This study investigates public perceptions of generative artificial intelligence (GenAI) in libraries through a large-scale analysis of posts on X (formerly Twitter). Using a mixed-method approach that combines temporal trend analysis, sentiment classification, and social network analysis, this paper explores how public discourse around GenAI and libraries has evolved over time, the emotional tones that dominate the conversation, and the key users or organizations driving engagement. The findings reveal that discussions are predominantly negative in tone, with surges linked to concerns about ethics and intellectual property. Furthermore, social network analysis identifies both institutional authority and individual bridge users who facilitate cross-domain engagement. The results in this paper contribute to the growing body of literature on GenAI in the library and GLAM (Galleries, Libraries, Archives, and Museums) sectors and offer a real-time, public-facing perspective on the emerging opportunities and concerns GenAI presents.