In many countries around the world (including Australia), the prescribing of opioid analgesic drugs is an increasing trend associated with significant increases in drug-related patient harm such as abuse, overdose, and death. In Australia, the Medicines Regulation and Quality Unit within Queensland Health maintains a database recording opioid analgesic drug prescriptions dispensed across the State (population 4.703 million). In this work, we propose the use of network visualisation and analysis as a tool for improved understanding of these data. Prescribing data for Fentanyl patches, a strong opioid with high potential for misuse and subsequent harm, across Queensland, Australia from 2011 to 2018 is analysed as an example of using network analysis, where prescribing patterns are viewed as a dynamic, bipartite graph of the interactions between patients and prescribers over time. The technique provides a global view of a large state-wide prescribing dataset, including the distribution of subgraph structures present. Local analysis is also carried out to demonstrate the clinical utility of the technique, including the dynamics of the graph structure over time. A variety of network statistics that measure network structural and dynamic properties are presented to reveal the characteristics and trends of drug seeking and prescribing behaviours. This approach has been recognised by healthcare professionals at Queensland Health as leading to new and useful insights on the relationship between patients and prescribers and supporting their advisory role to reduce patient harm from inappropriate use of prescription drugs.