They are used in image analysis (for feature identification) and for the segmentation of social networks.
The graph Ginzburg–Landau functional was introduced by Bertozzi and Flenner.
, 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 Allen–Cahn equation, where
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.