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 {\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. 
R. Burioni, D. Cassi, A. Vezzani