Higher-order interactions are increasingly recognized as a critical aspect in the modeling of complex systems. Higher-order networks provide a framework for studying the relationship between the structure of higher-order interactions and the function of the complex system. However, little is known about how higher-order interactions affect dynamic processes. In this thesis, we develop general frameworks of percolation aiming at understanding the interplay between higher-order network structures and the critical properties of dynamics. We reveal that degree correlations strongly affect the percolation threshold on higher-order networks and interestingly, the effect of correlations is different on ordinary percolation and higher-order percolation. We further elucidate the mechanisms responsible for the emergence of discontinuous transitions on higher-order networks. Moreover, we show that triadic regulatory interaction, as a general type of higher-order interaction found widely in nature, can turn percolation into a fully-fledged dynamic process that exhibits period doubling and a route to chaos. As an important example of dynamic processes, we further investigate the role of network topology on epidemic spreading. We show that higher-order interactions can induce a non-linear infection kernel in a pandemic, which results in a discontinuous phase transition, hysteresis, and superexponential spreading. Finally, we propose an epidemic model to evaluate the role of automated contact-and-tracing with mobile apps as a new containment measure to mitigate a pandemic. We reveal the non-linear effect on the reduction of the incidence provided by a certain fraction of app adoption in the population and we propose the optimal strategy to mitigate the pandemic with limited resources. Altogether, the thesis provides new insights into the interplay between the topology of higher-order networks and their dynamics. The results obtained may shed light on the research in other areas of interest such as brain functions and epidemic spreading
