A Wiener criterion at infinity for $p$-massiveness on weighted graphs

We study boundary value problems at infinity for the graph $p$-Laplacian on infinite, connected, locally finite weighted graphs. Our main result is a Wiener criterion for $p$-massiveness. Assuming volume doubling and a weak $(1,p)$-Poincaré inequality, we show that every infinite connected $p$-massive set satisfies a dyadic capacitary condition expressed through relative $p$-capacities in nested balls; under the additional $(p_0)$ condition, the converse also holds. This yields a nonlinear criterion at the point at infinity in a rough weighted-graph setting and extends the Wiener viewpoint to a nonlinear discrete framework. We also prove, without these geometric assumptions, that $p$-massiveness is equivalent to a strengthened nonuniqueness property for exterior Dirichlet problems. As a further consequence, bounded nonconstant $p$-harmonic functions are characterized by the existence of two disjoint massive sets. In this way, the Wiener criterion is placed in a broader and more flexible picture of exterior boundary behavior and Liouville-type phenomena on weighted graphs.