# The Nonlocal *p*-Laplacian Evolution Problem on Graphs: The Continuum Limit

## Abstract

The non-local *p*-Laplacian evolution equation, governed by given kernel, has various applications to model diffusion phenomena, in particular in signal and image processing. In practice, such an evolution equation is implemented in discrete form (in space and time) as a numerical approximation to a continuous problem, where the kernel is replaced by an adjacency matrix of graph. The natural question that arises is to understand the structure of solutions to the discrete problem, and study their continuous limit. This is the goal pursued in this work. Combining tools from graph theory and non-linear evolution equations, we give a rigorous interpretation to the continuous limit of the discrete *p*-Laplacian on graphs. More specifically, we consider a sequence of deterministic simple/weighted graphs converging to a so-called *graphon*. The continuous *p*-Laplacian evolution equation is then discretized on this graph sequence both in space and time. We therefore prove that the solutions of the sequence of discrete problems converge to the solution of the continuous evolution problem governed by the graphon, when the number of graph vertices grows to infinity. We exhibit the corresponding convergence rates for different graph models, and point out the role of the graphon geometry and the parameter *p*.

## Keywords

Nonlocal diffusion p-Laplacian Graphs Graph limits Numerical approximation## References

- 1.Andreu-Vaillo, F., Mazón, J.M., Rossi, J.D., Toledo-Melero, J.J.: Nonlocal Diffusion Problems American Mathematical Society Mathematical Surveys and Monographs, vol. 165 (2010)Google Scholar
- 2.Lovasz, L., Sezgedy, B.: Limits of dense graph sequences. J. Combin. Theory Ser. B
**96**, 933–957 (2006)MathSciNetCrossRefGoogle Scholar - 3.Hafiene, Y., Fadili, M.J., Elmoataz, A.: Nonlocal p- laplacian evolution problems on graphs. Technical report arXiv :1612.07156 (2016)
- 4.Borgs, C., Chayes, J., Lovász, L., Sós, V., Vesztergombi, K.: Limits of randomly grown graph sequences. Eur. J. Comb.
**32**, 985–999 (2011)MathSciNetCrossRefGoogle Scholar - 5.Medvedev, G.S.: The nonlinear heat equation on dense graphs. SIAM J. Math. Anal.
**46**, 2743–2766 (2014)MathSciNetCrossRefGoogle Scholar - 6.Kindermann, S., Osher, S., Jones, P.W.: Deblurring and denoising of images by nonlocal functionals. SIAM J. Multiscale Model. Simul.
**4**, 1091–1115 (2005)MathSciNetCrossRefGoogle Scholar