An exact formula for percolation on higher-order cycles

We present exact solutions for the size of the giant connected component (GCC) of graphs composed of higher-order homogeneous cycles, including weak cycles and cliques, following bond percolation. We use our theoretical result to find the location of the percolation threshold of the model, providing analytical solutions where possible. We expect the results derived here to be useful to a wide variety of applications including graph theory, epidemiology, percolation and lattice gas models as well as fragmentation theory. We also examine the Erd\H{o}s-Gallai theorem as a necessary condition on the graphicality of configuration model networks comprising higher-order clique sub-graphs.