A novel analytical tool for complex propagation processes in networks: High-order dynamic equation.

Controlling the spread of epidemics in complex networks has always been an important research problem in the field of network science and has been widely studied by many scholars so far. One of the key problems in the transmission process of epidemics in complex networks is the transmission mechanism. At present, the transmission mechanism in complex networks can be divided into simple transmission and complex transmission. Simple transmission has been widely studied and the theory is relatively mature, while complex transmission still has many questions to answer. In fact, in the complex transmission process, the higher-order structure of the network plays a very important role, which can affect the transmission speed, final scale, and transmission path of the epidemic by strengthening the mechanism. However, due to the lack of complex dynamic analysis tools, the measurement of influence on propagation is still at the low-dimensional node level. Therefore, in this paper, we propose a set of closed dynamic higher-order structure equations to gain insight into the complex propagation process in the network, which breaks the inherent thinking and enables us to reexamine the complex dynamic behavior more clearly from the higher-order level rather than just from the node level, opening up a new way to analyze the higher-order interaction on the dynamic network. We apply the proposed high-order dynamic equations to a complex susceptible-infection-recovery epidemiological model on two real and synthetic networks, and extensive numerical simulation results demonstrate the effectiveness of the proposed approach. Our research results help to deepen the understanding of the relationship between complex propagation mechanisms and higher-order structures and develop a complete set of complex dynamic analysis tools that can be extended to higher-order forms to help in-depth understanding of the propagation rules and mechanisms in complex propagation processes, providing an important theoretical basis for predicting, analyzing, and controlling complex propagation processes.