The dirichlet problem for ∞-harmonic functions on a network

We define the notion of ∞-harmonic functions on a network as a discrete version of that on a euclidean domain, and show some properties of such functions. We discuss the Dirichlet problems for discrete ∞-harmonic functions. We also show that limits of discrete p-harmonic functions as p → ∞ are in factdiscrete ∞-harmonic.