The Evolution of Hyperedge Cardinalities and Bose-Einstein Condensation in Hypernetworks

TL;DR

This paper introduces a Bose-Einstein hypernetwork model to analyze hyperedge cardinality evolution, revealing conditions for Bose-Einstein condensation and providing analytical and simulation validation.

Contribution

It is the first to study hyperedge cardinality in hypernetworks and links Bose-Einstein condensation as a special case within this framework.

Findings

Derived the characteristic equation for hyperedge cardinalities.

Established an analytical expression for stationary hyperedge distribution.

Confirmed theoretical results with numerical simulations.

Abstract

To depict the complex relationship among nodes and the evolving process of a complex system, a Bose-Einstein hypernetwork is proposed in this paper. Based on two basic evolutionary mechanisms, growth and preference jumping, the distribution of hyperedge cardinalities is studied. The Poisson process theory is used to describe the arrival process of new node batches. And, by using the Poisson process theory and a continuity technique, the hypernetwork is analyzed and the characteristic equation of hyperedge cardinalities is obtained. Additionally, an analytical expression for the stationary average hyperedge cardinality distribution is derived by employing the characteristic equation, from which Bose-Einstein condensation in the hypernetwork is obtained. The theoretical analyses in this paper agree with the conducted numerical simulations. This is the first study on the hyperedge cardinality in hypernetworks, where Bose-Einstein condensation can be regarded as a special case of hypernetworks. Moreover, a condensation degree is also discussed with which Bose-Einstein condensation can be classified.