Hyperedge approximation for stochastic processes on higher-order networks
Anzhi Sheng1,2,3, Alex McAvoy4,5, Ye Tian1
1Department of Decision and Control Systems, KTH Royal Institute of Technology, Stockholm 10044, Sweden.
Summary
This study introduces a new analytical framework for analyzing complex interactions in groups, extending traditional graph theory to hypergraphs. This method models multiway interactions and group dynamics, offering insights into cooperation and contagion spread in structured populations.
Area of Science:
- Mathematical modeling
- Network theory
- Evolutionary dynamics
Background:
- Traditional graph theory models pairwise interactions.
- Many real-world systems involve group interactions among three or more agents.
- Existing frameworks struggle to capture these higher-order interactions.
Purpose of the Study:
- To develop a novel analytical framework for stochastic processes on regular hypergraphs.
- To accommodate higher-order interactions determining payoffs and state-updating processes.
- To extend pair approximation methods from graphs to hypergraphs.
Main Methods:
- Development of the k-hyperedge approximation framework.
- Analysis of evolutionary games on hypergraphs, including donation and public goods games.
- Application to neutral complex contagions and payoff-biased contagions.
Main Results:
- Generalization of the k-rule for cooperation to the k-player donation game.
- Determination of critical benefit-to-cost ratios for the nonlinear k-player public goods game.
- Derivation of a closed-form fixation probability for complex contagions, governed by a single complexity parameter.
Conclusions:
- The k-hyperedge approximation successfully extends pair approximation to hypergraphs.
- The framework accommodates multiway interactions and group-level inheritance.
- This unified model provides a powerful tool for studying complex contagions in structured populations.
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