  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
{\it recurrence} and {\it transience} {\it on 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.
