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Phase-field models on graphs are a discrete analogue to phase-field models, defined on a
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
. They are used in
image analysis Image analysis or imagery analysis is the extraction of meaningful information from images; mainly from digital images by means of digital image processing techniques. Image analysis tasks can be as simple as reading bar coded tags or as sophi ...
(for feature identification) and for the segmentation of
social network A social network is a social structure made up of a set of social actors (such as individuals or organizations), sets of dyadic ties, and other social interactions between actors. The social network perspective provides a set of methods for an ...
s.


Graph Ginzburg–Landau functional

For a graph with vertices ''V'' and edge weights \omega_, the graph Ginzburg–Landau functional of a map u:V\to \mathbb is given by :F_\varepsilon(u) = \frac\varepsilon2 \sum_ \omega_ (u_i-u_j)^2 + \frac1\varepsilon \sum_ W(u_i), where ''W'' is a double well potential, for example the quartic potential ''W''(''x'') = ''x''2(1 − ''x''2). The graph Ginzburg–Landau functional was introduced by Bertozzi and Flenner. In analogy to continuum phase-field models, where regions with ''u'' close to 0 or 1 are models for two phases of the material, vertices can be classified into those with ''u''''j'' close to 0 or close to 1, and for small \varepsilon, minimisers of F_\varepsilon will satisfy that ''u''''j'' is close to 0 or 1 for most nodes, splitting the nodes into two classes.


Graph Allen–Cahn equation

To effectively minimise F_\varepsilon, a natural approach is by gradient flow (
steepest descent In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the ...
). This means to introduce an artificial time parameter and to solve the graph version of the
Allen–Cahn equation The Allen–Cahn equation (after John W. Cahn and Sam Allen) is a reaction–diffusion equation of mathematical physics which describes the process of phase separation in multi-component alloy systems, including order-disorder transitions. The eq ...
, :\frac u_j = -\varepsilon (\Delta u)_j-\frac1\varepsilon W'(u_j), where \Delta is the
graph Laplacian In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Named after Pierre-Simon Laplace, the graph Lapl ...
. The ordinary continuum Allen–Cahn equation and the graph Allen–Cahn equation are natural counterparts, just replacing ordinary calculus by calculus on graphs. A convergence result for a numerical graph Allen–Cahn scheme has been established by Luo and Bertozzi. It is also possible to adapt other computational schemes for
mean curvature flow In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold (for example, smooth surfaces in 3-dimensional Euclidean space). Intuitively, a family of surf ...
, for example schemes involving thresholding like the Merriman–Bence–Osher scheme, to a graph setting, with analogous results.van Gennip, Yves. Graph Ginzburg–Landau: discrete dynamics, continuum limits, and applications. An overview. In


See also

*
Graph cuts in computer vision As applied in the field of computer vision, graph cut optimization can be employed to efficiently solve a wide variety of low-level computer vision problems (''early vision''), such as image smoothing, the stereo correspondence problem, image segm ...


References

{{reflist Graph theory Mathematical modeling