While there is broad agreement that the structure and composition of interorganizational networks affect innovation, disentangling the two is challenging. Existing approaches to this issue typically conceptualize structure (e.g. brokerage) and composition (e.g. a nodal attribute) as distinct properties and interact the two to empirically assess their effects on innovation. We demonstrate that this “interaction” approach masks heterogeneity within the network in the processes of knowledge access and transfer that drive innovation. We propose a “configuration” approach that partitions firms’ networks into distinct triadic configurations, each embodying an intersection of structural and compositional attributes associated with different knowledge access and transfer properties. Applying our approach to brokerage in cross-national knowledge networks, we identify three configurations based on the distribution of the broker and its partners across or within countries: all foreign (AF), all domestic (AD), and a mix of foreign and domestic (MX). Using a sample of R&D alliances in biotechnology and a matching methodology, we find that certain configurations affect innovation volume (the productivity of innovation) while others affect innovation radicalness (the path breaking nature of the innovation). An interaction approach fails to explain these two innovation outcomes. Our research provides a deeper understanding of how networks (structure) and institutions (composition) jointly influence innovation outcomes.