Artificial intelligence (AI) research plays an increasingly important role in society, impacting key aspects of human life. From face recognition algorithms aiding national security in airports, to software that advises judges in criminal cases, and medical staff in healthcare, AI research is shaping critical facets of our experience in the world. But who are the people and institutional bodies behind this influential research? What are the predictors of influence of AI researchers and research organizations? We study this question using social network analysis, in an exploration of the structural characteristics, i.e., network topology, of research organizations that shape modern AI. In a sample of 149 organizations with 9, 987 affiliated authors of published papers in a major AI conference (NeurIPS) and two major conferences that specifically focus on societal impacts of AI (FAccT and AIES), we find that both industry and academic research organizations with influential authors are more interdisciplinary, have a greater fraction of women, are more hierarchical, and less clustered, even when controlling for the size of the organizations. The influence is operationalized as betweenness centrality in co-authorship networks, i.e., how often an author is on the shortest path connecting any pair of authors, acting as a bridge connecting otherwise distant (or even disconneted) members of the network, such as their own co-authors who are not each other's co-author themselves. Using this operationalization, we also find that women have less influence in the AI community, determined as lower betweenness centrality in co-authorship networks. These results suggest that while diverse AI institutions are more influential, the individuals contributing to the increased diversity are marginalized in the AI field. We discuss these results in the context of current events with important societal implications.