Phase-field models on graphs are a discrete analogue to phase-field models, defined on a graph. They are used in image analysis (for feature identification) and for the segmentation of social networks.
For a graph with vertices V and edge weights , the graph GinzburgâÂÂLandau functional of a map is given by
where W is a double well potential, for example the quartic potential W(x) = x<sup>2</sup>(1 â x<sup>2</sup>). The graph GinzburgâÂÂLandau 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<sub>j</sub> close to 0 or close to 1, and for small , minimisers of will satisfy that u<sub>j</sub> is close to 0 or 1 for most nodes, splitting the nodes into two classes.
To effectively minimise , a natural approach is by gradient flow (steepest descent). This means to introduce an artificial time parameter and to solve the graph version of the AllenâÂÂCahn equation,
where is the graph Laplacian. The ordinary continuum AllenâÂÂCahn equation and the graph AllenâÂÂCahn equation are natural counterparts, just replacing ordinary calculus by calculus on graphs. A convergence result for a numerical graph AllenâÂÂCahn scheme has been established by Luo and Bertozzi.
It is also possible to adapt other computational schemes for mean curvature flow, for example schemes involving thresholding like the MerrimanâÂÂBenceâÂÂOsher scheme, to a graph setting, with analogous results.