Γ-convergence of Graph Ginzburg–landau Functionals

We study Γ-convergence of graph-based Ginzburg–Landau functionals, both the limit for zero diffusive interface parameter ε → 0 and the limit for infinite nodes in the graph m→∞. For general graphs we prove that in the limit ε → 0 the graph cut objective function is recovered. We show that the continuum limit of this objective function on 4-regular graphs is related to the total variation seminorm and compare it with the limit of the discretized Ginzburg–Landau functional. For both functionals we also study the simultaneous limit ε → 0 and m → ∞, by expressing ε as a power of m and taking m → ∞. Finally we investigate the continuum limit for a nonlocal means-type functional on a completely connected graph.