(1) Background: This study introduces a novel computational approach to examine government capabilities in information intervention for risk management, influential agents in a global information network, and the socioeconomic factors of information-sharing behaviors of the public across regions during the COVID-19 pandemic. (2) Methods: Citation network analysis was employed to gauge the online visibility of governmental health institutions across regions. A bipartite exponential random graph modeling (ERGM) procedure was conducted to measure network dynamics. (3) Results: COVID-19 response agencies in Europe had the highest web impact, whereas health agencies in North America had the lowest. Various stakeholders, such as businesses, non-profit organizations, governments, and educational institutions played a key role in sharing the COVID-19 response by agencies’ information given on their websites. Income inequality and GDP per capita were associated with the high online visibility of governmental health agencies. Other factors, such as population size, an aging population, death rate, and case percentage, did not contribute to the agencies’ online visibility, suggesting that demographic characteristics and health status are not predictors of sharing government resources. (4) Conclusions: A combination of citation network analysis and ERGM helps reveal information flow dynamics and understand the socioeconomic consequences of sharing the government’s COVID-19 information during the pandemic.