Nonlinear random walks on hypergraphs characterized by higher-order interactions

Random walks, as one of the classical dynamics on networks, are capable of extracting information on the structure of interacting systems. While existing studies have extended classical network random walks to higher-order networks, prevailing models primarily assume linear relationships between transition probabilities and node hyperdegrees or hyperedge sizes, which inadequately represent the nonlinear properties of higher-order interactions. To address this, we propose a nonlinear random walk on hypergraphs that explicitly considers higher-order collaborative structures and nonlinear dynamics. By introducing a nonlinear mapping between transition probabilities and node hyperdegrees, we go beyond the linear assumption constraints of traditional random walks. Specifically, the probability of a node selecting a hyperedge is inversely proportional to a power function of its hyperdegree, where the power exponent is determined by the interaction order. We first conduct qualitative analysis of the model on star-clique structures, comparing it with classical random walk and linear random walk to reveal the mechanism by which higher-order interactions influence node importance rankings. Subsequently, we conducted node removal experiments on three large-scale datasets to validate the effectiveness of the model by comparing three structural integrity metrics. The results indicate that the proposed model consistently outperforms both Classical and Linear models across all datasets and metrics. These findings confirm that accounting for the nonlinear characteristics of higher-order interactions is essential for both accurately identifying critical nodes and understanding system robustness.