In all of science, the authors of publications depend on the knowledge presented by the previous publications. Thus they "stand on the shoulders of giants" and there is a flow of knowledge from previous publications to more recent ones. The dominating paradigm for tracking this flow of knowledge is to count the number of direct citations, but this neglects the fact that beneath the first layer of citations there is a full body of literature. In this study, we go underneath the "shoulders" by investigating the cumulative knowledge creation process in a citation network of around 35 million publications. In particular, we study stylized models of persistent influence and diffusion that take into account all the possible chains of citations. When we study the persistent influence values of publications and their citation counts, we find that the publications related to Nobel prizes, i.e., Nobel papers have higher ranks in terms of persistent influence than that due to citations, and that the most outperforming publications are typically early works leading to hot research topics of their time. The diffusion model reveals a significant variation in the rates at which different fields of research share knowledge. We find that these rates have been increasing systematically for several decades, which can be explained by the increase in the publication volumes. Overall, our results suggest that analyzing cumulative knowledge creation on a global scale can be useful in estimating the type and scale of scientific influence of individual publications and entire research areas as well as yielding insights that could not be discovered by using only the direct citation counts.