Quasi-bounded and singular functions in an infinite network

In a hyperbolic network X, the properties of non-negative quasi-bounded harmonic functions and singular harmonic functions are considered by using the notion of reduced functions (balayage). Later some extensions of these results are given with respect to the class of non-negative quasi-bounded super-harmonic functions and the class of non-negative q-harmonic functions defined by discrete Schrödinger operators on X.