In this manuscript, we analyze the interaction network on Twitter among members of the 117th U.S. Congress to assess the visibility of political leaders and explore how systemic properties and node attributes influence the formation of legislative connections. We employ descriptive social network statistical methods, the exponential random graph model (ERGM), and the stochastic block model (SBM) to evaluate the relative impact of network systemic properties, as well as institutional and personal traits, on the generation of online relationships among legislators. Our findings reveal that legislative networks on social media platforms like Twitter tend to reinforce the leadership of dominant political actors rather than diminishing their influence. However, we identify that these leadership roles can manifest in various forms. Additionally, we highlight that online connections within legislative networks are influenced by both the systemic properties of the network and institutional characteristics.