Ordinary Percolation with Discontinuous Transitions

We study percolation on hierarchical networks using generating functions and renormalization group techniques. Our exact results show the presence of novel features in these networks including the existence of non-trivial critical points, three distinct regimes in the phase diagram and, most importantly, a discontinuity in the formation of the extensive cluster at a critical point $p_{c}<1$ . At $p_{c}$, the order parameter $P_{\infty}$ describing the probability of any node to be a part of the largest cluster, jumps instantly to a finite value. We present simple examples of small-world networks with various hierarchies of long range bonds, indicating that the presence of discontinuous transitions is generic.\\[4pt] [1] S. Boettcher, V. Singh, and R.M. Ziff. Ordinary Percolation with Discontinuous Transitions. Arxiv preprint arXiv:1110.4288 (2):2 5, 2011.