We consider various models of randomly grown graphs. In these models the vertices and the edges accumulate within time according to certain rules. We study a phase transition in these models along a parameter which refers to the mean life-time of an edge. Although deleting old edges in the uniformly grown graph changes abruptly the properties of the model, we show that some of the macro-characteristics of the graph vary continuously. In particular, our results yield a lower bound for the size of the largest connected component of the uniformly grown graph.
"Continuity of the percolation threshold in randomly grown graphs.." Electron. J. Probab. 12 1036 - 1047, 2007. https://doi.org/10.1214/EJP.v12-436