Biomedical research increasingly relies on the analysis and visualization of a wide range of collected data. However, for certain research questions, such as those investigating the interconnectedness of biological elements, the sheer quantity and variety of data results in rather uninterpretable--this is especially true for network visualization, as a large and dense biological network is often compared to spaghetti or a hairball. The contents of this dissertation detail three major studies and a number of associated analysis studies that extend those studies. First, the challenges faced by researchers who analyze and visualize biological networks are elucidated, followed by a systematic review that analyzes and characterizes network figures from peer-reviewed bioinformatics literature. The systematic review dataset is further supplemented with an analysis of task completability, and the combination of the two are analyzed via Random Forest to provide insight into the varying importance of visual encodings in context of graph-based tasks. Next, a small theoretical framework that is valuable for framing network visualization research questions is detailed, followed by a description of visual encoding exploration software built on the framework. The final study included in this dissertation details the design and execution of a task-based perception study, where several visual encodings are estimated as functions of the measured task. Through these studies, I contribute to the understanding of network-related visualization challenges encountered by researchers, a measure of the status quo of network visualization, a conceptualization of a method to usefully frame research questions related to network visualization, visual encoding software that affords systematic and reproducible explorations of the visual encoding set space, and finally a set of functional estimates describing how numerous visual encodings are a related to one's ability to visually scan a network. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com.bibliotheek.ehb.be/en-US/products/dissertations/individuals.shtml.]