The ability to understand, process and evaluate arguments made by others and ourselves is important in many personal and professional spheres, such as political debates. Analysis typically appears in written form, but a growing number of tools support analysis through diagram-based graphical representations. These UIs might support better argument analysis because arguments have non-linear structures that are difficult to convey through linear text. However, there is little empirical evidence on the advantages or mechanisms that might make graph UIs superior to traditional textual documents. We ran and analyzed an extensive study with twenty participants who used text and graph editors to analyze political debates. Our quantitative and qualitative findings demonstrate the tradeoffs between the two approaches and explain some of the key mechanisms that support the analysis in both media.