Brand effect versus competitiveness in hypernetworks

A few of evolving models in hypernetworks have been proposed based on uniform growth. In order to better depict the growth mechanism and competitive aspect of real hypernetworks, we propose a model in term of the non-uniform growth. Besides hyperdegrees, the other two important factors are introduced to underlie preferential attachment. One dimension is the brand effect and the other is the competitiveness. Our model can accurately describe the evolution of real hypernetworks. The paper analyzes the model and calculates the stationary average hyperdegree distribution of the hypernetwork by using Poisson process theory and a continuous technique. We also address the limit in which this model has a condensation. The theoretical analyses agree with numerical simulations. Our model is universal, in that the standard preferential attachment, the fitness model in complex networks and scale-free model in hypernetworks can all be seen as degenerate cases of the model.

💡 Research Summary

The paper addresses a notable gap in the modeling of hypernetworks, which are structures where a single hyperedge can simultaneously connect multiple vertices, by moving beyond the conventional uniform‑growth assumption. Traditional hypernetwork models typically add a fixed number of hyperedges at each discrete time step, and the attachment probability of a vertex is governed solely by its hyperdegree (the number of hyperedges it belongs to). While such models capture the emergence of scale‑free degree distributions, they fail to reflect two salient features observed in many real‑world systems: (1) the arrival of hyperedges is highly heterogeneous in size and timing, and (2) vertices possess intrinsic attributes that influence their attractiveness beyond mere degree.

To incorporate these aspects, the authors propose a non‑uniform growth model that combines two distinct mechanisms: a brand effect and a competitiveness (fitness) effect. The brand effect is proportional to a vertex’s current hyperdegree, scaled by a global parameter (b). The competitiveness effect assigns each vertex a fitness value (\eta_i) and introduces a non‑linear exponent (\alpha) that modulates how strongly fitness interacts with degree. Consequently, the probability that vertex (i) is selected when a new hyperedge is formed is

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