Continuous and discontinuous phase transitions in hypergraph processes

Let V denote a set of N vertices. To construct a hypergraph process, create a new hyperedge at each event time of a Poisson process; the cardinality K of this hyperedge is random, with generating function ρ(x) def = ∑ ρkx , where P (K = k) = ρk; given K = k, the k vertices appearing in the new hyperedge are selected uniformly at random from V . Assume ρ1 + ρ2 > 0. Hyperedges of cardinality 1 are called patches, and serve as a way of selecting root vertices. Identifiable vertices are those which are reachable from these root vertices, in a strong sense which generalizes the notion of graph component. Hyperedges are called identifiable if all of their vertices are identifiable. We use “fluid limit” scaling: hyperedges arrive at rate N , and we study structures of size O(1) and O(N). After division by N , numbers of identifiable vertices and hyperedges exhibit phase transitions, which may be continuous or discontinuous depending on the shape of the structure function − log(1− x)/ρ′(x), x ∈ (0, 1). Both the case ρ1 > 0, and the case ρ1 = 0 < ρ2 are considered; for the latter, a single extraneous patch is added to mark the root vertex.