Properties of Recycling Hypergraphs
Open AccessHypergraph modeling is a growing area at the intersection of probabilitytheory, statistics and discrete mathematics with many applications in computer science, social sciences, medical science, genetics, engineering, marketing and business. We investigate the properties of recycling hypergraphs, which are structures with a fixed number of vertices. These vertices are sampled repeatedly and independently to form sets of fixed size---the hyperedge size---and appear in sets with probabilities called the popularity numbers. We build a probability model and derive some properties of the containment level of specific vertices and the number of vertices contained at a specific level. In addition, we conducted simulations for three cases with different scenarios to validate and explore the properties of recycling hypergraphs.In the probability model, we construct a probability triple: a sample space, a sigma-field and a probability measure. We define the popularity number of a vertex as the sum of probabilities of all combinations containing this vertex in the sample space. Note that while we can obtain the popularity numbers from the probability measure, the measure cannot be uniquely constructed from the popularity numbers.For the containment level of a specific vertex, we obtain the exact mean and variance of the containment level of that vertex, as well as its covariance with other vertices.Consequently, we derive a multivariate Gaussian limit law for the joint containment levels of the various vertices. As for the containment at a specific level, we obtain its exact mean and analytically find the containment level with the largest average number of vertices analytically in the uniform case. We also conducted simulations to examine the properties obtained under the uniform recycling hypergraph and looked at the average maximum level of some non-uniform recycling hypergraphs in practical situations.
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