We modify the usual Erdos-Renyi random graph evolution by letting connected clusters 'burn down' (i.e. fall apart to disconnected single sites) due to a Poisson flow of lightnings. In a range of the intensity of rate of lightnings the system sticks to a permanent.
"Erdos-Renyi random graphs + forest fires = self-organized criticality." Electron. J. Probab. 14 1290 - 1327, 2009. https://doi.org/10.1214/EJP.v14-653