Fig. 2: Cardinality distribution and Black Swan nodes.

a The probability distribution of the cardinality in real-world network hypergraphs based on different dynamic models. The green solid line signifies the fitting condition after the exclusion of outliers, resulting in a considerably improved fit for the local data. Consequently, it can be posited that the local data conforms to a power-law distribution, yet when accounting for outliers, the global data adheres to a Lévy distribution. Different networks exhibit varying degrees of resilience to perturbations due to their inherent characteristics, and they demonstrate similar properties under varying disturbances. b Hyperedge cardinality distributions for three types of network models under varying parameters. In both random and scale-free networks, the number of connections increases with the gradual rise in parameters, leading to an increased probability of the emergence of Black Swan nodes. This is attributed to disturbances and flow being able to propagate through an increasing number of pathways, thereby exerting a larger impact within the system. Since the average degree of small-world networks does not change with parameter increases, the probability of Black Swan nodes arising in these networks does not exhibit significant variation.