Later, another powerful method – known as the *potential-theoretic approach* to metastability based on the analogy between Markov processes and electric networks – was developed in Bovier, Eckhoff, Gayrard and Klein [@BEGK00], [@BEGK01], [@BEGK02], [@BEGK04]. This was shown in Bovier and Manzo [@BM02], Bovier, den Hollander and Spitoni [@BdHS10] to lead to a considerable sharpening of earlier results. For other approaches to metastability, as well as further examples of metastable stochastic dynamics and relevant literature, we refer the reader to the monographs by Olivieri and Vares [@OV05], Bovier and den Hollander [@BdH15].

Recently, there has been interest in the Ising model on *random graphs* (Dembo and Montanari [@DM10], Dommers, Giardinà and van der Hofstad [@DGvdH10], Mossel and Sly [@MS13]). The only results known to date about metastability subject to Glauber spin-flip dynamics are valid for $r$-regular random graphs (Dommers [@Dpr]). In the present paper we investigate what can be said for more general degree distributions. Metastability is much more challenging on random graphs than on lattices. Moreover, we need to capture the metastable behaviour for a *generic realisation* of the random graph.

In Section \[IsingGlauber\] we define the Ising model on a random multigraph subject to Glauber spin-flip dynamics. We 