Percolation in random graphs with higher-order clustering

Percolation theory can be used to describe the structural properties of complex networks using the generating function formulation. This mapping assumes that the network is locally treelike and does not contain short-range loops between neighbors. In this paper we use the generating function formulation to examine clustered networks that contain simple cycles and cliques of any order. We use the natural generalization to the Molloy-Reed criterion for these networks to describe their critical properties and derive an approximate analytical description of the size of the giant component, providing solutions for Poisson and power-law networks. We find that networks comprising larger simple cycles behave increasingly more treelike. Conversely, clustering composed of larger cliques increasingly deviate from the treelike solution, although the behavior is strongly dependent on the degree-assortativity.