Why do some species successfully invade new environments? Which of these invasive species will alter or even reshape their new environment? The answers to these questions are simultaneously critical and complex. They are critical because invasive species can spectacularly alter their new environment, leading to native species extinctions or loss of important ecosystem functions that fundamentally reduce environmental and societal services. They are complex because invasion success in a novel environment is influenced by various attributes embedded in natural landscapes—biogeographical landscape properties, abiotic environmental characteristics, and the relationship between the invasive species and the resident species present in the new environment. We explore whether a condensed record of the relationships among species, in the form of a network, contains the information needed to understand and predict invasive species success and subsequent impacts. Applying network theory to study invasive species is a relatively novel approach. For this reason, much research will be needed to incorporate existing ecological properties into a network framework and to identify which network features hold the information needed to understand and predict whether or not an invasive species is likely to establish or come to dominate a novel environment. This paper asks and begins to answer the 14 most important questions that biologists must address to integrate network analysis into the study of invasive species. Answering these questions can help ecologists produce a practical monitoring scheme to identify invasive species before they substantially alter native environments or to provide solutions to mitigate their harmful impacts.