A universal representation for quantifying the significance of higher-order structures within networks under complex propagation mechanisms

With the increasing complexity of epidemic propagation, traditional metrics for assessing critical components of networks often fail to adequately capture the higher-order structural characteristics of networks under complex propagation mechanisms. These higher-order features significantly influence the speed, scale, and pathways of propagation through synergistic reinforcement mechanisms. Based on existing complex propagation models, this study introduces the concept of “multiple infections” and constructs a universal representation method to quantify the importance of higher-order structures through the closure approximation of lower-order structures. The core innovation of this method lies in the introduction of an enhancement factor, which explicitly characterizes the nonlinear growth of propagation rates when infected nodes participate in higher-order interactions. Additionally, a probabilistic closure framework is proposed to transform the dynamic characteristics of higher-order structures into computable mathematical expressions, supporting the dimensionality reduction representation of higher-order structures at arbitrary scales. We specifically apply the proposed representation for quantifying the importance of higher-order structures to links and fully connected triplets (i.e., triangles), comparing it with existing metrics for link and triangle importance. By analyzing the degree of suppression of infection scale and complex propagation rates, the accuracy of identification, the efficiency of disrupting higher-order structures, and monotonicity, the superiority of the proposed quantification method is validated. The findings of this research contribute to a deeper understanding of the structural characteristics and dynamic behaviors of networks in the context of complex transmission. By introducing a generalized dynamic representation of higher-order structures, we provide researchers in fields such as network science, social network analysis, and transmission modeling with new insights and tools, thereby promoting further advancements in related areas and offering more reliable theoretical support for practical applications.