relevant for graph MBO dynamics, but also appears in the variational formulation of a discrete time graph mean curvature flow.We prove estimates showing that the dynamics are trivial for both MBO and AC evolutions if the parameters (the timestep and diffuse interface scale, respectively) are sufficiently small (a phenomenon known as "freezing" or "pinning") and also that the dynamics for MBO are nontrivial if the time step is large enough. These bounds are in terms of graph quantities such as the spectrum of the graph Laplacian and the graph curvature. Adapting a Lyapunov functional for the continuum

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Abstract. In the continuum, close connections exist between mean curvature flow, the Allen-Cahn (AC) partial differential equation, and the Merriman-Bence-Osher (MBO) threshold dynamics scheme. Graph analogues of these processes have recently seen a rise in popularity as relaxations of NP-complete combinatorial problems, which demands deeper theoretical underpinnings of the graph processes. The aim of this paper is to introduce these graph processes in the light of their continuum counterparts, provide some background, prove the first results connecting them, illustrate these processes with examples and identify open questions for future study.We derive a graph curvature from the graph cut function, the natural graph counterpart of total variation (perimeter). This derivation and the resulting curvature definition differ from those in earlier literature, where the continuum mean curvature is simply discretized, and bears many similarities to the continuum nonlocal curvature or nonlocal means formulation. This new graph curvature is not only relevant for graph MBO dynamics, but also appears in the variational formulation of a discrete time graph mean curvature flow.We prove estimates showing that the dynamics are trivial for both MBO and AC evolutions if the parameters (the time-step and diffuse interface scale, respectively) are sufficiently small (a phenomenon known as "freezing" or "pinning") and also that the dynamics for MBO are nontrivial if the time step is large enough. These bounds are in terms of graph quantities such as the spectrum of the graph Laplacian and the graph curvature. Adapting a Lyapunov functional for the continuum MBO scheme to graphs, we prove that the graph MBO scheme converges to a stationary state in a finite number of iterations. Variations on this scheme have recently become popular in the literature as ways to minimize (continuum) nonlocal total variation.Mathematics Subject Classification (2010). 34B45, 35R02, 53C44, 53A10, 49K15, 49Q05, 35K05.

We consider variations of the Rudin-Osher-Fatemi functional which are particularly well-suited to denoising and deblurring of 2D bar codes. These functionals consist of an anisotropic total variation favoring rectangles and a fidelity term which measure the L 1 distance to the signal, both with and without the presence of a deconvolution operator. Based upon the existence of a certain associated vector field, we find necessary and sufficient conditions for a function to be a minimizer. We apply these results to 2D bar codes to find explicit regimes -in terms of the fidelity parameter and smallest length scale of the bar codes -for which the perfect bar code is attained via minimization of the functionals. Via a discretization reformulated as a linear program, we perform numerical experiments for all functionals demonstrating their denoising and deblurring capabilities.

In this article we identify social communities among gang members in the Hollenbeck policing district in Los Angeles, based on sparse observations of a combination of social interactions and geographic locations of the individuals. This information, coming from LAPD Field Interview cards, is used to construct a similarity graph for the individuals. We use spectral clustering to identify clusters in the graph, corresponding to communities in Hollenbeck, and compare these with the LAPD's knowledge of the individuals' gang membership. We discuss different ways of encoding the geosocial information using a graph structure and the influence on the resulting clusterings. Finally we analyze the robustness of this technique with respect to noisy and incomplete data, thereby providing suggestions about the relative importance of quantity versus quality of collected data.