Algebraic Connectivity Characterization of Ensemble Random Hypergraphs

Abstract

Random hypergraph is a broad concept used to describe probability distributions over hypergraphs, which are mathematical structures with applications in various fields, e.g., complex systems in physics, computer science, social sciences, and network science. Ensemble methods, on the other hand, are crucial both in physics and machine learning. In physics, ensemble theory helps bridge the gap between the microscopic and macroscopic worlds, providing a statistical framework for understanding systems with a vast number of particles. In machine learning, ensemble methods are valuable because they improve predictive accuracy, reduce overfitting, lower prediction variance, mitigate bias, and capture complex relationships in data. However, there is limited research on applying ensemble methods to a set of random hypergraphs. This work aims to study the connectivity behavior of an ensemble of random hypergraphs. Specifically, it focuses on quantifying the random behavior of the algebraic connectivity of these ensembles through tail bounds. We utilize Laplacian tensors to represent these ensemble random hypergraphs and establish mathematical theorems, such as Courant-Fischer and Lieb-Seiringer theorems for tensors, to derive tail bounds for the algebraic connectivity. We derive three different tail bounds, i.e., Chernoff, Bennett, and Bernstein bounds, for the algebraic connectivity of ensemble hypergraphs with respect to different random hypergraphs assumptions.