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Coronavirus disease 2019 (COVID-19) has attracted research interests from all fields. Phylogenetic and social network analyses based on connectivity between either COVID-19 patients or geographic regions and similarity between syndrome coronavirus 2 (SARS-CoV-2) sequences provide unique angles to answer public health and pharmaco-biological questions such as relationships between various SARS-CoV-2 mutants, the transmission pathways in a community and the effectiveness of prevention policies. This paper serves as a systematic review of current phylogenetic and social network analyses with applications in COVID-19 research. Challenges in current phylogenetic network analysis on SARS-CoV-2 such as unreliable inferences, sampling bias and batch effects are discussed as well as potential solutions. Social network analysis combined with epidemiology models helps to identify key transmission characteristics and measure the effectiveness of prevention and control strategies. Finally, future new directions of network analysis motivated by COVID-19 data are summarized. [ABSTRACT FROM AUTHOR]