The type-problem on the average for random walks on graphs

When averages over all starting points are considered, the type problem for the recurrence or transience of a simple random walk on an inhomogeneous n etwork in general differs from the usual "local" type problem. This differe nce leads to a new classification of inhomogeneous discrete structures in t erms of recurrence and transience oa the average, describing their large sc ale topology from a ''statistical" point of view. In this paper we analyze this classification and the properties connected to it, showing how the ave rage behavior affects the thermodynamic properties of statistical models on graphs.