Dense Percolation in Large-Scale Mean-Field Random Networks Is Provably “Explosive”

Recent reports suggest that evolving large-scale networks exhibit “explosive percolation”: a large fraction of nodes suddenly becomes connected when sufficiently many links have formed in a network. This phase transition has been shown to be continuous (second-order) for most random network formation processes, including classical mean-field random networks and their modifications. We study a related yet different phenomenon referred to as dense percolation, which occurs when a network is already connected, but a large group of nodes must be dense enough, i.e., have at least a certain minimum required percentage of possible links, to form a “highly connected” cluster. Such clusters have been considered in various contexts, including the recently introduced network modularity principle in biological networks. We prove that, contrary to the traditionally defined percolation transition, dense percolation transition is discontinuous (first-order) under the classical mean-field network formation process (with no modifications); therefore, there is not only quantitative, but also qualitative difference between regular and dense percolation transitions. Moreover, the size of the largest dense (highly connected) cluster in a mean-field random network is explicitly characterized by rigorously proven tight asymptotic bounds, which turn out to naturally extend the previously derived formula for the size of the largest clique (a cluster with all possible links) in such a network. We also briefly discuss possible implications of the obtained mathematical results on studying first-order phase transitions in real-world linked systems.