M. Hambly, B. et Jordan, Jonathan, A random hierarchical lattice: the series-parallel graph and its properties, Advances in applied probability , 36(2), 2004, pp. 824-838
We consider a sequence of random graphs constructed by a hierarchical procedure. The construction replaces existing edges by pairs of edges in series or parallel with probability p. We investigate the effective resistance across the graphs, first-passage percolation on the graphs and the Cheeger constants of the graphs as the number of edges tends to infinity. In each case we find a phase transition at p=1/2