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 network in general differs from the usual "local" type problem. This difference leads to a new classification of inhomogeneous discrete structures in terms of recurrence and transience oa the average, describing their large scale topology from a ''statistical" point of view. In this paper we analyze this classification and the properties connected to it, showing how the average behavior affects the thermodynamic properties of statistical models on graphs.