It is now well-known that multiclass networks may be unstable even under the «usual conditions» of stability (when the loads are less than one at all queues), but the proofs of transience (in the Markovian case) generally require a complex work based on the dynamics of an associated «fluid model». Here we develop a sample-path argument introduced in a previous paper, which provides new ergodicity conditions for networks ruled by {\em priorities}; when one of these conditions is violated, the network diverges at linear speed. Our approach is based on the identification of the {\em essential faces}, which are the sets of classes that can be simultaneously occupied in stationary regime. A graph is associated with the network, the existence of unessential faces being equivalent to the presence of {\em cycles} in this graph, in which case the usual conditions are not sufficient conditions of stability. As a by-product of our results, we recover the stability conditions and complete the analysis of two exemplary models, the Rybko-Stolyar network and the Lu-Kumar network.