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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 = � ρkxk, where P (K = k) = ρk; given K = k, the k vertices appearing in the new h ..."
Abstract

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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 = � ρkxk, 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.