Non-uniform Evolving Hypergraphs and Weighted Evolving Hypergraphs.

Area of Science:

• Complex systems
• Network science
• Statistical physics

Background:

• Evolving hypergraphs model complex relationships beyond pairwise interactions.
• Existing models often assume uniform hyperedge sizes and simple attachment rules.
• Understanding the dynamics of large-scale, evolving networks is crucial for various scientific domains.

Purpose of the Study:

• To propose a non-uniform evolving hypergraph model with nonlinear preferential attachment and attractiveness.
• To develop a model for weighted evolving hypergraphs incorporating node and hyperedge dynamics.
• To analyze the emergent properties, specifically scale-free distributions, of these models.

Main Methods:

• Development of a non-uniform evolving hypergraph model with batch node arrivals via a Poisson process.
• Mathematical derivation of the characteristic equation for hyperdegrees and analysis of hyperdegree changes.
• Construction of a coupled model for weighted evolving hypergraphs, integrating weight evolution.
• Application of Poisson process theory and derived equations to obtain stationary distributions.

Main Results:

• The non-uniform evolving hypergraph model exhibits a stationary average hyperdegree distribution.
• The weighted evolving hypergraph model demonstrates scale-free behavior for both hyperdegree and hyperstrength distributions.
• The size of each hyperedge is non-uniform due to random variables governing node and existing node selection.

Conclusions:

• The proposed non-uniform evolving hypergraph model provides a more realistic representation of complex systems.
• The weighted evolving hypergraph model successfully captures scale-free properties, common in real-world networks.
• This work offers a theoretical framework for analyzing dynamic and weighted higher-order networks.