start from the configuration where all spins point down, and allow spins to flip up or down according to a Metropolis dynamics at positive temperature. We are interested in the time it takes the system to reach the configuration where all spins point up. In Section \[metastability\] we introduce certain geometric quantities that play a central role in the description of the metastable behaviour of the system, and state three general theorems that are valid under a certain key hypothesis. These theorems concern the average transition time, the distribution of the transition time, and the gate of saddle point configurations for the crossover, all in the limit of low temperature. They involve certain key quantities associated with the random graph. Our goal is to study the scaling behaviour of these quantities as the size of the graph tends to infinity.

In Section \[literature\] we describe four examples to which the three general theorems apply: three refer to regular lattices, while one refers to the Erdős-Rényi random graph. In Section \[configurationmodel\] we recall the definition of the Configuration Model, which is an example of a random graph with a non-trivial geometric structure. In Section \[maintheorems\] we state our main metastability results 