Competitive higher-order network evolution model

A simplicial complex that evolves over time is called a higher-order network, which has recently found applications in social, technological, and biological contexts. The higher-order network is a generalization of traditional networks, and its evolution models are generally more difficult to analyze than those of traditional networks. This paper proposes a higher-order network evolution model with competitiveness. We introduce the method of difference equation analysis into the study of higher-order networks to enable more rigorous analysis. This approach avoids relying on the assumption of continuity in node degree, which is commonly made in traditional network analysis. Applying Poisson process theory, we not only prove the existence of a stationary higher-order degree distribution but also derive an analytical expression for this distribution. Our results show that the scale-free behavior of (d-1)-dimensional simplex is controlled by the competitiveness in the d-dimensional simplicial complex. As competitiveness increases, the d-order degree distribution of (d-1)-dimensional simplices exhibits curvature in log-log coordinates, whereas for d-order degree of the e-dimensional simplices (where e\le d-2) follows a scale-free distribution. We further extract competitiveness parameters under different probability distributions to simulate the model, and find that simulation results agree with the theoretical analysis.