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The use of network analysis as a tool has increased exponentially as more clinical researchers see the benefits of network data for modeling of infectious disease transmission or translational activities in a variety of areas, including patient-caregiving teams, provider networks, patient-support networks, and adoption of health behaviors or treatments, to name a few. Yet, relational data such as network data carry a higher risk of deductive disclosure. Cases of reidentification have occurred and this is expected to become more common as computational ability increases. Recent data sharing policies aim to promote reproducibility, support replicability, and protect federal investment in the effort to collect these research data by making them available for secondary analyses. However, typical practices to protect individual-level clinical research data may not be sufficiently protective of participant privacy in the case of network data, nor in some cases do they permit secondary data analysis. When sharing data, researchers must balance security, accessibility, reproducibility, and adaptability (suitability for secondary analyses). Here, we provide background about applying network analysis to health and clinical research, describe the pros and cons of applying typical practices for sharing clinical data to network data, and provide recommendations for sharing network data.