The infinite network \(\mathcal {N}\) is a discrete space, with a collection of finite or a countable infinite number of vertices that has a graphical structure provided by a set of edges (finite or countable infinite in number). In many cases, these varying graph structures (connectivity-type problems) are fascinating and important. The study of the intricate role of functions on \(\mathcal {N}\) is essential for some important examples (example: potential functions, effective resistance, Kirchhoff problem in electrical networks, and escape probability, Dirichlet functions, hitting time in random walks). In this article, we review a part of the function theory developed by some researchers in this field and present a cohesive narrative. We have placed special emphasis on different discrete versions of the Dirichlet problem, the Neumann problem, and the Poisson equation.
