Citations in the scientific literature system do not simply reflect relationships between knowledge but are influenced by non-objective and societal factors. Citation bias, irresponsible citation, and citation manipulation are widespread and have become a serious and growing problem. However, it has been difficult to assess the consequences of mixing societal factors into the literature system because there was no observable literature system unmixed with societal factors for comparison. In this paper, we construct a mathematical theorem network, representing a logic-based and objective knowledge system, to address this problem. By comparing the mathematical theorem network and the scientific citation networks, we find that these two types of networks are significantly different in their structure and function. In particular, the reward function in citation networks is impaired: The scientific citation network fails to provide more recognition for more disruptive results, while the mathematical theorem network can achieve. We develop a network generation model that can create two types of links$\unicode{x2014}$logical and societal$\unicode{x2014}$to account for these differences. The model parameter $q$, which we call the human influence factor, can control the number of societal links and thus regulate the degree of mixing of societal factors in the networks. Under this design, the model successfully reproduces the differences among real networks. These results suggest that the presence of societal factors undermines the function of the scientific reward system. To improve the status quo, we advocate for reforming the reference list format in papers, urging journals to require authors to separately disclose logical references and social references.