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Mean Curvature, Threshold Dynamics, and Phase Field Theory on Finite Graphs


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.


  1. R. A. Adams, Sobolev spaces, first ed., Pure and applied mathematics; a series of monographs and textbooks, vol. 65, Academic Press, Inc, New York, 1975.

  2. S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica 27 (1979), no. 6, 1085–1095.

    Google Scholar 

  3. F. Almgren, J. E. Taylor, and L. Wang, Curvature-driven flows: a variational approach, SIAM J. Control Optim. 31 (1993), no. 2, 387–438.

  4. L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems, first ed., Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000.

  5. G. Barles and C. Georgelin, A simple proof of convergence for an approximation scheme for computing motions by mean curvature, SIAM J. Numer. Anal. 32 (1995), no. 2, 484–500.

    Google Scholar 

  6. G. Barles, H. M. Soner, and P. E. Souganidis, Front propagation and phase field theory, SIAM J. Control Optim. 31 (1993), no. 2, 439–469.

  7. A. Barrat, M. Barthelemy, R. Pastor-Satorras, and A. Vespignani, The architecture of complex weighted networks, Proceedings of the National Academy of Sciences of the United States of America 101 (2004), no. 11, 3747–3752.

  8. P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal. 150 (1999), no. 4, 281–305.

  9. A. L. Bertozzi and A. Flenner, Diffuse interface models on graphs for analysis of high dimensional data, Multiscale Modeling and Simulation 10 (2012), no. 3, 1090–1118.

  10. T. Biyikoglu, J. Leydold, and P. F. Stadler, Laplacian eigenvectors of graphs, Springer, 2007.

  11. A. Björner, L. Lovász, and P. W. Shor, Chip-firing games on graphs, European J. Combin 12 (1991), no. 4, 283–291.

  12. T. Bühler and M. Hein, Spectral clustering based on the graph p-laplacian, Proceedings of the 26th Annual International Conference on Machine Learning, ACM, 2009, 81–88.

  13. M. Bradonjić and I. Saniee, Bootstrap percolation on random geometric graphs, Probability in the Engineering and Informational Sciences 28, (2014), no. 2, 169–181 (to appear).

  14. A. Braides, Γ-convergence for beginners, first ed., Oxford Lecture Series inMathematics and its Applications, vol. 22, Oxford University Press, Oxford, 2002.

  15. K. A. Brakke, The motion of a surface by its mean curvature, Mathematical Notes, vol. 20, Princeton University Press, Princeton, N.J., 1978.

  16. L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Differential Equations 90 (1991), no. 2, 211–237.

    Google Scholar 

  17. L. Caffarelli, J.-M. Roquejoffre, and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010), no. 9, 1111–1144.

  18. L. A. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation, Arch. Ration. Mech. Anal. 195 (2010), no. 1, 1–23.

    Google Scholar 

  19. G. Caginalp and E. Socolovsky, Phase field computations of single-needle crystals, crystal growth, and motion by mean curvature, SIAM Journal on Scientific Computing 15 (1994), no. 1, 106–126.

    Google Scholar 

  20. M. Calle, Mean curvature flow and minimal surfaces, Dissertation Department of Mathematics, New York University, UMI Number: 3283351 (2007).

  21. O. Candogan, I. Menache, A. Ozdaglar, and P. A. Parrilo, Flows and decompositions of games: harmonic and potential games, Math. Oper. Res. 36 (2011), no. 3, 474–503.

  22. M. C. Caputo and N. Guillen, Regularity for non-local almost minimal boundaries and applications, arXiv preprint 1003.2470 (2010).

  23. Chalupa J., Leath P. L., Reich G. R.: Bootstrap percolation on a Bethe lattice. J. Phys. C 12, L31–L35 (1979)

    Article  Google Scholar 

  24. A. Chambolle, Total variation minimization and a class of binary MRF models, Energy minimization methods in computer vision and pattern recognition, Springer, 2005, 136– 152.

  25. A. Chambolle and J. Darbon, On total variation minimization and surface evolution using parametric maximum flows, International Journal of Computer Vision 84 (2009), no. 3, 288–307.

    Google Scholar 

  26. A. Chambolle and M. Novaga, Convergence of an algorithm for the anisotropic and crystalline mean curvature flow, SIAM J. Math. Anal. 37 (2006), no. 6, 1978–1987.

    Google Scholar 

  27. T. F. Chan, S. Esedoḡlu, and M. Nikolova, Algorithms for finding global minimizers of image segmentation and denoising models, SIAM J. Appl. Math. 66 (2006), no. 5, 1632–1648.

    Google Scholar 

  28. R. Choksi, Y. van Gennip, and A. Oberman, Anisotropic total variation regularized L 1 approximation and denoising/deblurring of 2D bar codes, Inverse Probl. Imaging 5 (2011), no. 3, 591–617.

  29. F. R. K. Chung, Spectral graph theory, AMS, 1997.

  30. T. H. Colding and W. P. Minicozzi II, Minimal surfaces and mean curvature flow, arXiv preprint 1102.1411 (2011).

  31. G. Dal Maso, An introduction to Γ-convergence, first ed., Progress in Nonlinear Differential Equations and Their Applications, vol. 8, Birkhäuser, Boston, 1993.

  32. X. Desquesnes, A. Elmoataz, and O. Lézoray, PDEs level sets on weighted graphs, Image Processing (ICIP), 2011 18th IEEE International Conference on, IEEE, 2011, 3377–3380.

  33. X. Desquesnes, A. Elmoataz, and O. Lézoray, Eikonal equation adaptation on weighted graphs: Fast geometric diffusion process for local and non-local image and data processing, Journal of Mathematical Imaging and Vision (2012), 1–20.

  34. X. Desquesnes, A. Elmoataz, and O. Lézoray, Generalized fronts propagation on weighted graphs, Proceedings of ALGORITMY, 2012, 371–381.

  35. X. Desquesnes, A. Elmoataz, and O. Lézoray, PdEs-based morphology on graphs for cytological slides segmentation and clustering, Biomedical Imaging (ISBI), 2012 9th IEEE International Symposium on, IEEE, 2012, 1619–1622.

  36. X. Desquesnes, A. Elmoataz, O. Lézoray, and Vinh-Thong Ta, Efficient algorithms for image and high dimensional data processing using eikonal equation on graphs, Advances in Visual Computing, Springer, 2010, 647–658.

  37. R. Durrett, Random graph dynamics, vol. 20, Cambridge university press, 2007.

  38. K. Ecker, Regularity theory for mean curvature flow, vol. 57, Springer, 2004.

  39. I. Ekeland and R. Temam, Convex analysis and variational problems, North-Holland Publishing Co., Amsterdam, 1976, Translated from the French, Studies in Mathematics and its Applications, Vol. 1.

  40. A. Elmoataz, X. Desquesnes, and O. Lézoray, Non-local morphological PDEs and p- Laplacian equation on graphs with applications in image processing and machine learning, IEEE Journal of Selected Topics in Signal Processing 6 (2012), no. 7, 764–779.

    Google Scholar 

  41. A. Elmoataz, O. Lezoray, and S. Bougleux, Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing, Image Processing, IEEE Transactions on 17 (2008), no. 7, 1047–1060.

  42. S. Esedoḡlu and F. Otto, Threshold dynamics for networks with arbitrary surface tensions (to appear).

  43. S. Esedoḡlu, S. J. Ruuth, and R. Tsai, Threshold dynamics for high order geometric motions, Interfaces and Free Boundaries 10 (2008), no. 3, 263–282.

  44. S. Esedoḡlu, S. Ruuth, and R. Tsai, Diffusion generated motion using signed distance functions, Journal of Computational Physics 229 (2010), no. 4, 1017–1042.

  45. S. Esedoḡlu and Y.-H. R. Tsai, Threshold dynamics for the piecewise constant Mumford-Shah functional, Journal of Computational Physics 211 (2006), no. 1, 367– 384.

  46. L. C. Evans, Convergence of an algorithm for mean curvature motion, Indiana Univ. Math. J. 42 (1993), no. 2, 533–557.

  47. L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, first ed., Studies in Advanced Mathematics, CRC Press LLC, Boca Raton, Florida, 1992.

  48. Fiedler M.: Algebraic connectivity of graphs. Czech. Math. J. 23, 298–305 (1973)

    MathSciNet  Google Scholar 

  49. J. Friedman and J. P. Tillich, Calculus on graphs, arXiv preprint cs/0408028 (2004).

  50. T. Funaki and H. Spohn, Motion by mean curvature from the Ginzburg-Landau ∇ϕ interface model, Comm. Math. Phys. 185 (1997), no. 1, 1–36.

    Google Scholar 

  51. C. Garcia-Cardona, E. Merkurjev, A. L. Bertozzi, A. Flenner, and A. Percus, Fast multiclass segmentation using diffuse interface methods on graphs, submitted (2013).

  52. R. Ghosh, K. Lerman, T. Surachawala, K. Voevodski, and S. Teng, Non-conservative diffusion and its application to social network analysis, arXiv preprint 1102.4639 (2011).

  53. R. Ghosh and K. Lerman, Rethinking centrality: the role of dynamical processes in social network analysis, arXiv preprint 1209.4616 (2012).

  54. G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling & Simulation 7 (2009), no. 3, 1005–1028.

  55. E. Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics 80, Springer Science+Business Media, New York, 1984.

  56. L. J. Grady and J. R. Polimeni, Discrete calculus, Springer-Verlag London Ltd., 2010.

  57. S. Guattery and G. L. Miller, On the quality of spectral separators, SIAM J. Matrix Anal. Appl. 19 (1998), no. 3, 701–719.

  58. Hele-Shaw H. S.: Flow of water. Nature 58, 520 (1898)

    Article  Google Scholar 

  59. R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge University Press, 1990.

  60. H. Hu, T. Laurent, M. A. Porter, and A. L. Bertozzi, A method based on total variation for network modularity optimization using the MBO scheme, submitted (2013).

  61. H. J. Hupkes, D. Pelinovsky, and B. Sandstede, Propagation failure in the discrete Nagumo equation, Proc. Amer. Math. Soc. 139 (2011), no. 10, 3537–3551.

  62. C. Imbert, Level set approach for fractional mean curvature flows, Interfaces and Free Boundaries 11 (2009), no. 1, 153–176.

  63. S. Janson, T. Łuczak, T. Turova, and T. Vallier, Bootstrap percolation on the random graph G n,p , Ann. Appl. Probab. 22 (2012), no. 5, 1989–2047.

  64. R. Kannan, S. Vempala, and A. Vetta, On clusterings—good, bad and spectral, 41st Annual Symposium on Foundations of Computer Science (Redondo Beach, CA, 2000), IEEE Comput. Soc. Press, Los Alamitos, CA, 2000, 367–377.

  65. R. Kannan, S. Vempala, and A. Vetta, On clusterings: good, bad and spectral, J. ACM 51 (2004), no. 3, 497–515.

  66. M. A. Katsoulakis and P. E. Souganidis, Interacting particle systems and generalized evolution of fronts, Archive for rational mechanics and analysis 127 (1994), no. 2, 133–157.

  67. M. A. Katsoulakis and P. E. Souganidis, Stochastic ising models and anisotropic front propagation, Journal of statistical physics 87 (1997), no. 1-2, 63–89.

  68. S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations 3 (1995), no. 2, 253–271.

  69. R. Lyons, The ising model and percolation on trees and tree-like graphs, Communications in Mathematical Physics 125 (1989), no. 2, 337–353.

  70. R. Lyons, Phase transitions on nonamenable graphs. (english summary), J. Math. Phys 41 (2000), no. 3, 1099–1126.

  71. J. J. Manfredi, A. M. Oberman, and A. P. Sviridov, Nonlinear elliptic partial differential equations and p-harmonic functions on graphs, Preprint (2012).

  72. P. Mascarenhas, Diffusion generated motion by mean curvature, UCLA Department of Mathematics CAM report CAM 92–33 (1992).

  73. Meeks W.H. III, Pérez J.: The classical theory of minimal surfaces. Bull. Amer. Math. Soc. 48, 325–407 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  74. E. Merkurjev, T. Kostić, and A. L. Bertozzi, An MBO scheme on graphs for segmentation and image processing, submitted (2012).

  75. B. Merriman, J. K. Bence, and S. Osher, Diffusion generated motion by mean curvature, UCLA Department of Mathematics CAM report CAM 06–32 (1992).

  76. B. Merriman, J.K. Bence, and S. Osher, Diffusion generated motion by mean curvature, AMS Selected Letters, Crystal Grower’s Workshop (1993), 73–83.

  77. B. Merriman, J. K. Bence, and S. J. Osher, Motion of multiple functions: a level set approach, J. Comput. Phys. 112 (1994), no. 2, 334–363.

    Google Scholar 

  78. L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal. 98 (1987), no. 2, 123–142.

  79. L. Modica and S. Mortola, Un esempio di Γ-convergenza, Bollettino U.M.I. 5 (1977), no. 14-B, 285–299.

  80. B. Mohar, The Laplacian spectrum of graphs, Graph Theory, Combinatorics, and Applications, vol. 2, Wiley, 1991, 871–898.

  81. W. W. Mullins and R. F. Sekerka, Stability of a planar interface during solidification of a dilute binary alloy, Journal of Applied Physics 35 (1964), no. 2, 444–451.

    Google Scholar 

  82. A. Ng, M. Jordan, and Y. Weiss, On spectral clustering: Analysis and an algorithm, Dietterich, T., Becker, S., Ghahramani, Z. (eds.) Advances in Neural Information Processing Systems 14, MIT Press, Cambridge, 2002, 849–856.

  83. Oberman A. M.: A convergent monotone difference scheme for motion of level sets by mean curvature. Numer. Math. 99, 365–279 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  84. R. Olfati-Saber, A. Fax, and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proceedings of the IEEE 95 (2007), no. 1, 215–233.

  85. S. J. Osher and R. Fedkiw, Level set methods and dynamic implicit surfaces, vol. 153, Springer Verlag, 2003.

  86. B. Osting, C. Brune, and S. Osher, Enhanced statistical rankings via targeted data collection, JMLR, W&CP 28 (2013), no. 1, 489–497.

  87. R. L. Pego, Front migration in the nonlinear cahn-hilliard equation, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 422 (1989), no. 1863, 261–278.

  88. N. Peters, Turbulent combustion, Cambridge university press, 2000.

  89. M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, 1979.

  90. L. I. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, 60 (1992), no. 1–4, 259–268.

  91. S. J. Ruuth, Efficient algorithms for diffusion-generated motion by mean curvature, ProQuest LLC, Ann Arbor, MI, 1996, Thesis (Ph.D.)–The University of British Columbia (Canada).

  92. S. J. Ruuth, A diffusion-generated approach to multiphase motion, Journal of Computational Physics 145 (1998), no. 1, 166–192.

  93. S. J. Ruuth, Efficient algorithms for diffusion-generated motion by mean curvature, Journal of Computational Physics 144 (1998), no. 2, 603–625.

  94. D. Sha, Gossip algorithms, Foundations and Trends in Networking 3 (2008), no. 1, 1–125.

  95. J. Shi and J. Malik, Normalized cuts and image segmentation, IEEE Transactions on Pattern Analysis and Machine Intelligence 22 (2000), no. 8, 888–905.

    Google Scholar 

  96. L. M. Smith, K. Lerman, C. Garcia-Cardona, A. G. Percus, and R. Ghosh, Spectral clustering with epidemic diffusion, arXiv preprint 1303.2663 (2013).

  97. H. M. Soner and N. Touzi, A stochastic representation for mean curvature type geometric flows, The Annals of probability 31 (2003), no. 3, 1145–1165.

  98. D. A. Spielman and S.-H. Teng, Spectral partitioning works: planar graphs and finite element meshes, 37th Annual Symposium on Foundations of Computer Science (Burlington, VT, 1996), IEEE Comput. Soc. Press, Los Alamitos, CA, 1996, 96–105.

  99. D. A. Spielman and S.-H. Teng, Spectral partitioning works: planar graphs and finite element meshes, Linear Algebra Appl. 421 (2007), no. 2-3, 284–305.

  100. J. Sun, S. Boyd, L. Xiao, and P. Diaconis, The fastest mixing Markov process on a graph and a connection to a maximum variance unfolding problem, SIAM Review 48 (2004), 2006.

  101. J. Szarski, Differential inequalities, PWN Warsaw, 1965.

  102. A. Szlam and X. Bresson, Total variation and cheeger cuts, Proceedings of the 27th International Conference on Machine Learning (ICML-10) (Haifa, Israel) (Johannes Fürnkranz and Thorsten Joachims, eds.), Omnipress, June 2010, 1039–1046.

  103. V.-T. Ta, A. Elmoataz, and O. Lézoray, Partial difference equations over graphs: Morphological processing of arbitrary discrete data, Computer Vision–ECCV 2008, Springer, 2008, 668–680.

  104. V.-T. Ta, A. Elmoataz, and O. Lézoray, Nonlocal PDEs-based morphology on weighted graphs for image and data processing, Image Processing, IEEE Transactions on 20 (2011), no. 6, 1504–1516.

  105. J. E. Taylor, II—mean curvature and weighted mean curvature, Acta Metallurgica et Materialia 40 (1992), no. 7, 1475–1485.

  106. R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, second ed., Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997.

  107. Y. van Gennip and A. L. Bertozzi, Γ-convergence of graph Ginzburg-Landau functionals, Adv. Differential Equations 17 (2012), no. 11–12, 1115–1180.

  108. U. von Luxburg, A tutorial on spectral clustering, Statistics and Computing 17 (2007), no. 4, 395–416.

  109. D. Wagner and F. Wagner, Between min cut and graph bisection, Mathematical foundations of computer science 1993 (Gdańsk, 1993), Lecture Notes in Comput. Sci., vol. 711, Springer, Berlin, 1993, 744–750.

  110. D. J. Watts and S. H. Strogatz, Collective dynamics of ’small-world’ networks., Nature 393 (1998), no. 6684, 440–442.

  111. J. Xin and Y. Yu, Analysis and comparison of large time front speeds in turbulent combustion models, arXiv preprint 1105.5607 (2011).

  112. B. A Zalesky, Network flow optimization for restoration of images, Journal of Applied Mathematics 2 (2002), no. 4, 199–218.

    Google Scholar 

  113. J. Zhang and Q. Du, Numerical studies of discrete approximations to the Allen-Cahn equation in the sharp interface limit, SIAM Journal on Scientific Computing 31 (2009), no. 4, 3042–3063.

    Google Scholar 

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Correspondence to Andrea L. Bertozzi.

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Lecture given by A. Bertozzi in the conference “Mathematics in a Complex World”, Politecnico di Milano, Feb. 28-March 1, 2013

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van Gennip, Y., Guillen, N., Osting, B. et al. Mean Curvature, Threshold Dynamics, and Phase Field Theory on Finite Graphs. Milan J. Math. 82, 3–65 (2014).

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Mathematics Subject Classification (2010)

  • 34B45
  • 35R02
  • 53C44
  • 53A10
  • 49K15
  • 49Q05
  • 35K05


  • Spectral graph theory
  • Allen-Cahn equation
  • Ginzburg-Landau functional
  • Merriman-Bence-Osher threshold dynamics
  • graph cut function
  • total variation
  • mean curvature flow
  • nonlocal mean curvature
  • gamma convergence
  • graph coarea formula